Theory and Applications of Categories, Vol. 10, No. 1, 2002, pp. 1–70.

A SURVEY OF DEFINITIONS OF n-CATEGORY

TOM LEINSTER

ABSTRACT. Many people have proposed definitions of ‘weak n-category’. Ten of them are presented here. Each definition is given in two pages, with a further two pages on what happens when n ≤ 2. The definitions can be read independently. Chatty bibliography follows.

Contents

Introduction  2
Background  7
Definition Tr  10
Definition P  14
Definitions B  18
Definitions L  22
Definition L′  26
Definition Si  30
Definition Ta  34
Definition J  38
Definition St  42
Definition X  46
Further Reading 50

Received by the editors 2001 August 9 and, in revised form, 2001 December 22. Transmitted by Ross Street. Published on 2002 January 14. 2000 Mathematics Subject Classification: 18D05, 18D50, 18F99, 18A99. Key words and phrases: n-category, higher-dimensional category, higher categorical structure. \begin{matrix}c \\ ◯ \end{matrix} Tom Leinster, 2002. Permission to copy for private use granted.

Introduction

Lévy … once remarked to me that reading other mathematicians’ research gave him actual physical pain.

—J. L. Doob on the probabilist Paul Lévy, Statistical Science 1, no. 1, 1986.

Hell is other people.

—Jean-Paul Sartre, Huis Clos.

The last five years have seen a vast increase in the literature on higher-dimensional categories. Yet one question of central concern remains resolutely unanswered: what exactly is a weak n-category? There have, notoriously, been many proposed definitions, but there seems to be a general perception that most of these definitions are obscure, difficult and long. I hope that the present work will persuade the reader that this is not the case, or at least does not need to be: that while no existing approach is without its mysteries, it is quite possible to state the definitions in a concise and straightforward way.

What’s in here, and what’s not. The sole purpose of this paper is to state several possible definitions of weak n-category. In particular, I have made no attempt to compare the proposed definitions with one another (although certainly I hope that this work will help with the task of comparison). So the definitions of weak n-category that follow may or may not be ‘equivalent’; I make no comment. Moreover, I have not included any notions of weak functor or equivalence between weak n-categories, which would almost certainly be required before one could make any statement such as ‘Professor Yin’s definition of weak n-category is equivalent to Professor Yang’s’.

I have also omitted any kind of motivational or introductory material. The ‘Further Reading’ section lists various texts which attempt to explain the relevance of n-categories and other higher categorical structures to mathematics at large (and to physics and computer science). I will just mention two points here for those new to the area. Firstly, it is easy to define strict n-categories (see ‘Preliminaries’), and it is true that every weak 2-category is equivalent to a strict 2-category, but the analogous statement fails for n-categories when n > 2: so the difference between the theories of weak and strict n-categories is nontrivial. Secondly, the issue of comparing definitions of weak n-category is a slippery one, as it is hard to say what it even means for two such definitions to be equivalent. For instance, suppose you and I each have in mind a definition of algebraic variety and of morphism of varieties; then we might reasonably say that our definitions of variety are ‘equivalent’ if your category of varieties is equivalent to mine. This makes sense because the structure formed by varieties and their morphisms is a category. It is widely held that the structure formed by weak n-categories and the functors, transformations, … between them should be a weak (n + 1)-category; and if this is the case then the question is whether your weak (n + 1)-category of weak n-categories is equivalent to mine—but whose definition of weak (n + 1)-category are we using here… ?

A SURVEY OF DEFINITIONS OF n-CATEGORY  3

This paper gives primary importance to n-categories, with other higher categorical structures only mentioned where they have to be. In writing it this way I do not mean to imply that n-categories are the only interesting structures in higher-dimensional category theory: on the contrary, I see the subject as including a whole range of interesting structures, such as operads and multicategories in their various forms, double and ntuple categories, computads and string diagrams, homotopy-algebras, n-vector spaces, and structures appropriate for the study of braids, knots, graphs, cobordisms, proof nets, flowcharts, circuit diagrams, …. Moreover, consideration of n-categories seems inevitably to lead into consideration of some of these other structures, as is borne out by the definitions below. However, n-categories are here allowed to upstage the other structures in what is probably an unfair way.

Finally, I do not claim to have included all the definitions of weak n-category that have been proposed by people; in fact, I am aware that I have omitted a few. They are omitted purely because I am not familiar with them. More information can be found under ‘Further Reading’.

Layout. The first section is ‘Background’. This is mainly for reference, and it is not really recommended that anyone starts reading here. It begins with a page on ordinary category theory, recalling those concepts that will be used in the main text and fixing some terminology. Everything here is completely standard, and almost all of it can be found in any introductory book or course on the subject; but only a small portion of it is used in each definition of weak n-category. There is then a page each on strict n-categories and bicategories, again recalling widely-known material.

Next come the ten definitions of weak n-category. They are absolutely independent and self-contained, and can be read in any order. This means that there is a certain amount of redundancy: in particular, sizeable passages occur identically in definitions B and L, and similarly in definitions Si and Ta. (The reasons for the names are explained below.) No significance should be attached to the order in which they are presented; I tried to arrange them so that definitions with common themes were grouped together in the sequence, but that is all. (Some structures just don’t fit naturally into a single dimension.)

Each definition of weak n-category is given in two pages, so that if this is printed double-sided then the whole definition will be visible on a double-page spread. This is followed, again in two pages, by an explanation of the cases n = 0, 1, 2. We expect weak 0-categories to be sets, weak 1-categories to be categories, and weak 2-categories to be bicategories—or at least, to resemble them to some reasonable degree—and this is indeed the case for all of the definitions as long as we interpret the word ‘reasonable’ generously. Each main definition is given in a formal, minimal style, but the analysis of n ≤ 2 is less formal and more explanatory; partly the analysis of n ≤ 2 is to show that the proposed definition of n-category is a reasonable one, but partly it is for illustrative purposes. The reader who gets stuck on a definition might therefore be helped by looking at n ≤ 2.

Taking a definition of weak n-category and performing a rigorous comparison between the case n = 2 and bicategories is typically a long and tedious process. For this reason, I have not checked all the details in the n ≤ 2 sections. The extent to which I feel confident in my assertions can be judged from the number of occurrences of phrases such as ‘probably’ and ‘it appears that’, and by the presence or absence of references under ‘Further Reading’.

There are a few exceptions to this overall scheme. The section labelled B consists, in fact, of two definitions of weak n-category, but they are so similar in their presentation that it seemed wasteful to give them two different sections. The same goes for definition L, so we have definitions of weak n-category called B1, B2, L1 and L2. A variant for definition St is also given (in the n ≤ 2 section), but this goes nameless. However, definition X is not strictly speaking a mathematical definition at all: I was unable to find a way to present it in two pages, so instead I have given an informal version, with one sub-definition (opetopic set) done by example only. The cases n ≤ 2 are clear enough to be analysed precisely.

Another complicating factor comes from those definitions which include a notion of weak ω-category (= weak ∞-category). There, the pattern is very often to define weak ω-category and then to define a weak n-category as a weak ω-category with only trivial cells in dimensions > n. This presents a problem when one comes to attempt a precise analysis of n ≤ 2, as even to determine what a weak 0-category is involves considering an infinite-dimensional structure. For this reason it is more convenient to redefine weak ncategory in a way which never mentions cells of dimension > n, by imitating the original definition of weak ω-category. Of course, one then has to show that the two different notions of weak n-category are equivalent, and again I have not always done this with full rigour (and there is certainly not the space to give proofs here). So, this paper actually contains significantly more than ten possible definitions of weak n-category.

‘Further Reading’ is the final section. To keep the definitions of n-category brief and self-contained, there are no citations at all in the main text; so this section is a combination of reference list, historical notes, and general comments, together with a few pointers to literature in related areas.

Overview of the definitions. Table 1 shows some of the main features of the definitions of weak n-category. Each definition is given a name such as A or Z, according to the name of the author from whom the definition is derived. (Definition X is a combination of the work of many people, principally Baez, Dolan, Hermida, Makkai and Power.) The point of these abbreviations is to put some distance between the definitions as proposed by those authors and the definitions as stated below. At the most basic level, I have in all cases changed some notation and terminology. Moreover, taking what is often a long paper and turning it into a two-page definition has seldom been just a matter of leaving out words; sometimes it has required a serious reshaping of the concepts involved. Whether the end result (the definition of weak n-category) is mathematically the same as that of the original author is not something I always know: on various occasions there have been passages in the source paper that have been opaque to me, so I have guessed at the author’s intended meaning. Finally, in several cases only a definition of weak ω-category was explicitly given, leaving me to supply the definition of weak n-category for finite n.

Definition Author(s) Shapes used  A/the  ω?
Tr  Trimble  globular  the  x
P  Penon  globular  the  ✓
B  Batanin  globular  the (B1), a (B2) ✓
L  Leinster  globular  the (L1), a (L2) ✓
L′  Leinster  globular  a  ✓
Si  Simpson  simplicial/globular a  x
Ta  Tamsamani simplicial/globular a  x
J  Joyal  globular/simplicial a  ✓
St  Street  simplicial  a  ✓
X  see text  opetopic  a  x

Table 1: Some features of the definitions

[figure: •; •; •; •; •; •; •; •; •; •]
globular  simplicial  opetopic

Figure 1: Shapes used in the definitions

In summary, then, I do believe that I have given ten reasonable definitions of weak ncategory, but I do not guarantee that they are the same as those of the authors listed in Table 1; ultimately, the responsibility for them is mine.

The column headed ‘shapes used’ refers to the different shapes of m-cell (or ‘m-arrow’, or ‘m-morphism’) employed in the definitions. These are shown in Figure 1.

It has widely been observed that the various definitions of n-category fall into two groups, according to the attitude one takes to the status of composition. This distinction can be explained by analogy with products. Given two sets A and B, one can define a product of A and B to be a triple (P, p_{1}, p_{2}) where P is a set and p_{1} : P \overline{}A, p_{2} : P \overline{}B are functions with the usual universal property. This is of course the standard thing to do in category theory, and in this context one can strictly speaking never refer to the product of A and B. On the other hand, one could define the product of A and B to be the set A × B of ordered pairs (a, b) = {{a}, {a, b}} with a ∈ A and b ∈ B; this has the virtue of being definite and allowing one to speak of the product in the customary way, but involves a wholly artificial construction. Similarly, in some of the proposed definitions of weak n-category, one can never speak of the composite of morphisms g and f, only of a composite (of which there may be many, all equally valid); but in some of the definitions one does have definite composites g^{∘}f, the composite of g and f. (The use of the word ‘the’ is not meant to imply strictness, e.g. the three-fold composite h^{∘}(g^{∘}f) will in general be different from the three-fold composite (h^{∘}g)^{∘}f.) So this is the meaning of the column headed ‘a/the’; it might also have been headed ‘indefinite/definite’, ‘relational/functional’, ‘universal/coherent’, or even ‘geometric/algebraic’.

All of the sections include a definition of weak n-category for natural numbers n, but some also include a definition of weak ω-category (in which there are m-cells for all natural m). This is shown in the last column.

Finally, I warn the reader that the words ‘contractible’ and ‘contraction’ occur in many of the definitions, but mean different things from definition to definition. This is simply to save having to invent new words for concepts which are similar but not identical, and to draw attention to the common idea.

Acknowledgements. I first want to thank Eugenia Cheng and Martin Hyland. Their involvement in this project has both made it much more pleasurable for me and provided a powerful motivating force. Without them, I suspect it would still not be done.

I prepared for writing this by giving a series of seminars (one definition per week) in Cambridge in spring 2001, and am grateful to the participants: the two just mentioned, Mario Cáccamo, Marcelo Fiore, and Joe Templeton. I would also like to thank those who have contributed over the years to the many Cambridge Category Theory seminars on the subject of n-categories, especially Jeff Egger (who introduced me to Tamsamani’s definition), Peter Johnstone, and Craig Snydal (with whom I have also had countless interesting conversations on the subject).

Todd Trimble was generous enough to let me publish his definition for the first time, and to cast his eye over a draft of what appears below as definition Tr—though all errors, naturally, are mine.

I am also grateful to the other people with whom I have had helpful communications, including Michael Batanin (who told me about Penon’s definition), David Carlton, Jack Duskin, Keith Harbaugh, Anders Kock, Bill Lawvere, Peter May, Carlos Simpson, Ross Street, Bertrand Toen, Dominic Verity, Marek Zawadowski (who told me about Joyal’s definition), and surely others, whose names I apologize for omitting.

Many of the diagrams were drawn using Paul Taylor’s commutative diagrams package. It is a pleasure to thank St John’s College, Cambridge, where I hold the Laurence

Goddard Fellowship, for their support.

Background

Category Theory. Here is a summary of the categorical background and terminology needed in order to read the entire paper. The reader who isn’t familiar with everything below shouldn’t be put off: each individual Definition only uses some of it.

I assume familiarity with categories, functors, natural transformations, adjunctions, limits, and monads and their algebras. Limits include products, pullbacks (with the pullback of a diagram X \overline{}Z \underline{}Y sometimes written X ×_{Z} Y), and terminal objects (written 1, especially for the terminal set {∗}); we also use initial objects. A monad (T, η, μ) is often abbreviated to T.

I make no mention of the difference between sets and classes (‘small and large collections’). All the Definitions are really of small weak n-category.

Let C be a category. X ∈ C means that X is an object of C, and C(X, Y) is the set of morphisms (or maps, or arrows) from X to Y in C. If f ∈ C(X, Y) then X is the domain or source of f, and Y the codomain or target.

Set is the category (sets + functions), and Cat is (categories + functors). A set is just a discrete category (one in which the only maps are the identities).

C^{op} is the opposite or dual of a category C. [C, D] is the category of functors from C to D and natural transformations between them. Any object X of C induces a functor C(X, −) : C \overline{}Set, and a natural transformation from C(X, −) to F : C \overline{}Set is the same thing as an element of F X (the Yoneda Lemma); dually for C(−, X) : C^{op} \overline{}Set.

A functor F : C \overline{}D is an equivalence if these equivalent conditions hold: (i) F is full, faithful and essentially surjective on objects; (ii) there exist a functor G : D \overline{}C (a pseudo-inverse to F) and natural isomorphisms η : 1 \overline{}GF, ε : F G \overline{}1 ; (iii) as (ii), but with (F, G, η, ε) also being an adjunction.

Any set D_{0} of objects of a category C determines a full subcategory D of C, with objectset D_{0} and D(X, Y) = C(X, Y). Every category C has a skeleton: a subcategory whose inclusion into C is an equivalence and in which no two distinct objects are isomorphic. If F, G : C \overline{}Set, GX ⊆ F X for each X ∈ C, and F and G agree on morphisms of C, then G is a subfunctor of F.

A total order on a set I is a reflexive transitive relation ≤ such that if i ≠ j then exactly one of i ≤ j and j ≤ i holds. (I, ≤) can be seen as a category with object-set I in which each hom-set has at most one element. An order-preserving map (I, ≤) \overline{}(I′, ≤′) is a function f such that i ≤ j ⇒ f(i) ≤′ f(j).

Let Δ be the category with objects [k] = {0, …, k} for k ≥ 0, and order-preserving functions as maps. A simplicial set is a functor Δ^{op} \overline{}Set. Every category C has a nerve (the simplicial set NC : [k] ⟼ Cat([k], C)), giving a full and faithful functor N : Cat \overline{}[Δ^{op}, Set]. So Cat is equivalent to the full subcategory of [Δ^{op}, Set] with objects {X | X ≅ NC for some C}; there are various characterizations of such X, but we come to that in the main text.

Leftovers: a monoid is a set (or more generally, an object of a monoidal category) with an associative binary operation and a two-sided unit. Cat is monadic over the category of directed graphs. The natural numbers start at 0.

Strict n-Categories. If V is a category with finite products then there is a category V-Cat of V-enriched categories and V-enriched functors, and this itself has finite products. (A V-enriched category is just like an ordinary category, except that the ‘hom-sets’ are now objects of V.) Let 0-Cat = Set and, for n ≥ 0, let (n + 1)-Cat = (n-Cat)-Cat; a strict n-category is an object of n-Cat. Note that 1-Cat = Cat.

Any finite-product-preserving functor U : V \overline{}W induces a finite-product-preserving functor U_{∗} : V-Cat \overline{}W-Cat, so we can define functors U_{n} : (n + 1)-Cat \overline{}n-Cat by taking U_{0} to be the objects functor and U_{n+1} = (U_{n})_{∗}. The category ω-Cat of strict ω-categories is the limit of the diagram

⋯ ^{U_{n+1}} (n + 1)-Cat ^{U_{n}} ⋯ ^{U_{1}} 1-Cat = Cat ^{U_{0}} 0-Cat = Set.

Alternatively: a globular set (or ω-graph) A consists of sets and functions

⋯ [figure: s; t] A_{m} [figure: s; t] A_{m−1} [figure: s; t] ⋯ [figure: s; t] A_{0}

such that for m ≥ 2 and α ∈ A_{m}, ss(α) = st(α) and ts(α) = tt(α). An element of A_{m} is called an m-cell, and we draw a 0-cell a as _{a}^{•}, a 1-cell f as _{a}_{•} ^{^{f} \overline{}}_{b}_{•} (where

a = s(f), b = t(f)), a 2-cell α as _{a}_{•} [figure: f; g; α] _{b}_{•} , etc. For m > p ≥ 0, write A_{m} ×_{A_{p}} A_{m} =

{(α′, α) ∈ A_{m} × A_{m} | s^{m−p}(α′) = t^{m−p}(α)}.

A strict ω-category is a globular set A together with a function ^{∘}_{p} : A_{m}×_{A_{p}}A_{m} \overline{}A_{m} (composition) for each m > p ≥ 0 and a function i : A_{m} \overline{}A_{m+1} (identities, usually written i(α) = 1_{α}) for each m ≥ 0, such that

i. if m > p ≥ 0 and (α′, α) ∈ A_{m} ×_{A_{p}} A_{m} then

s(α′ ^{∘}_{p} α) = s(α)  and t(α′ ^{∘}_{p} α) = t(α′)  for m = p + 1
s(α′ ^{∘}_{p} α) = s(α′)^{∘}_{p} s(α) and t(α′ ^{∘}_{p} α) = t(α′)^{∘}_{p} t(α) for m ≥ p + 2

ii. if m ≥ 0 and α ∈ A_{m} then s(i(α)) = α = t(i(α))

iii. if m > p ≥ 0 and α ∈ A_{m} then i^{m−p}(t^{m−p}(α))^{∘}_{p} α = α = α^{∘}_{p} i^{m−p}(s^{m−p}(α)); if also α′, α′′ are such that (α′′, α′), (α′, α) ∈ A_{m}×_{A_{p}} A_{m}, then (α′′ ^{∘}_{p} α′)^{∘}_{p} α = α′′ ^{∘}_{p} (α′ ^{∘}_{p} α)

iv. if p > q ≥ 0 and (f′, f) ∈ A_{p} ×_{A_{q}} A_{p} then i(f′)^{∘}_{q} i(f) = i(f′ ^{∘}_{q} f); if also m > p and α, α′, β, β′ are such that (β′, β), (α′, α) ∈ A_{m}×_{A_{p}}A_{m} and (β′, α′), (β, α) ∈ A_{m}×_{A_{q}}A_{m}, then (β′ ^{∘}_{p} β)^{∘}_{q} (α′ ^{∘}_{p} α) = (β′ ^{∘}_{q} α′)^{∘}_{p} (β^{∘}_{q} α).

The composition ^{∘}_{p} is ‘composition of m-cells by gluing along p-cells’. Pictures for (m, p) = (2, 1), (1, 0), (2, 0) are in the Bicategories section below.

Strict n-categories are defined similarly, but with the globular set only going up to A_{n}. Strict n- and ω-functors are maps of globular sets preserving composition and identities; the categories n-Cat and ω-Cat thus defined are equivalent to the ones defined above. The comments below on the two alternative definitions of bicategory give an impression of how this equivalence works.

Bicategories. Bicategories are the traditional and best-known formulation of ‘weak 2-category’. A bicategory B consists of

• a set B_{0}, whose elements a are called 0-cells or objects of B and drawn _{a}^{•}

• for each a, b ∈ B_{0}, a category B(a, b), whose objects f are called 1-cells and drawn

a_{•} ^{^{f} \overline{}}_{b}_{•} , whose arrows α : f \overline{}g are called 2-cells and drawn _{a}_{•} [figure: f; g; α] _{b}_{•} , and

whose composition _{a}_{•} [figure: f; g; h; α; β]_{b}_{•} ⟼ _{a}_{•} [figure: f; h; β^{∘}α] _{b}_{•} is called vertical composition of 2-cells

• for each a ∈ B_{0}, an object 1_{a} ∈ B(a, a) (the identity on a); and for each a, b, c ∈ B_{0}, a functor B(b, c)×B(a, b) \overline{}B(a, c), which on objects is called 1-cell composition, a_{•} ^{^{f} \overline{}}_{b}_{•} ^{^{g} \overline{}}_{c}_{•} ⟼ _{a}_{•} \overline{}^{g^{∘}f} _{c}_{•} , and on arrows is called horizontal composition of 2-cells,

[figure: a•; f; g; α; a′ •; f′; g′; α′; a′′ •; ⟼; a•; f′^{∘}f; g′^{∘}g; α′∗α; a′′ •]

• coherence 2-cells: for each f ∈ B(a, b), g ∈ B(b, c), h ∈ B(c, d), an associativity isomorphism ξ_{h,g,f} : (h^{∘}g)^{∘}f \overline{}h^{∘}(g^{∘}f); and for each f ∈ B(a, b), unit isomorphisms λ_{f} : 1_{b}^{∘}f \overline{}f and ρ_{f} : f^{∘}1_{a} \overline{}f

satisfying the following coherence axioms:

• ξ_{h,g,f} is natural in h, g and f, and λ_{f} and ρ_{f} are natural in f

• if f ∈ B(a, b), g ∈ B(b, c), h ∈ B(c, d), k ∈ B(d, e), then ξ_{k,h,g^{∘}f} ^{∘} ξ_{k^{∘}h,g,f} = (1_{k} ∗ ξ_{h,g,f})^{∘} ξ_{k,h^{∘}g,f} ^{∘} (ξ_{k,h,g} ∗ 1_{f}) (the pentagon axiom); and if f ∈ B(a, b), g ∈ B(b, c), then ρ_{g} ∗ 1_{f} = (1_{g} ∗ λ_{f})^{∘} ξ_{g,1_{b},f} (the triangle axiom).

An alternative definition is that a bicategory consists of a diagram of sets and functions

B_{2} [figure: s; t] B_{1} [figure: s; t] B_{0} satisfying ss = st and ts = tt, together with functions determining

composition, identities and coherence cells (in the style of the second definition of strict ω-category above). The idea is that B_{m} is the set of m-cells and that s and t give the source and target of a cell. Strict 2-categories can be identified with bicategories in which the coherence 2-cells are all identities.

A 1-cell _{a}_{•} ^{^{f} \overline{}}_{b}_{•} in a bicategory B is called an equivalence if there exists a 1-cell b_{•} ^{^{g} \overline{}}_{a}_{•} such that g^{∘}f ≅ 1_{a} and f^{∘}g ≅ 1_{b}.

A monoidal category can be defined as a bicategory with only one 0-cell, ⋆ say: for such a bicategory is just a category B(⋆, ⋆) equipped with an object I, a functor ⊗ : B(⋆, ⋆)^{2} \overline{}B(⋆, ⋆), and associativity and unit isomorphisms satisfying coherence.

We can consider strict functors of bicategories, in which composition etc is preserved strictly; more interesting are weak functors F, in which there are isomorphisms F g^{∘}F f \overline{}F (g^{∘}f) and 1_{F a} \overline{}F (1_{a}) satisfying coherence axioms.

Definition Tr

Topological Background

Spaces. Let Top be the category of topological spaces and continuous maps. Recall that compact spaces are exponentiable in Top: that is, if K is compact then the set Z^{K} of continuous maps from K to a space Z can be given a topology (namely, the compactopen topology) in such a way that there is an isomorphism Top(Y, Z^{K}) ≅ Top(K × Y, Z) natural in Y, Z ∈ Top.

Operads. A (non-symmetric, topological) operad D is a sequence (D(k))_{k≥0} of spaces together with an element (the identity) of D(1) and for each k, r_{1}, …, r_{k} ≥ 0 a map

D(k) × D(r_{1}) × ⋯ × D(r_{k}) \overline{}D(r_{1} + ⋯ + r_{k})

(composition), obeying unit and associativity laws. (Example: fix an object M of a monoidal category M, and define D(k) = M(M^{⊗k}, M).)

The All-Important Operad. There is an operad E in which E(k) is the space of continuous endpoint-preserving maps from [0, 1] to [0, k]. (‘Endpoint-preserving’ means that 0 maps to 0 and 1 to k.) The identity element of E(1) is the identity map, and composition in the operad is by substitution.

Path Spaces. For any space X and x, x′ ∈ X, a path from x to x′ in X is a map p : [0, 1] \overline{}X satisfying p(0) = x and p(1) = x′. There is a space X(x, x′) of paths from x to x′, a subspace of the exponential X^{[0,1]}.

Operad Action on Path Spaces. Fix a space X. For any k ≥ 0 and x_{0}, …, x_{k} ∈ X, there is a canonical map

act_{x_{0},...,x_{k}} : E(k) × X(x_{0}, x_{1}) × ⋯ × X(x_{k−1}, x_{k}) \overline{}X(x_{0}, x_{k}).

These maps are compatible with the composition and identity of the operad E, and the construction is functorial in X.

Path-Components. Let Π_{0} : Top \overline{}Set be the functor assigning to each space its set of path-components, and note that Π_{0} preserves finite products.

The Definition

We will define inductively, for each n ≥ 0, a category Wk-n-Cat with finite products and a functor Π_{n} : Top \overline{}Wk-n-Cat preserving finite products. A weak n-category is an object of Wk-n-Cat. (Maps in Wk-n-Cat are to be thought of as strict n-functors.)

Base Case. Wk-0-Cat = Set, and Π_{0} : Top \overline{}Set is as above.

A SURVEY OF DEFINITIONS OF n-CATEGORY  11

Objects of Wk-(n + 1)-Cat. Inductively, a weak (n+1)-category (A, γ) consists of • a set A_{0}

• a family (A(a, a′))_{a,a′∈A_{0}} of weak n-categories

• for each k ≥ 0 and a_{0}, …, a_{k} ∈ A_{0}, a map

γ_{a_{0},...,a_{k}} : Π_{n}(E(k)) × A(a_{0}, a_{1}) × ⋯ × A(a_{k−1}, a_{k}) \overline{}A(a_{0}, a_{k}) in Wk-n-Cat,

such that the γ_{a_{0},...,a_{k}}’s satisfy compatibility axioms of the same form as those satisfied by the act_{x_{0},...,x_{k}}’s. (All this makes sense because Π_{n} preserves finite products and Wk-n-Cat has them.) Maps in Wk-(n + 1)-Cat. A map (A, γ) \overline{}(B, δ) in Wk-(n + 1)-Cat consists of

• a function F_{0} : A_{0} \overline{}B_{0}

• for each a, a′ ∈ A_{0}, a map F_{a,a′} : A(a, a′) \overline{}B(F_{0}a, F_{0}a′) of weak n-categories, satisfying the axiom

F_{a_{0},a_{k}} ^{∘} γ_{a_{0},...,a_{k}} = δ_{F_{0}a_{0},...,F_{0}a_{k}} ^{∘} (1_{Π_{n}(E(k))} × F_{a_{0},a_{1}} × ⋯ × F_{a_{k−1},a_{k}})

for all k ≥ 0 and a_{0}, …, a_{k} ∈ A_{0}. Composition and Identities in Wk-(n + 1)-Cat. Obvious. Π_{n+1} on Objects. For a space X we define Π_{n+1}(X) = (A, γ), where

• A_{0} is the underlying set of X

• A(x, x′) = Π_{n}(X(x, x′))

• for x_{0}, …, x_{k} ∈ X, the map γ_{x_{0},...,x_{k}} is the composite

Π_{n}(E(k)) × Π_{n}(X(x_{0}, x_{1})) × ⋯ × Π_{n}(X(x_{k−1}, x_{k}))

^{∼}  Π_{n}(E(k) × X(x_{0}, x_{1}) × ⋯ × X(x_{k−1}, x_{k}))
^{Π_{n}(act_{x_{0},...,x_{k}})} Π_{n}(X(x_{0}, x_{k})).

Π_{n+1} on Maps. The functor Π_{n+1} is defined on maps in the obvious way. Finite Products Behave. It is easy to show that Wk-(n + 1)-Cat has finite products and that Π_{n+1} preserves finite products: so the inductive definition goes through.

Definition Tr for n ≤ 2

First observe that the space E(k) is contractible for each k (being, in a suitable sense, convex). In particular this tells us that E(k) is path-connected, and that the path space E(k)(θ, θ′) is path-connected for every θ, θ′ ∈ E(k).

n = 0

By definition, Wk-0-Cat = Set and Π_{0} : Top \overline{}Set is the path-components functor.

n = 1

The Category Wk-1-Cat. A weak 1-category (A, γ) consists of

• a set A_{0}

• a set A(a, a′) for each a, a′ ∈ A_{0}

• for each k ≥ 0 and a_{0}, …, a_{k} ∈ A_{0}, a function

γ_{a_{0},...,a_{k}} : Π_{0}(E(k)) × A(a_{0}, a_{1}) × ⋯ × A(a_{k−1}, a_{k}) \overline{}A(a_{0}, a_{k})

such that these functions satisfy certain axioms. So a weak 1-category looks something like a category: A_{0} is the set of objects, A(a, a′) is the set of maps from a to a′, and γ provides some kind of composition. Since E(k) is path-connected, we may strike out Π_{0}(E(k)) from the product above; and then we may suggestively write

(f_{k}^{∘} ⋯^{∘}f_{1}) = γ_{a_{0},...,a_{k}}(f_{1}, …, f_{k}).

The axioms on these ‘k-fold composition functions’ mean that a weak 1-category is, in fact, exactly a category. Maps in Wk-1-Cat are just functors, and so Wk-1-Cat is equivalent to Cat.

The Functor Π_{1}. For a space X, the (weak 1-)category Π_{1}(X) = (A, γ) is given by

• A_{0} is the underlying set of X

• A(x, x′) is the set of path-components of the path-space X(x, x′): that is, the set of homotopy classes of paths from x to x′

• Let x_{0} ^{p_{1}} ⋯ ^{p_{k}} x_{k} be a sequence of paths in X, and write [p] for the homotopy class of a path p. Then

([p_{k}]^{∘} ⋯^{∘}[p_{1}]) = [act_{x_{0},...,x_{k}}(θ, p_{1}, … p_{k})]

where θ is any member of E(k)—it doesn’t matter which. In other words, composition of paths is by laying them end to end.

Hence Π_{1}(X) is the usual fundamental groupoid of X, and indeed Π_{1} : Top \overline{}Cat is the usual fundamental groupoid functor. n = 2

A weak 2-category (A, γ) consists of

• a set A_{0}

• a category A(a, a′) for each a, a′ ∈ A_{0}

• for each k ≥ 0 and a_{0}, …, a_{k} ∈ A_{0}, a functor

γ_{a_{0},...,a_{k}} : Π_{1}(E(k)) × A(a_{0}, a_{1}) × ⋯ × A(a_{k−1}, a_{k}) \overline{}A(a_{0}, a_{k})

such that these functors satisfy axioms expressing compatibility with the composition and identity of the operad E.

By the description of Π_{1} and the initial observations of this section, the category Π_{1}(E(k)) is indiscrete (i.e. all hom-sets have one element) and its objects are the elements of E(k). So γ assigns to each θ ∈ E(k) and a_{i} ∈ A_{0} a functor

\overline{θ} : A(a_{0}, a_{1}) × ⋯ × A(a_{k−1}, a_{k}) \overline{}A(a_{0}, a_{k}),

and to each θ, θ′ ∈ E(k) and a_{i} ∈ A_{0} a natural isomorphism

ω_{θ,θ′} : \overline{θ} ^{∼} \overline{θ′}.

(Really we should add ‘a_{0}, …, a_{k}’ as a subscript to \overline{θ} and to ω_{θ,θ′}.) Functoriality of γ_{a_{0},...,a_{k}} says that

ω_{θ,θ} = 1, ω_{θ,θ′′} = ω_{θ′,θ′′} ^{∘} ω_{θ,θ′}.

The ‘certain axioms’ say firstly that

\overline{θ^{∘} (θ_{1}, …, θ_{k})} = \overline{θ}^{∘} (\overline{θ_{1}} × ⋯ × \overline{θ_{k}}), \overline{1} = 1

for θ ∈ E(k) and θ_{i} ∈ E(r_{i}), where the left-hand sides of the two equations refer respectively to composition and identity in the operad E; and secondly that the natural isomorphisms ω_{θ,θ′} fit together in a coherent way.

So a weak 2-category is probably not a structure with which we are already familiar. However, it nearly is. For define tr(k) to be the set of k-leafed rooted trees which are ‘unitrivalent’ (each vertex has either 0 or 2 edges coming up out of it); and suppose we replaced Π_{1}(E(k)) by the indiscrete category with object-set tr(k), so that the θ’s above would be trees. A weak 2-category would then be exactly a bicategory: e.g. if θ = _{•}∨ then \overline{θ} is binary composition, and if (θ, θ′) = ( _{•}^{•}∨ ∨, _{•}^{•} ∨∨) then ω_{θ,θ′} is the associativity isomorphism. And in some sense, a k-leafed tree might be thought of as a discrete version of an endpoint-preserving map [0, 1] \overline{}[0, k].

With this in mind, any weak 2-category (A, γ) gives rise to a bicategory B (although the converse process seems less straightforward). First pick at random an element θ_{2} of E(2), and let θ_{0} be the unique element of E(0). Then take B_{0} = A_{0}, B(a, a′) = A(a, a′), binary composition to be \overline{θ_{2}}, identities to be \overline{θ_{0}}, the associativity isomorphism to be ω_{θ_{2}^{∘}(1,θ_{2}),θ_{2}^{∘}(θ_{2},1)}, and similarly units. The coherence axioms on B follow from the coherence axioms on ω: and so we have a bicategory.

Definition P

Some Globular Structures

Reflexive Globular Sets. Let ℝ be the category whose objects are the natural numbers 0, 1, …, and whose arrows are generated by

⋯  m + 1 [figure: σ_{m+1}; τ_{m+1}; ι_{m+1}] m [figure: σ_{m}; τ_{m}; ι_{m}]  ⋯  [figure: σ_{1}; τ_{1}; ι_{1}] 0

subject to the equations

σ_{m}^{∘}σ_{m+1} = σ_{m}^{∘}τ_{m+1},  τ_{m}^{∘}σ_{m+1} = τ_{m}^{∘}τ_{m+1},  σ_{m}^{∘}ι_{m} = 1 = τ_{m}^{∘}ι_{m}

(m ≥ 1). A functor A : ℝ \overline{}Set is called a reflexive globular set. I will write s for A(σ_{m}), and t for A(τ_{m}), and 1_{a} for (A(ι_{m}))(a) when a ∈ A(m − 1).

Strict ω-Categories, and ω-Magmas. A strict ω-category is a reflexive globular set S together with a function (composition) ^{∘}_{p} : S(m) ×_{S(p)} S(m) \overline{}S(m) for each m > p ≥ 0, satisfying

• axioms determining the source and target of a composite (part (i) in the Preliminary section ‘Strict n-Categories’)

• strict associativity, unit and interchange axioms (parts (iii) and (iv)).

An ω-magma is like a strict ω-category, but only satisfying the first group of axioms ((i)) and not necessarily the second ((iii), (iv)). A map of ω-magmas is a map of reflexive globular sets which commutes with all the composition operations. (A strict ωfunctor between strict ω-categories is, therefore, just a map of the underlying ω-magmas.)

Contractions

Let ϕ : A \overline{}B be a map of reflexive globular sets. For m ≥ 1, define

V_{ϕ}(m) = {(f_{0}, f_{1}) ∈ A(m) × A(m) | s(f_{0}) = s(f_{1}), t(f_{0}) = t(f_{1}), ϕ(f_{0}) = ϕ(f_{1})},

and define

V_{ϕ}(0) = {(f_{0}, f_{1}) ∈ A(0) × A(0) | ϕ(f_{0}) = ϕ(f_{1})}.

A contraction γ on ϕ is a family of functions

(γ_{m} : V_{ϕ}(m) \overline{}A(m + 1))_{m≥0}

such that for all m ≥ 0 and (f_{0}, f_{1}) ∈ V_{ϕ}(m),

s(γ_{m}(f_{0}, f_{1})) = f_{0}, t(γ_{m}(f_{0}, f_{1})) = f_{1}, ϕ(γ_{m}(f_{0}, f_{1})) = 1_{ϕ(f_{0})}(= 1_{ϕ(f_{1})}),

and for all m ≥ 0 and f ∈ A(m),

γ_{m}(f, f) = 1_{f}.

M \underline{◁}  ω-magma
\underline{◁} map of ; ω-magmas
Sπ_{}\underline{◁}  strict ω- ; category

a [figure: f_{0}; f_{1}] b  ⟼  a [figure: f_{0}; f_{1}; γ_{1}(f_{0},f_{1})] b_{π(f_{0}) = π(f_{1})  π(γ_{1}(f_{0}, f_{1})) = 1_{π(f_{0})}}

Figure 2: An object of Q, with γ shown for m = 1

The Mysterious Category Q

Objects. An object of Q (see Fig. 2) is a quadruple (M, S, π, γ) in which

• M is an ω-magma

• S is a strict ω-category

• π is a map of ω-magmas from M to (the underlying ω-magma of) S

• γ is a contraction on π.

Maps. A map (M, S, π, γ) \overline{}(M′, S′, π′, γ′) in Q is a pair (M ^{^{χ}} M′, S ^{^{ζ}} S′) commuting with everything in sight. That is, χ is a map of ω-magmas, ζ is a strict ω-functor, π′^{∘}χ = ζ^{∘}π, and γ′_{m}(χ(f_{0}), χ(f_{1})) = χ(γ_{m}(f_{0}, f_{1})) for all (f_{0}, f_{1}) ∈ V_{M}(m).

Composition and Identities. These are defined in the obvious way.

The Definition

An Adjunction. Let U : Q \overline{}[ℝ, Set] be the functor sending (M, S, π, γ) to the underlying reflexive globular set of the ω-magma M. It can be shown that U has a left adjoint: so there is an induced monad T on [ℝ, Set].

Weak ω-Categories. A weak ω-category is a T -algebra.

Weak n-Categories. Let n ≥ 0. A reflexive globular set A is n-dimensional if for all m ≥ n, the map A(ι_{m+1}) : A(m) \overline{}A(m + 1) is an isomorphism (and so s = t = (A(ι_{m+1}))^{−1}). A weak n-category is a weak ω-category whose underlying reflexive globular set is n-dimensional.

Definition P for n ≤ 2

Direct Interpretation

The Left Adjoint in Low Dimensions. Here is a description of what the left adjoint F to U does in dimensions ≤ 2. It is perhaps not obvious that F as described does form the left adjoint; we come to that later. For a reflexive globular set A, write

F (A) = (\begin{matrix}A^{#} \\ A^{∗} & π_{A}_{} \end{matrix}, γ_{A}) .

A^{∗} is, in fact, relatively easy to describe: it is the free strict ω-category on A, in which an m-cell is a formal pasting-together of cells of A of dimension ≤ m.

Dimension 0 We have A^{#}(0) = A^{∗}(0) = A(0) and (π_{A})_{0} = id.

Dimension 1 Next, A^{∗}(1) is the set of formal paths of 1-cells in A, where we identify each identity cell 1_{a} with the identity path on a. The set A^{#}(1) and the functions s, t : A^{#}(1) \overline{}A(0) are generated by the following recursive clauses:

• if a_{0} ^{^{f}} a_{1} is a 1-cell in A then A^{#}(1) contains an element called f, with

s(f) = a_{0} and t(f) = a_{1}

• if w, w′ ∈ A^{#}(1) with t(w) = s(w′) then A^{#}(1) contains an element called

(w′ ^{•}_{0} w), with s(w′ ^{•}_{0} w) = s(w) and t(w′ ^{•}_{0} w) = t(w′).

The identities map A(0) \overline{}A^{#}(1) sends a to 1_{a} ∈ A(1) ⊆ A^{#}(1); the map π_{A} removes parentheses and sends ^{•}_{0} to ^{∘}_{0} ; the contraction γ_{A} is given by γ_{A}(a, a) = 1_{a}.

Dimension 2 A^{∗}(1) is the set of formal pastings of 2-cells in A, again respecting the identities. A^{#}(2) and s, t : A^{#}(2) \overline{}A^{#}(1) are generated by:

• if α is a 2-cell in A then A^{#}(2) has an element called α, with the evident source and target

• if a [figure: w_{0}; w_{1}] b in A^{#}(1) with π_{A}(w_{0}) = π_{A}(w_{1}) then A^{#}(2) has an element called

γ_{A}(w_{0}, w_{1}), with source w_{0} and target w_{1}

• if x, x′ ∈ A^{#}(2) with t(x) = s(x′) then A^{#}(2) has an element called (x′ ^{•}_{1} x), with source s(x) and target t(x′)

• if x, x′ ∈ A^{#}(2) with tt(x) = ss(x′) then A^{#}(2) has an element called (x′ ^{•}_{0} x), with source s(x′)^{•}_{0} s(x) and target t(x′)^{•}_{0} t(x);

furthermore, if f ∈ A(1) then 1_{f} (from the first clause) is to be identified with γ_{A}(f, f) (from the second). The identity map A^{#}(1) \overline{}A^{#}(2) sends w to γ_{A}(w, w). The map π_{A} sends cells of the form γ_{A}(w_{0}, w_{1}) to identity cells, and otherwise acts as in dimension 1. The contraction γ_{A} is defined in the way suggested by the notation.

Adjointness. We now have to see that this F is indeed left adjoint to U. First observe that there is a natural embedding of A(m) into A^{#}(m) (for m ≤ 2); this gives the unit of the adjunction. Adjointness then says: given (M, S, π, γ) ∈ Q and a map A ^{^{ϕ}} M of reflexive globular sets, there’s a unique map

(χ, ζ) : (\begin{matrix}A^{#} \\ A^{∗} & π_{A}_{} \end{matrix}) \overline{}(\begin{matrix}M \\ S & π_{} \end{matrix})

in Q such that χ extends ϕ. This can be seen from the description above.

Weak 2-Categories. A weak 2-category consists of a 2-dimensional reflexive globular set A together with:

• (a map A^{#}(0) \overline{}A(0) obeying axioms—which force it to be the identity)

• a map A^{#}(1) \overline{}A(1) obeying axioms, which amounts to a binary composition on the 1-cells of A (not obeying any axioms)

• similarly, vertical and horizontal binary compositions of 2-cells, not obeying any axioms ‘yet’

• for each string · ^{f_{1}} ⋯ ^{f_{k}} · of 1-cells, and each pair τ, τ′ of k-leafed binary trees, a 2-cell ω_{τ,τ′} : ^{∘}_{τ}(f_{1}, …, f_{k}) \overline{}^{∘}_{τ′}(f_{1}, …, f_{k}), where^{∘}_{τ} indicates the iterated composition dictated by the shape of τ

• amongst other things in dimension 3: whenever we have some 2-cells (α_{i}), and two different ways of composing all the α_{i}’s and some ω_{τ,τ′}’s to obtain new 2-cells β and β′ respectively, and these satisfy s(β) = s(β′) and t(β) = t(β′), then there is assigned a 3-cell β \overline{}β′. Since ‘the only 3-cells of A are equalities’, we get β = β′.

Analysing this precisely, we find that the category of weak 2-categories is equivalent to the category of bicategories and strict functors. And more easily, a weak 1-category is just a category and a weak 0-category is just a set.

Indirect Interpretation

An alternative way of handling weak n-categories is to work only with n-dimensional (not infinite-dimensional) structures throughout: e.g. reflexive globular sets A in which A(m) is only defined for m ≤ n. We then only speak of contractions on a map ϕ if (f_{0}, f_{1}) ∈ V_{ϕ}(n) ⇒ f_{0} = f_{1} (and in particular, the map π must satisfy this in order for (M, S, π, γ) to qualify as an object of Q). Our new category of weak n-categories appears to be equivalent to the old one, taking algebra maps as the morphisms in both cases.

The analysis of n = 2 is easier now: we can write down the left adjoint F explicitly, and so get an explicit description of the monad T on the category of ‘reflexive 2-globular sets’. This monad is presumably the free bicategory monad.

Definitions B

Globular Operads and their Algebras

Globular Sets. Let 𝔾 be the category whose objects are the natural numbers 0, 1, …, and whose arrows are generated by σ_{m}, τ_{m} : m \overline{}m − 1 for each m ≥ 1, subject to equations

σ_{m−1}^{∘}σ_{m} = σ_{m−1}^{∘}τ_{m}, τ_{m−1}^{∘}σ_{m} = τ_{m−1}^{∘}τ_{m}

(m ≥ 2). A functor A : 𝔾 \overline{}Set is called a globular set; I will write s for A(σ_{m}), and t for A(τ_{m}).

The Free Strict ω-Category Monad. Any (small) strict ω-category has an underlying globular set A, in which A(m) is the set of m-cells and s and t are the source and target maps. We thus obtain a forgetful functor U from the category of strict ω-categories and strict ω-functors to the category [𝔾, Set] of globular sets. U has a left adjoint, so there is an induced monad (T, id ^{^{η}} T, T^{2} ^{^{μ}} T) on [𝔾, Set].

Collections. We define a monoidal category Coll of collections. Let 1 be the terminal globular set. A (globular) collection is a map C ^{^{d}} T 1 into T 1 in [𝔾, Set]; a map of collections is a commutative triangle. The tensor product of collections C ^{^{d}} T 1, C′ ^{d′} T 1 is the composite along the top row of

[figure: C ⊗ C′; T C′; T d′; T^{2}1; μ_{1}; T 1; C; d; T 1,; T!]

where the right-angle symbol means that the square containing it is a pullback, and ! denotes the unique map to 1. The unit for the tensor is 1 ^{η_{1}} T 1.

Globular Operads. A (globular) operad is a monoid in the monoidal category Coll; a map of operads is a map of monoids.

Algebras. Any operad C induces a monad C ·− on [𝔾, Set]. For an object A of [𝔾, Set], this is defined by pullback:

[figure: C · A; T A; C; d; T 1.; T!]

The multiplication and unit of the monad come from the multiplication and unit of the operad. A C-algebra is an algebra for the monad C · −. Note that every C-algebra has an underlying globular set.

Contractions and Systems of Composition

Contractions. Let C ^{^{d}} T 1 be a collection. For m ≥ 0 and ν ∈ (T 1)(m), write C(ν) = {θ ∈ C(m) | d(θ) = ν}. For m ≥ 1 and ν ∈ (T 1)(m), define

Q_{C}(ν) = {(θ_{0}, θ_{1}) ∈ C(ν) × C(ν) | s(θ_{0}) = s(θ_{1}) and t(θ_{0}) = t(θ_{1})},

and for ν ∈ (T 1)(0), define Q_{C}(ν) = C(ν) × C(ν). Part of the strict ω-category structure on T 1 is that each element ν ∈ (T 1)(m) gives rise to an element 1_{ν} ∈ (T 1)(m + 1). A contraction on C is a family of functions

(γ_{ν} : Q_{C}(ν) \overline{}C(1_{ν}))_{m≥0,ν∈(T 1)(m)}

satisfying

s(γ_{ν}(θ_{0}, θ_{1})) = θ_{0}, t(γ_{ν}(θ_{0}, θ_{1})) = θ_{1}

for every m ≥ 0, ν ∈ (T 1)(m) and (θ_{0}, θ_{1}) ∈ Q_{C}(ν).

Systems of Compositions. The map η_{1} : 1 \overline{}T 1 picks out an element η_{1,m} of (T 1)(m) for each m ≥ 0. The strict ω-category structure on T 1 then gives an element

β_{p}^{m} = η_{1,m} ^{∘}_{p} η_{1,m} ∈ (T 1)(m)

for each m > p ≥ 0; also put β_{m}^{m} = η_{1,m}. Defining B(m) = {β_{p}^{m} | m ≥ p ≥ 0} ⊆ (T 1)(m), we obtain a collection B ^{⊂} ^{}T 1.

Also, the elements β_{m}^{m} = η_{1,m} ∈ (T 1)(m) determine a map 1 \overline{}B.

A system of compositions in an operad C is a map B \overline{}C of collections such that the composite 1 \overline{}B \overline{}C is the unit of the operad C.

Initial Object. Let OCS be the category in which an object is an operad equipped with both a contraction and a system of compositions, and in which a map is a map of operads preserving both the specified contraction and the specified system of compositions. Then OCS can be shown to have an initial object, whose underlying operad will be written K.

The Definitions

Definition B1. A weak ω-category is a K-algebra.

Definition B2. A weak ω-category is a pair (C, A), where C is an operad satisfying C(0) ≅ 1 and on which there exist a contraction and a system of compositions, and A is a C-algebra.

Weak n-Categories. Let n ≥ 0. A globular set A is n-dimensional if for all m ≥ n,

s = t : A(m + 1) \overline{}A(m)

and this map is an isomorphism. A weak n-category is a weak ω-category whose underlying globular set is n-dimensional. This can be interpreted according to either B1 or B2.

Definitions B for n ≤ 2

Definition B1

An alternative way of handling weak n-categories is to work with only n-dimensional (not infinite-dimensional) structures throughout. So we replace 𝔾 by its full subcategory 𝔾_{n} with objects 0, …, n, and T by the free strict n-category monad T_{n}, to obtain definitions of n-collection, n-operads, and their algebras. Contractions are defined as before, except that we only speak of contractions on C if

∀ν ∈ (T_{n}1)(n), (θ_{0}, θ_{1}) ∈ Q_{C}(ν) ⇒ θ_{0} = θ_{1}.  (∗)

There is an initial n-operad K_{n} equipped with a contraction and a system of compositions, and the category of weak n-categories turns out to be equivalent to the category of K_{n}- algebras. The latter is easier to analyse.

n = 0. We have [𝔾_{0}, Set] ≅ Set, T_{0} = id, and 0-Coll ≅ Set; a 0-operad C is a monoid, and a C-algebra is a set with a C-action. By (∗), the only 0-operad with a contraction is the one-element monoid, so a weak 0-category is just a set.

n = 1. [𝔾_{1}, Set] is the category of directed graphs and T_{1} is the free category monad. K_{1} is the terminal 1-operad, by arguments similar to those under ‘n = 2’ below. It follows that K_{1} · − is just T_{1}, and so a weak 1-category is just a T_{1}-algebra, that is, a category.

n = 2. A functor A : 𝔾_{2} \overline{}Set consists of a set of 0-cells (drawn _{a}^{•}), a set of 1-cells

(_{a}_{•} ^{^{f} \overline{}}_{b}_{•} ), and a set of 2-cells (_{a}_{•} [figure: f; g; α] _{b}_{•} ). A 2-collection C consists of a set C(0), a set

C(ν_{k}) for each k ≥ 0 (where ν_{k} indicates the ‘1-pasting diagram’ _{•}  \overline{}…  \overline{}_{•} with

k arrows), and a set C(π) for each ‘2-pasting diagram’ π such as the π_{i} in Fig. 3, together with source and target functions.

A 2-operad is a 2-collection C together with ‘composition’ functions such as

[figure: C(ν_{3}) × [C(ν_{2}) ×_{C(0)} C(ν_{1}) ×_{C(0)} C(ν_{2})]; C(ν_{5}),; C(π_{1}) × [C(π_{2}) ×_{C(ν_{2})} C(π_{3})]; C(π_{4}).]

In the first, the point is that there are 3 terms 2, 1, 2 and their sum is 5. This makes sense if an element of C(ν_{k}) is regarded as an operation which turns a string of k 1-cells into a single 1-cell. (The ×_{C(0)}’s denote pullbacks.) Similarly for the second; see Fig. 3. There are also identities. A C-algebra is a functor A : 𝔾_{2} \overline{}Set together with functions

\overline{ψ} : A(0) \overline{}A(0) for each ψ ∈ C(0),
\overline{ϕ} : \{diagrams _{a_{0}}_{•} ^{^{f_{1}}\overline{}}⋯ ^{^{f_{k}}\overline{}}_{a_{k}}_{•} in A\} \overline{}A(1) for each ϕ ∈ C(ν_{k}),
\overline{θ} : \{diagrams _{a}_{•} [figure: f; g; h; α; β]_{b}_{•} [figure: l; m; γ] _{c}_{•} in A\} \overline{}A(2) for each θ ∈ C [figure: (; •; •; •; )]

[figure: •; •; •; π_{2}; •; •; •; π_{3}; •; •; π_{1}; •; •; ⇝; •; •; •; π_{4}; •; •; ▽; △; ▽; ▽; θ_{1}; θ_{2}; θ; θ^{∘}(θ_{1}, θ_{2})]

Figure 3: Composition of operations in a globular operad

(etc), all compatible with the source, target, composition and identities in C.

K_{2} is generated from the empty collection by adding in the minimal amount to obtain a 2-operad with contraction and system of compositions. We have the identity 1 ∈ K_{2}(0). Then, contraction gives an element (‘1-cell identities’) of K_{2}(ν_{0}), the system of compositions gives an element (‘1-cell composition’) of K_{2}(ν_{2}), and composition in K_{2} gives one element of K_{2}(ν_{k}) for each k-leafed tree in which every vertex has 0 or 2 edges coming up out of it. Contraction at the next level gives associativity and unit isomorphisms and identity 2-cells; the system of compositions gives vertical and horizontal 2-cell composition. Condition (∗) gives coherence axioms. Thus a weak 2-category is exactly a bicategory.

Definition B2

Definition B2 of weak n-category refers to infinite-dimensional globular operads. So in order to do a concrete analysis of n ≤ 2, we redefine a weak n-category as a pair (C, A) where C is a n-operad admitting a contraction and a system of compositions and with C(0) ≅ 1, and A is a C-algebra. (Temporarily, call such a C good.) I do not know to what extent this is equivalent to B2, but the spirit, at least, is the same.

n = 0. From ‘n = 0’ above we see that a weak 0-category is just a set.

n = 1. The only good 1-operad is the terminal 1-operad, so by ‘n = 1’ above, a weak 1-category is just a category.

n = 2. A (non-symmetric) classical operad D is a sequence (D(k))_{k≥0} of sets together with an element (the identity) of D(1) and for each k, r_{1}, …, r_{k} ≥ 0 a map D(k)×D(r_{1})× ⋯ × D(r_{k}) ^{}D(r_{1} + ⋯ + r_{k}) (composition), obeying unit and associativity laws. It turns out that good 2-operads C correspond one-to-one with classical operads D such that D(k) ≠ ∅ for each k, via D(k) = C(ν_{k}). A C-algebra is then something like a 2-category or bicategory, with one way of composing a string of k 1-cells for each element of D(k), and all the appropriate coherence 2-cells. E.g. if D = 1 then a C-algebra is a 2-category; if D(k) is the set of k-leafed trees in which each vertex has either 0 or 2 edges coming up out of it then a C-algebra is a bicategory. C can, therefore, be regarded as a theory of (more or less weak) 2-categories, and A as a model for such a theory.

Definitions L

Globular Operads and their Algebras

Globular Sets. Let 𝔾 be the category whose objects are the natural numbers 0, 1, …, and whose arrows are generated by σ_{m}, τ_{m} : m \overline{}m − 1 for each m ≥ 1, subject to equations

σ_{m−1}^{∘}σ_{m} = σ_{m−1}^{∘}τ_{m}, τ_{m−1}^{∘}σ_{m} = τ_{m−1}^{∘}τ_{m}

(m ≥ 2). A functor A : 𝔾 \overline{}Set is called a globular set; I will write s for A(σ_{m}), and t for A(τ_{m}).

The Free Strict ω-Category Monad. Any (small) strict ω-category has an underlying globular set A, in which A(m) is the set of m-cells and s and t are the source and target maps. We thus obtain a forgetful functor U from the category of strict ω-categories and strict ω-functors to the category [𝔾, Set] of globular sets. U has a left adjoint, so there is an induced monad (T, id ^{^{η}} T, T^{2} ^{^{μ}} T) on [𝔾, Set].

Collections. We define a monoidal category Coll of collections. Let 1 be the terminal globular set. A (globular) collection is a map C ^{^{d}} T 1 into T 1 in [𝔾, Set]; a map of collections is a commutative triangle. The tensor product of collections C ^{^{d}} T 1, C′ ^{d′} T 1 is the composite along the top row of

[figure: C ⊗ C′; T C′; T d′; T^{2}1; μ_{1}; T 1; C; d; T 1,; T!]

where the right-angle symbol means that the square containing it is a pullback, and ! denotes the unique map to 1. The unit for the tensor is 1 ^{η_{1}} T 1.

Globular Operads. A (globular) operad is a monoid in the monoidal category Coll; a map of operads is a map of monoids.

Algebras. Any operad C induces a monad C ·− on [𝔾, Set]. For an object A of [𝔾, Set], this is defined by pullback:

[figure: C · A; T A; C; d; T 1.; T!]

The multiplication and unit of the monad come from the multiplication and unit of the operad. A C-algebra is an algebra for the monad C · −. Note that every C-algebra has an underlying globular set.

Contractions

Contractions. Let C ^{^{d}} T 1 be a collection. For m ≥ 0 and ν ∈ (T 1)(m), write C(ν) = {θ ∈ C(m) | d(θ) = ν}. For m ≥ 2 and π ∈ (T 1)(m), define

P_{C}(π) = {(θ_{0}, θ_{1}) ∈ C(s(π)) × C(t(π)) | s(θ_{0}) = s(θ_{1}) and t(θ_{0}) = t(θ_{1})},

and for π ∈ (T 1)(1), define P_{C}(π) = C(s(π)) × C(t(π)). A contraction γ on C is a family of functions

(γ_{π} : P_{C}(π) \overline{}C(π))_{m≥1,π∈(T 1)(m)}

satisfying

s(γ_{π}(θ_{0}, θ_{1})) = θ_{0}, t(γ_{π}(θ_{0}, θ_{1})) = θ_{1}

for every m ≥ 1, π ∈ (T 1)(m) and (θ_{0}, θ_{1}) ∈ P_{C}(π).

Initial Object. Let OC be the category in which an object is an operad equipped with a contraction and a map is a map of operads preserving the specified contraction. Then OC can be shown to have an initial object, whose underlying operad will be written L.

The Definitions

Definition L1. A weak ω-category is an L-algebra. (Maps of L-algebras should be regarded as strict ω-functors.)

Definition L2. A weak ω-category is a pair (C, A), where C is an operad on which there exists a contraction and satisfying C(0) ≅ 1, and A is a C-algebra.

Weak n-Categories. Let n ≥ 0. A globular set A is n-dimensional if for all m ≥ n,

s = t : A(m + 1) \overline{}A(m)

and this map is an isomorphism. A weak n-category is a weak ω-category whose underlying globular set is n-dimensional. This can be interpreted according to either L1 or L2.

Definitions L for n ≤ 2

Definition L1

An alternative way of handling weak n-categories is to work with only n- (not infinite-) dimensional structures throughout. So we replace 𝔾 by its full subcategory 𝔾_{n} with objects 0, …, n, and T by the free strict n-category monad T_{n}, to obtain definitions of n-collection, n-operads, and their algebras. Contractions are defined as before, except that we only speak of contractions on C if

∀ν ∈ (T_{n}1)(n), ∀θ_{0}, θ_{1} ∈ C(ν), s(θ_{0}) = s(θ_{1}) & t(θ_{0}) = t(θ_{1}) ⇒ θ_{0} = θ_{1}  (†)

(taking C(−1) = 1 to understand this when n = 0). There is an initial n-operad L_{n} equipped with a contraction, and the category of weak n-categories turns out to be equivalent to the category of L_{n}-algebras. The latter is easier to analyse.

n = 0. We have [𝔾_{0}, Set] ≅ Set, T_{0} = id, and 0-Coll ≅ Set; a 0-operad C is a monoid, and a C-algebra is a set with a C-action. By (†), the only 0-operad with a contraction is the one-element monoid, so a weak 0-category is just a set.

n = 1. [𝔾_{1}, Set] is the category of directed graphs and T_{1} is the free category monad. L_{1} is the terminal 1-operad, by arguments similar to those under ‘n = 2’ below. It follows that L_{1} · − is just T_{1}, and so a weak 1-category is just a T_{1}-algebra, that is, a category.

n = 2. A functor A : 𝔾_{2} \overline{}Set consists of a set of 0-cells (drawn _{a}^{•}), a set of 1-cells

(_{a}_{•} ^{^{f} \overline{}}_{b}_{•} ), and a set of 2-cells (_{a}_{•} [figure: f; g; α] _{b}_{•} ). A 2-collection C consists of a set C(0), a set

C(ν_{k}) for each k ≥ 0 (where ν_{k} indicates the ‘1-pasting diagram’ _{•}  \overline{}…  \overline{}_{•} with

k arrows), and a set C(π) for each ‘2-pasting diagram’ π such as the π_{i} in Fig. 4, together with source and target functions.

A 2-operad is a 2-collection C together with ‘composition’ functions such as

[figure: C(ν_{3}) × [C(ν_{2}) ×_{C(0)} C(ν_{1}) ×_{C(0)} C(ν_{2})]; C(ν_{5}),; C(π_{1}) × [C(π_{2}) ×_{C(ν_{2})} C(π_{3})]; C(π_{4}).]

In the first, the point is that there are 3 terms 2, 1, 2 and their sum is 5. This makes sense if an element of C(ν_{k}) is regarded as an operation which turns a string of k 1-cells into a single 1-cell. (The ×_{C(0)}’s denote pullbacks.) Similarly for the second; see Fig. 4. There are also identities. A C-algebra is a functor A : 𝔾_{2} \overline{}Set together with functions

\overline{ψ} : A(0) \overline{}A(0) for each ψ ∈ C(0),
\overline{ϕ} : \{diagrams _{a_{0}}_{•} ^{^{f_{1}}\overline{}}⋯ ^{^{f_{k}}\overline{}}_{a_{k}}_{•} in A\} \overline{}A(1) for each ϕ ∈ C(ν_{k}),
\overline{θ} : \{diagrams _{a}_{•} [figure: f; g; h; α; β]_{b}_{•} [figure: l; m; γ] _{c}_{•} in A\} \overline{}A(2) for each θ ∈ C(_{•} [figure]_{•} [figure] _{•})

[figure: •; •; •; π_{2}; •; •; •; π_{3}; •; •; π_{1}; •; •; ⇝; •; •; •; π_{4}; •; •; ▽; △; ▽; ▽; θ_{1}; θ_{2}; θ; θ^{∘}(θ_{1}, θ_{2})]

Figure 4: Composition of operations in a globular operad

(etc), all compatible with the source, target, composition and identities in C.

L_{2} is generated from the empty collection by adding in the minimal amount to obtain a 2-operad-with-contraction. We have the identity 1 ∈ L_{2}(0). Then, contraction gives an element of L_{2}(ν_{k}) for each k, so that composition in L_{2} gives an element of L_{2}(ν_{k}) for each k-leafed tree. Contraction at the next level (with (†)) says that if π is a 2-pasting diagram of width k then L_{2}(π) = L_{2}(ν_{k}) × L_{2}(ν_{k}). So L_{2}(0) = {1}, L_{2}(ν_{k}) = {k-leafed trees}, L_{2}(π) = {k-leafed trees}^{2}. An L_{2}-algebra is, then, an ‘unbiased bicategory’: that is, just like a bicategory except that there is specified k-fold composition for every k ≥ 0 rather than just k = 2 (binary composition) and k = 0 (identities). Since these are essentially the same as ordinary bicategories, so too are weak 2-categories.

Definition L2

Definition L2 of weak n-category refers to infinite-dimensional globular operads. So in order to do a concrete analysis of n ≤ 2, we redefine a weak n-category as a pair (C, A) where C is a n-operad admitting a contraction and with C(0) ≅ 1, and A is a C-algebra. (Temporarily, call such a C good.) I do not know to what extent this is equivalent to L2, but the spirit, at least, is the same.

n = 0. From ‘n = 0’ above we see that a weak 0-category is just a set.

n = 1. The only good 1-operad is the terminal 1-operad, so by ‘n = 1’ above, a weak 1-category is just a category.

n = 2. A (non-symmetric) classical operad D is a sequence (D(k))_{k≥0} of sets together with an element (the identity) of D(1) and for each k, r_{1}, …, r_{k} ≥ 0 a map D(k)×D(r_{1})× ⋯ × D(r_{k}) ^{}D(r_{1} + ⋯ + r_{k}) (composition), obeying unit and associativity laws. It turns out that good 2-operads C correspond one-to-one with classical operads D such that D(k) ≠ ∅ for each k, via D(k) = C(ν_{k}). A C-algebra is then something like a 2-category or bicategory, with one way of composing a string of k 1-cells for each element of D(k), and all the appropriate coherence 2-cells. E.g. if D = 1 then a C-algebra is a 2-category; if D(k) is the set of k-leafed trees in which each vertex has either 0 or 2 edges coming up out of it then a C-algebra is a bicategory. C can, therefore, be regarded as a theory of (more or less weak) 2-categories, and A as a model for such a theory.

Definition L′

Globular Multicategories

Globular Sets. Let 𝔾 be the category whose objects are the natural numbers 0, 1, …, and whose arrows are generated by σ_{m}, τ_{m} : m \overline{}m − 1 for each m ≥ 1, subject to equations

σ_{m−1}^{∘}σ_{m} = σ_{m−1}^{∘}τ_{m}, τ_{m−1}^{∘}σ_{m} = τ_{m−1}^{∘}τ_{m}

(m ≥ 2). A functor A : 𝔾 \overline{}Set is called a globular set; I will write s for A(σ_{m}), and t for A(τ_{m}).

The Free Strict ω-Category Monad. Any (small) strict ω-category has an underlying globular set A, in which A(m) is the set of m-cells and s and t are the source and target maps. We thus obtain a forgetful functor U from the category of strict ω-categories and strict ω-functors to the category [𝔾, Set] of globular sets. U has a left adjoint, so there is an induced monad (T, id ^{^{η}} T, T^{2} ^{^{μ}} T) on [𝔾, Set].

Globular Graphs. For each globular set A, we define a monoidal category Graph_{A}. An object of Graph_{A} is a (globular) graph on A: that is, a globular set R together with maps of globular sets

R
[figure: dom; cod]
T A  A.

A map (R, dom, cod) \overline{}(R′, dom′, cod′) of graphs on A is a map R \overline{}R′ making the evident triangles commute. The tensor product of graphs (R, dom, cod), (R′, dom′, cod′) is given by composing along the upper edges of the following diagram, in which the rightangle symbol means that the square containing it is a pullback:

R ⊗ R′ [figure]
[figure]
T R′  R
[figure: T dom′; T cod′; dom; cod]
T^{2}A  T A  A.
[figure: μ_{A}]
T A

[figure: q_{0}^{•}; r_{0}; r_{1}; ∃ρ; ·; ·; ·; ·; ·; ·; ·; q_{1}^{•}; ⟼^{d}; d(q_{0})^{•}; d(r_{0}); d(r_{1}); ϕ; d(q_{1})^{•}]

Figure 5: Part (ii) of the definition of contractibility, shown for m = 1

The unit for the tensor is the graph

A
[figure: η_{A}; 1]
T A  A.

Globular Multicategories. A globular multicategory is a globular set A together with a monoid in Graph_{A}. A globular multicategory A therefore consists of a globular set A, a graph (R, dom, cod) on A, and maps comp : R⊗R \overline{}R and ids : A \overline{}R compatible with dom and cod and obeying associativity and identity laws.

Contractible Maps

A map d : R \overline{}S of globular sets is contractible (Figure 5) if

i. the function d_{0} : R(0) \overline{}S(0) is bijective, and

ii. for every

m ≥ 0,

r_{0}, r_{1} ∈ R(m) with s(r_{0}) = s(r_{1}) and t(r_{0}) = t(r_{1}),

ϕ ∈ S(m + 1) with s(ϕ) = d_{m}(r_{0}) and t(ϕ) = d_{m}(r_{1}),

there exists ρ ∈ R(m + 1) with s(ρ) = r_{0}, t(ρ) = r_{1}, and d_{m+1}(ρ) = ϕ. In the case m = 0 we drop the (nonsensical) conditions that s(r_{0}) = s(r_{1}) and t(r_{0}) = t(r_{1}).

The Definition

Weak ω-Categories. A weak ω-category is a globular multicategory A = (A, R, dom, cod, comp, ids) such that dom : R \overline{}T A is contractible.

Weak n-Categories. Let n ≥ 0. A globular set A is n-dimensional if for all m ≥ n,

s = t : A(m + 1) \overline{}A(m)

and this map is an isomorphism. A weak n-category is a weak ω-category A such that the globular sets A and R are n-dimensional.

Definition L′ for n ≤ 2

An alternative way of handling weak n-categories is to work with only n-dimensional (not infinite-dimensional) structures throughout. Thus we replace 𝔾 by its full subcategory 𝔾_{n} with objects 0, …, n, replace T by the free strict n-category monad T_{n}, and so obtain a definition of n-globular multicategory. We also modify part (ii) of the definition of contractibility by changing ‘m ≥ 0’ to ‘n − 1 ≥ m ≥ 0’, and ‘there exists ρ’ to ‘there exists a unique ρ’ in the case m = n − 1. From these ingredients we get a new definition of weak n-category.

The new and old definitions give two different, but equivalent, categories of weak ncategories (with maps of multicategories as the morphisms); the analysis of n ≤ 2 is more convenient with the new definition.

n = 0. We have [𝔾_{0}, Set] ≅ Set and T_{0} = id, and the contractible maps are the bijections. So a weak 0-category is a category whose domain map is a bijection; that is, a discrete category; that is, a set.

n = 1. [𝔾_{1}, Set] is the category of directed graphs, T_{1} is the free category monad on it, and a map of graphs is contractible if and only if it is an isomorphism. So a weak 1-category is essentially a 1-globular multicategory whose underlying 1-globular graph looks like T_{1}A ^{1} T_{1}A ^{cod} A. Such a graph has at most one multicategory structure, and it has one if and only if cod is a T_{1}-algebra structure on A. So a weak 1-category is just a T_{1}-algebra, i.e., a category.

n = 2. The free 2-category T_{2}A on a 2-globular set A ∈ [𝔾_{2}, Set] has the same 0-cells as A; 1-cells of T_{2}A are formal paths ψ in A as in Fig. 6(a); and a typical 2-cell of T_{2}A is the diagram ϕ in Fig. 6(b).

Next, what is a 2-globular multicategory A = (A, R, dom, cod, comp, ids)? Since we ultimately want to consider just those A in which dom is contractible, let us assume immediately that R(0) = A(0). Then A consists of:

• a 2-globular set A ∈ [𝔾_{2}, Set]

• for each ψ and f as in Fig. 6(a), a set of cells r : ψ ⇒ f; such an r is a 1-cell of R, and can be regarded as a ‘reason why f is a composite of ψ’

• for each ϕ and g_{0}, g_{1}, α as in Fig. 6(b), a set of cells ρ : ϕ ⇛ α; such a ρ is a 2-cell of R, and can be regarded as a ‘reason why α is a composite of ϕ’

• source and target functions R(2) [figure] R(1), which, for instance, assign to ρ a reason s(ρ) why g_{0} is a composite of _{a_{0}}_{•} ^{^{f_{1}}\overline{}}_{a_{1}}_{•} ^{^{f_{5}}\overline{}}_{a_{2}}_{•} ^{^{f_{6}}\overline{}}_{a_{3}}_{•}

• composition and identities: given r as in Fig. 6(a) and similarly r_{i} : (f_{i}^{1}, …, f_{i}^{p_{i}}) ⇒ f_{i} for each i = 1, …, k, there is a composite r^{∘} (r_{1}, …, r_{k}) : (f_{1}^{1}, …, f_{k}^{p_{k}}) ⇒ f; and similarly for 2-cells and for identities,

ψ = _{a_{0}}_{•} ^{^{f_{1}}\overline{}}… ^{^{f_{k}}\overline{}}_{a_{k}}_{•} ⇓r^{_{a_{0}}_{•} ^{^{f} \overline{}}_{a_{k}}_{•}} (a)  ϕ = _{a_{0}}_{•} [figure: f_{1}; f_{2}; f_{3}; f_{4}; α_{1}; α_{2}; α_{3}]_{a_{1}}_{•} ^{^{f_{5}}\overline{}}_{a_{2}}_{•} [figure: f_{6}; f_{7}; f_{8}; α_{4}; α_{5}]_{a_{3}}_{•} ⇓ρ^{_{a_{0}}_{•} [figure: g_{0}; g_{1}; α] _{a_{3}}_{•}} (b)

Figure 6: (a) A 1-cell, and (b) a typical 2-cell, of R. Here a_{i}, f_{i}, f, g_{i}, α_{i} and α are all cells of A

such that the composition and identities satisfy associativity and identity axioms and are compatible with source and target.

Contractibility says that for each ψ as in Fig. 6(a) there is at least one pair (r, f) as in Fig. 6(a), and that for each ϕ as in Fig. 6(b) and each r_{0} : (f_{1}, f_{5}, f_{6}) ⇒ g_{0} and r_{1} : (f_{4}, f_{5}, f_{8}) ⇒ g_{1}, there is exactly one pair (ρ, α) as in Fig. 6(b) satisfying s(ρ) = r_{0} and t(ρ) = r_{1}. That is: every diagram of 1-cells has at least one composite, and every diagram of 2-cells has exactly one composite once a way of composing the 1-cells along its boundary has been chosen.

When ϕ = _{a_{0}}_{•} [figure: f_{0}; f_{1}; f_{2}; α_{1}; α_{2}]_{a_{1}}_{•} , the identity reasons for f_{0} and f_{2} give via contractibility a

composite α_{2}^{∘}α_{1} : f_{0} \overline{}f_{2}, and in this way the 1- and 2-cells between a_{0} and a_{1} form a category A(a_{0}, a_{1}). Now suppose that ψ is as in Fig. 6(a) and r : ψ ⇒ f, r′ : ψ ⇒ f′. Applying contractibility to the degenerate 2-cell diagram ϕ which looks exactly like ψ,

we obtain a 2-cell _{a_{0}}_{•} [figure: f; f′] _{a_{k}}_{•} ; and similarly the other way round; so by the uniqueness

property of the ρ’s, f ≅ f′ in A(a_{0}, a_{k}). Thus any two composites of a string of 1-cells are canonically isomorphic.

A weak 2-category is essentially what is known as an ‘anabicategory’. To see how one of these gives rise to a bicategory, choose for each _{a_{0}}_{•} ^{^{f} \overline{}}_{a_{1}}_{•} ^{^{g} \overline{}}_{a_{2}}_{•} in A a reason r_{f,g} :

(f, g) ⇒ h and write h = (g^{∘}f); and similarly for identities. Then, for instance, the hori-

zontal composite of 2-cells _{a_{0}}_{•} [figure: f_{0}; f_{1}]_{a_{1}}_{•} [figure: g_{0}; g_{1}] _{a_{2}}_{•} comes via contractibility from r_{f_{0},g_{0}} and r_{f_{1},g_{1}},

the associativity cells arise from the degenerate 2-cell diagram ϕ = _{a_{0}}_{•} ^{^{f_{1}}\overline{}}_{a_{1}}_{•} ^{^{f_{2}}\overline{}}_{a_{2}}_{•} ^{^{f_{3}}\overline{}}_{a_{3}}_{•} , and the coherence axioms come from the uniqueness of the ρ’s.

Definition Si

Simplicial Objects

The Simplicial Category. Let Δ be a skeleton of the category of nonempty finite totally ordered sets: that is, Δ has objects [k] = {0, …, k} for k ≥ 0, and maps are order-preserving functions (with respect to the usual ordering ≤).

Some Maps in Δ. Let σ, τ : [0] \overline{}[1] be the maps in Δ with respective values 0 and 1. Given k ≥ 0, let ι_{1}, …, ι_{k} : [1] \overline{}[k] denote the ‘embeddings’ of [1] into [k], defined by ι_{j}(0) = j − 1 and ι_{j}(1) = j.

The Segal Maps. Let k ≥ 0. Then the following diagram in Δ commutes:

[k]
[figure: ι_{1}; ι_{2}; …; ι_{k}]
[1]  [1]  …  [1]
[figure: τ; σ; τ; σ; τ; σ]
[0]  [0]  ⋯  [0].

Let X : Δ^{op} \overline{}E be a functor into a category E possessing finite limits, and write X[1] ×_{X[0]} ⋯ ×_{X[0]} X[1] (with k occurrences of X[1]) for the limit of the diagram

X[1]  X[1]  …  X[1]
[figure: Xτ; Xσ; Xτ; Xσ; Xτ; Xσ]
X[0]  X[0]  ⋯  X[0]

(with, again, k occurrences of X[1]) in E. Then by commutativity of the first diagram, there is an induced map in E—a Segal map—

X[k] \overline{}X[1] ×_{X[0]} ⋯ ×_{X[0]} X[1].  (‡)

Contractibility

Sources and Targets. If 0 ≤ p ≤ r, write I_{p} for the object (\underbrace{[1], …, [1]}_{p}, \underbrace{[0], …, [0]}_{r−p}) of

Δ^{r}. Let X : (Δ^{r})^{op} \overline{}Set, 0 ≤ p ≤ r, and x, x′ ∈ X(I_{p}). Then x, x′ are parallel if p = 0 or if p ≥ 1 and s(x) = s(x′) and t(x) = t(x′); here s and t are the maps

X(I_{p})

X(id, …, id, σ, id, …, id)
X(id, …, id, τ, id, …, id)^{} X(I_{p−1}).

[figure: x; .; .; .; .; .; .; .; .; .; .; .; .; g; x′; ϕ(x); h; \begin{aligned}& ϕ \\ ↓ \end{aligned}; ϕ(x′)]

Figure 7: Contractibility, shown for p = 0

Contractible Maps. Let r ≥ 1 and let X, Y : (Δ^{r})^{op} \overline{}Set. A natural transformation ϕ : X \overline{}Y is contractible (Figure 7) if

• the function ϕ_{I_{0}} : X(I_{0}) \overline{}Y (I_{0}) is surjective

• given p ∈ {0, …, r − 1}, parallel x, x′ ∈ X(I_{p}), and h ∈ Y (I_{p+1}) satisfying

s(h) = ϕ_{I_{p}}(x), t(h) = ϕ_{I_{p}}(x′),

there exists g ∈ X(I_{p+1}) satisfying

s(g) = x, t(g) = x′, ϕ_{I_{p+1}}(g) = h

• given parallel x, x′ ∈ X(I_{r}) satisfying ϕ_{I_{r}}(x) = ϕ_{I_{r}}(x′), then x = x′.

If r = 0 then X and Y are just sets and ϕ is just a function X \overline{}Y; call ϕ contractible if it is bijective.

The Definition Let n ≥ 0. A weak n-category is a functor A : (Δ^{n})^{op} \overline{}Set such that for each m ∈ {0, …, n − 1} and K = ([k_{1}], …, [k_{m}]) ∈ Δ^{m},

i. the functor A(K, [0], −) : (Δ^{n−m−1})^{op} \overline{}Set is constant, and ii. for each [k] ∈ Δ, the Segal map

A(K, [k], −) \overline{}A(K, [1], −) ×_{A(K,[0],−)} ⋯ ×_{A(K,[0],−)} A(K, [1], −) is contractible. (We are taking E = [(Δ^{n−m−1})^{op}, Set] and X[j] = A(K, [j], −) in the definition of Segal map.)

Definition Si for n ≤ 2

n = 0. Parts (i) and (ii) of the definition are vacuous, so a weak 0-category is just a functor (Δ^{0})^{op} \overline{}Set, that is, a set.

n = 1. A weak 1-category is a functor A : Δ^{op} \overline{}Set (that is, a simplicial set) satisfying (i) and (ii). Part (i) is always true, and (ii) says that for each k ≥ 0 the Segal map (‡) (with X = A) is a bijection—in other words, that A is a nerve. The category of nerves and natural transformations between them is equivalent to Cat, where a nerve A corresponds to a certain category with object-set A[0] and morphism-set A[1]. So a weak 1-category is essentially just a category.

n = 2. A weak 2-category is a functor A : (Δ^{2})^{op} \overline{}Set such that

i. the functor A([0], −) : Δ^{op} \overline{}Set is constant

ii. for each k ≥ 0, the Segal map

A([k], −) \overline{}A([1], −) ×_{A([0],−)} ⋯ ×_{A([0],−)} A([1], −)

is contractible, and for each k_{1}, k ≥ 0, the Segal map

A([k_{1}], [k]) \overline{}A([k_{1}], [1]) ×_{A([k_{1}],[0])} ⋯ ×_{A([k_{1}],[0])} A([k_{1}], [1])

is a bijection.

The second half of (ii) says that A([k_{1}], −) is a nerve for each k_{1}, so we can regard A as a functor Δ^{op} \overline{}Cat. Note that if X and Y are nerves then a natural transformation ϕ : X \overline{}Y is the same thing as a functor between the corresponding categories, and that ϕ is contractible if and only if this functor is full, faithful and surjective on objects. So a weak 2-category is a functor A : Δ^{op} \overline{}Cat such that

i. A[0] is a discrete category (i.e. the only morphisms are the identities)

ii. for each k ≥ 0, the Segal functor

A[k] \overline{}A[1] ×_{A[0]} ⋯ ×_{A[0]} A[1]

is full, faithful and surjective on objects.

I will now argue that a weak 2-category is essentially the same thing as a bicategory. First take a weak 2-category A : Δ^{op} \overline{}Cat, and let us construct a bicategory B.

The object-set of B is A[0]. The two functors s, t : A[1] \overline{}A[0] express the category A[1] as a disjoint union ∐_{a,b∈A[0]} B(a, b) of categories; the 1-cells from a to b are the objects of B(a, b), and the 2-cells are the morphisms. Vertical composition of 2-cells in B is composition in each B(a, b). To define horizontal composition of 1- and 2-cells, first choose for each k a pseudo-inverse A[1] ×_{A[0]} ⋯ ×_{A[0]} A[1] ^{ψ_{k}} A[k] to the Segal functor ϕ_{k} (an equivalence of categories), and natural isomorphisms η_{k} : 1 \overline{}ψ_{k} ^{∘} ϕ_{k}, ε_{k} : ϕ_{k} ^{∘} ψ_{k} \overline{}1. Horizontal composition is given by

A[1] ×_{A[0]} A[1] ^{ψ_{2}} A[2] ^{Aδ} A[1],

where δ : [1] \overline{}[2] is the injection whose image omits 1 ∈ [2]. The associativity isomorphisms are built up from η_{k}’s and ε_{k}’s, and the pentagon commutes just as long as the equivalence (ϕ_{k}, ψ_{k}, η_{k}, ε_{k}) was chosen to be an adjunction too (which is always possible). Identities work similarly.

Conversely, take a bicategory B and construct a weak 2-category A : (Δ^{2})^{op} \overline{}Set (its ‘2-nerve’) as follows. An element of A([j], [k]) is a quadruple

((a_{u})_{0≤u≤j}, (f_{uv}^{z}) \begin{matrix}0≤u<v≤j \\ 0≤z≤k \end{matrix} , (α_{uv}^{z}) \begin{matrix}0≤u<v≤j \\ 1≤z≤k \end{matrix} , (ι_{uvw}^{z}) \begin{matrix}0≤u<v<w≤j \\ 0≤z≤k \end{matrix} )

where

• a_{u} is an object of B

• f_{uv}^{z} : a_{u} \overline{}a_{v} is a 1-cell of B

• α_{uv}^{z} : f_{uv}^{z−1} \overline{}f_{uv}^{z} is a 2-cell of B

• ι_{uvw}^{z} : f_{vw}^{z}^{∘}f_{uv}^{z} ^{∼} f_{uw}^{z} is an invertible 2-cell of B

such that

• ι_{uvw}^{z} ^{∘} (α_{vw}^{z} ∗ α_{uv}^{z}) = α_{uw}^{z} ^{∘} ι_{uvw}^{z−1} whenever 0 ≤ u < v < w ≤ j, 1 ≤ z ≤ k

• ι_{uwx}^{z} ^{∘} (1_{f_{wx}^{z}} ∗ ι_{uvw}^{z})^{∘} (associativity isomorphism) = ι_{uvx}^{z} ^{∘} (ι_{vwx}^{z} ∗ 1_{f_{uv}^{z}}) whenever 0 ≤ u < v < w < x ≤ j, 0 ≤ z ≤ k.

This defines the functor A on objects of Δ^{2}; it is defined on maps by a combination of inserting identities and forgetting data.

To get a rough picture of A, consider the analogous construction for strict 2-categories, in which we insist that the isomorphisms ι_{uvw}^{z} are actually equalities. Then an element of A([j], [k]) is a grid of jk 2-cells, of width j and height k. (When j = 0 this is just a single object of B, regardless of k.) The bicategorical version is a suitable weakening of this construction.

Finally, passing from a bicategory to a weak 2-category and back again gives a bicategory isomorphic (in the category of bicategories and weak functors) to the original one. Passing from a weak 2-category to a bicategory and back again gives a weak 2-category which is ‘equivalent’ to the original one in a sense which we do not have quite enough vocabulary to make precise here.

Definition Ta

Simplicial Objects

The Simplicial Category. Let Δ be a skeleton of the category of nonempty finite totally ordered sets: that is, Δ has objects [k] = {0, …, k} for k ≥ 0, and maps are order-preserving functions (with respect to the usual ordering ≤).

Some Objects and Morphisms. Given k ≥ 0, let ι_{1}, …, ι_{k} : [1] \overline{}[k] denote the ‘embeddings’ of [1] into [k], defined by ι_{j}(0) = j −1 and ι_{j}(1) = j. Let σ, τ : [0] \overline{}[1] be the maps in Δ with respective values 0 and 1. Given p ≥ 0, write 0^{p} = ([0], …, [0]) ∈ Δ^{p} and 1^{p} = ([1], …, [1]) ∈ Δ^{p}. Let X : (Δ^{r})^{op} \overline{}Set, 0 ≤ p ≤ r, and x, x′ ∈ X(1^{p}, 0^{r−p}). Then x, x′ are parallel if p = 0 or if p ≥ 1 and s(x) = s(x′) and t(x) = t(x′); here s and t are the maps

X(1^{p}, 0^{r−p})

X(id, …, id, σ, id, …, id)
X(id, …, id, τ, id, …, id)^{} X(1^{p−1}, 0^{r−p+1}).

The Segal Maps. Let k ≥ 0. Then the following diagram in Δ commutes:

[k]
[figure: ι_{1}; ι_{2}; …; ι_{k}]
[1]  [1]  …  [1]
[figure: τ; σ; τ; σ; τ; σ]
[0]  [0]  ⋯  [0].

Let X : Δ^{op} \overline{}E be a functor into a category E possessing finite limits, and write X[1] ×_{X[0]} ⋯ ×_{X[0]} X[1] (with k occurrences of X[1]) for the limit of the diagram

X[1]  X[1]  …  X[1]
[figure: Xτ; Xσ; Xτ; Xσ; Xτ; Xσ]
X[0]  X[0]  ⋯  X[0]

(with, again, k occurrences of X[1]) in E. Then by commutativity of the first diagram, there is an induced map in E—a Segal map—

X[k] \overline{}X[1] ×_{X[0]} ⋯ ×_{X[0]} X[1].  (§)

Nerves. Call X : Δ^{op} \overline{}Set a nerve if for each k ≥ 0, the Segal map (§) is a bijection. The category of nerves and natural transformations is equivalent to Cat, where a nerve X corresponds to a category with object-set X[0] and morphism-set X[1]. Let QX be the set of isomorphism classes of objects of the category corresponding to X, and let π_{X} : X[0] \overline{}QX be the quotient map.

Truncatability

For each r ≥ 0, we define what it means for a functor X : (Δ^{r})^{op} \overline{}Set to be truncatable, writing r-Trunc for the category of truncatable functors (Δ^{r})^{op} \overline{}Set and natural transformations between them. We also define functors ob^{(r)}, Q^{(r)} : r-Trunc \overline{}Set and a natural transformation π^{(r)} : ob^{(r)} \overline{}Q^{(r)}.

The functor ob^{(r)} is given by ob^{(r)}X = X(0^{r}). All functors (Δ^{0})^{op} \overline{}Set are truncatable, and Q^{(0)} and π^{(0)} are identities. Inductively, when r ≥ 1, a functor X : (Δ^{r})^{op} \overline{}Set is truncatable if

• for each k ≥ 0, the functor X([k], −) : (Δ^{r−1})^{op} \overline{}Set is truncatable

• the functor X̂: Δ^{op} \overline{}Set defined by [k] ⟼ Q^{(r−1)}(X([k], −)) is a nerve.

If X is truncatable then we define Q^{(r)}(X) = Q(X̂) and π_{X}^{(r)} = π_{X} ^{∘} π_{X([0],−)}^{(r−1)}.

Equivalence

Internal Equivalence. Let 0 ≤ p ≤ r, let X : (Δ^{r})^{op} \overline{}Set be truncatable, and let x_{1}, x_{2} ∈ X(1^{p}, 0^{r−p}). We call x_{1} and x_{2} equivalent, and write x_{1} ∼ x_{2}, if x_{1} and x_{2} are parallel and π_{X(1^{p},−)}^{(r−p)}(x_{1}) = π_{X(1^{p},−)}^{(r−p)}(x_{2}).

External Equivalence. Let r ≥ 0. A natural transformation ϕ : X \overline{}Y of truncatable functors X, Y : (Δ^{r})^{op} \overline{}Set is called an equivalence if

• for each y ∈ Y (0^{r}) there is a unique-up-to-equivalence x ∈ X(0^{r}) such that ϕ_{0^{r}}(x) ∼ y

• for all 0 ≤ p ≤ r − 1, parallel x, x′ ∈ X(1^{p}, 0^{r−p}), and h ∈ Y (1^{p+1}, 0^{r−p−1}) satisfying

s(h) = ϕ_{(1^{p},0^{r−p})}(x), t(h) = ϕ_{(1^{p},0^{r−p})}(x′),

there is an element g ∈ X(1^{p+1}, 0^{r−p−1}), unique up to equivalence, satisfying

s(g) = x,  t(g) = x′,  ϕ_{(1^{p+1},0^{r−p−1})}(g) ∼ h.

The Definition

Let n ≥ 0. A weak n-category is a truncatable functor A : (Δ^{n})^{op} \overline{}Set such that for each m ∈ {0, …, n − 1} and K = ([k_{1}], …, [k_{m}]) ∈ Δ^{m},

i. the functor A(K, [0], −) : (Δ^{n−m−1})^{op} \overline{}Set is constant, and

ii. for each [k] ∈ Δ, the Segal map

A(K, [k], −) \overline{}A(K, [1], −) ×_{A(K,[0],−)} ⋯ ×_{A(K,[0],−)} A(K, [1], −)  (¶)

is an equivalence. (We are taking E = [(Δ^{n−m−1})^{op}, Set] and X[j] = A(K, [j], −) in the definition of Segal map, and we can check that both the domain and the codomain of (¶) are truncatable.)

Definition Ta for n ≤ 2

n = 0. Parts (i) and (ii) of the definition are vacuous, and truncatability is automatic, so a weak 0-category is just a functor (Δ^{0})^{op} \overline{}Set, that is, a set.

n = 1. Note that a functor A : Δ^{op} \overline{}Set is truncatable exactly when it is a nerve; that a functor X : (Δ^{0})^{op} \overline{}Set is merely a set, and two elements of X are equivalent just when they are equal; and that a map ϕ : X \overline{}Y of functors X, Y : (Δ^{0})^{op} \overline{}Set is an equivalence just when it is a bijection. A weak 1-category is a truncatable functor A : Δ^{op} \overline{}Set satisfying (i) and (ii). Part (i) is trivially true, and both truncatability and (ii) say that A is a nerve. So a weak 1-category is just a nerve, that is, a category.

n = 2. First note that if X : Δ^{op} \overline{}Set is a nerve then two elements of X[0] are equivalent just when they are isomorphic (as objects of the category corresponding to X), and two elements of X[1] are equivalent just when they are equal. Note also that a map ϕ : X \overline{}Y of nerves is an equivalence if and only if (regarded as a functor between the corresponding categories) it is full, faithful and essentially surjective on objects—that is, an equivalence of categories.

A weak 2-category is a truncatable functor A : (Δ^{2})^{op} \overline{}Set such that

i. the functor A([0], −) : Δ^{op} \overline{}Set is constant

ii. for each k ≥ 0, the Segal map

A([k], −) \overline{}A([1], −) ×_{A([0],−)} ⋯ ×_{A([0],−)} A([1], −)

is an equivalence, and for each k_{1}, k ≥ 0, the Segal map

A([k_{1}], [k]) \overline{}A([k_{1}], [1]) ×_{A([k_{1}],[0])} ⋯ ×_{A([k_{1}],[0])} A([k_{1}], [1])

is a bijection.

The second half of (ii) says that A([k_{1}], −) is a nerve for each k_{1}, so we can regard A as a functor A : Δ^{op} \overline{}Cat; then the first half of (ii) says that the Segal map (§) (with X = A) is an equivalence of categories. Truncatability of A says that the functor Δ^{op} \overline{}Set given by [k] ⟼ {isomorphism classes of objects of A[k]} is a nerve, which follows anyway from the other conditions. So a weak 2-category is a functor A : Δ^{op} \overline{}Cat such that

i. A[0] is a discrete category (i.e. the only morphisms are the identities)

ii. for each k ≥ 0, the Segal functor A[k] \overline{}A[1] ×_{A[0]} ⋯ ×_{A[0]} A[1] is an equivalence of categories.

It seems that a weak 2-category is essentially just a bicategory.

First take a weak 2-category A : Δ^{op} \overline{}Cat, and let us construct a bicategory B. The object-set of B is A[0]. The two functors s, t : A[1] \overline{}A[0] express the category A[1] as a disjoint union ∐_{a,b∈A[0]} B(a, b) of categories; the 1-cells from a to b are the objects of B(a, b), and the 2-cells are the morphisms. Vertical composition of 2-cells in B is composition in each B(a, b). To define horizontal composition of 1- and 2-cells, first choose for each k a pseudo-inverse A[1] ×_{A[0]} ⋯ ×_{A[0]} A[1] ^{ψ_{k}} A[k] to the Segal functor ϕ_{k}, and choose natural isomorphisms η_{k} : 1 \overline{}ψ_{k} ^{∘} ϕ_{k}, ε_{k} : ϕ_{k} ^{∘} ψ_{k} \overline{}1. Horizontal composition is then given as

A[1] ×_{A[0]} A[1] ^{ψ_{2}} A[2] ^{Aδ} A[1],

where δ : [1] \overline{}[2] is the injection whose image omits 1 ∈ [2]. The associativity isomorphisms are built up from η_{k}’s and ε_{k}’s, and the pentagon commutes just as long as the equivalence (ϕ_{k}, ψ_{k}, η_{k}, ε_{k}) was chosen to be an adjunction too (which is always possible). Identities work similarly.

Conversely, take a bicategory B and construct a weak 2-category A : (Δ^{2})^{op} \overline{}Set (its ‘2-nerve’) as follows. An element of A([j], [k]) is a quadruple

((a_{u})_{0≤u≤j}, (f_{uv}^{z}) \begin{matrix}0≤u<v≤j \\ 0≤z≤k \end{matrix} , (α_{uv}^{z}) \begin{matrix}0≤u<v≤j \\ 1≤z≤k \end{matrix} , (ι_{uvw}^{z}) \begin{matrix}0≤u<v<w≤j \\ 0≤z≤k \end{matrix} )

where

• a_{u} is an object of B

• f_{uv}^{z} : a_{u} \overline{}a_{v} is a 1-cell of B

• α_{uv}^{z} : f_{uv}^{z−1} \overline{}f_{uv}^{z} is a 2-cell of B

• ι_{uvw}^{z} : f_{vw}^{z}^{∘}f_{uv}^{z} ^{∼} f_{uw}^{z} is an invertible 2-cell of B

such that

• ι_{uvw}^{z} ^{∘} (α_{vw}^{z} ∗ α_{uv}^{z}) = α_{uw}^{z} ^{∘} ι_{uvw}^{z−1} whenever 0 ≤ u < v < w ≤ j, 1 ≤ z ≤ k

• ι_{uwx}^{z} ^{∘} (1_{f_{wx}^{z}} ∗ ι_{uvw}^{z})^{∘} (associativity isomorphism) = ι_{uvx}^{z} ^{∘} (ι_{vwx}^{z} ∗ 1_{f_{uv}^{z}}) whenever 0 ≤ u < v < w < x ≤ j, 0 ≤ z ≤ k.

This defines the functor A on objects of Δ^{2}; it is defined on maps by a combination of inserting identities and forgetting data.

To get a rough picture of A, consider the analogous construction for strict 2-categories, in which we insist that the isomorphisms ι_{uvw}^{z} are actually equalities. Then an element of A([j], [k]) is a grid of jk 2-cells, of width j and height k. (When j = 0 this is just a single object of B, regardless of k.) The bicategorical version is a suitable weakening of this construction.

Finally, it appears that passing from a bicategory to a weak 2-category and back again gives a bicategory isomorphic (by weak functors) to the original one, and that passing from a weak 2-category to a bicategory and back again gives a weak 2-category equivalent to the original one.

Definition J

Disks

Disks. A disk D is a diagram of sets and functions

⋯  D_{m} [figure: p_{m}; u_{m}; v_{m}] D_{m−1}  ⋯  [figure: p_{2}; u_{2}; v_{2}] D_{1} [figure: p_{1}; u_{1}; v_{1}] D_{0} = 1

equipped with a total order on the fibre p_{m}^{−1}(d) for each m ≥ 1 and d ∈ D_{m−1}, such that for each m ≥ 1 and d ∈ D_{m−1},

• u_{m}(d) and v_{m}(d) are respectively the least and greatest elements of p_{m}^{−1}(d)

• u_{m}(d) = v_{m}(d) ⟺ d ∈ image(u_{m−1}) ∪ image(v_{m−1}).

When m = 1, the second condition is to be interpreted as saying that u_{1} ≠ v_{1} (or equivalently, that D_{1} has at least two elements).

A map D ^{^{ψ}} D′ of disks is a family of functions (D_{m} ^{ψ_{m}} D′_{m})_{m≥0} commuting with the p’s, u’s and v’s and preserving the order in each fibre. (The last condition means that if d ∈ D_{m−1} and b, c ∈ p_{m}^{−1}(d) with b ≤ c, then ψ_{m}(b) ≤ ψ_{m}(c) ∈ p_{m}^{′−1}(ψ_{m−1}(d)).) Call ψ a surjection if each ψ_{m} is a surjection.

Interiors, Volume, Dimension. Let D be a disk. For m ≥ 1, define

ιD_{m} = D_{m}\(image(u_{m}) ∪ image(v_{m})),

(the interior of D_{m}), and define ιD_{0} = D_{0}. If the set ∐_{m≥1} ιD_{m} is finite then we call D finite and define the volume |D| of D to be its cardinality. In this case we may also define the dimension of D to be the largest m ≥ 0 for which ιD_{m} ≠ ∅.

Finite Disks. Write 𝔻 for a skeleton of the category of finite disks and maps between them. In other words, take the category of all finite disks and choose one object in each isomorphism class; the objects of 𝔻 are all these chosen objects, and the morphisms in 𝔻 are all disk maps between them. Thus 𝔻 is equivalent to the category of finite disks and no two distinct objects of 𝔻 are isomorphic.

Faces and Horns, Cofaces and Cohorns

Cofaces. Let D ∈ 𝔻. A (covolume 1) coface of D is a surjection D ^{^{ϕ}} E in 𝔻 where |E| = |D| − 1. We call ϕ an inner coface of D if ϕ_{m}(ιD_{m}) ⊆ ιE_{m} for all m ≥ 0.

Cohorns. For each D ∈ 𝔻 and coface D ^{^{ϕ}} E of 𝔻, define the cohorn

Λ_{ϕ}^{D} : 𝔻 \overline{}Set

by

Λ_{ϕ}^{D}(C) = {ψ ∈ 𝔻(D, C) | ψ factors through some coface of D other than ϕ}.

That is, a map ψ : D \overline{}C is a member of Λ_{ϕ}^{D}(C) if and only if there is a coface

(D ^{ϕ′} E′) ≠ (D ^{^{ϕ}} E) of D and a map χ : E′ \overline{}C such that

D  \overline{ϕ′ ^{}}E′
[figure: ψ]
Cχ_{}

commutes. There is an inclusion Λ_{ϕ}^{D}(C)^{⊂} ^{}𝔻(D, C) for each C, and Λ_{ϕ}^{D} is thus a subfunctor of 𝔻(D, −). Write i_{ϕ}^{D} : Λ_{ϕ}^{D} ^{⊂} ^{}𝔻(D, −) for the inclusion.

Fillers. Let A : 𝔻 \overline{}Set, let D ∈ 𝔻, and let ϕ be a coface of D. A (D, ϕ)-cohorn in A is a natural transformation h : Λ_{ϕ}^{D} \overline{}A; if ϕ is an inner coface then h is an inner cohorn in A.

A filler for a (D, ϕ)-cohorn h in A is a natural transformation \overline{h} : 𝔻(D, −) \overline{}A making the following diagram commute:

Λ_{ϕ}^{D} ^{_{⊂} i_{ϕ}^{D}^{}}𝔻(D, −)
[figure: h]
A.\overline{h}_{}

The Definition

Weak ω-Categories. A weak ω-category is a functor A : 𝔻 \overline{}Set such that there exists a filler for every inner cohorn in A.

Weak n-Categories. Let n ≥ 0. A functor A : 𝔻 \overline{}Set is n-dimensional if, whenever ψ : D \overline{}E is a map in 𝔻 such that

• D has dimension n

• ψ_{m} is a bijection for every m ≤ n,

then A(ψ) is a bijection. A weak n-category is an n-dimensional weak ω-category.

Definition J for n ≤ 2

Let n ≥ 0. An n-disk is defined in the same way as a disk, except that D_{m}, p_{m}, u_{m} and v_{m} are now only defined for m ≤ n: so an n-disk is essentially the same thing as a disk of dimension ≤ n. Write 𝔻_{n} for a skeleton of the category of finite n-disks. An n-dimensional functor 𝔻 \overline{}Set is determined by its effect on disks of dimension ≤ n, and conversely any functor 𝔻_{n} \overline{}Set extends uniquely to become an n-dimensional functor 𝔻 \overline{}Set. So the category of n-dimensional functors 𝔻 \overline{}Set is equivalent to [𝔻_{n}, Set].

Take an n-dimensional functor A : 𝔻 \overline{}Set and its restriction Ã : 𝔻_{n} \overline{}Set. Then there is automatically a unique filler for every cohorn of dimension ≥ n + 2 in A (that is, cohorn Λ_{ϕ}^{D} \overline{}A where D has dimension ≥ n + 2). Moreover, there exists a filler for every inner cohorn of dimension n+1 in A if and only if there is at most one filler for every inner cohorn of dimension n in Ã. (We do not prove this, but the idea of the method is in the last sentence of ‘n = 2’.) So: a weak n-category is a functor 𝔻_{n} \overline{}Set such that every inner cohorn has a filler, unique when the cohorn is of dimension n.

n = 0. 𝔻_{0} is the terminal category 1. The unique 0-disk has no cofaces, so a weak 0-category is merely a functor 1 \overline{}Set, that is, a set.

n = 1. An interval is a totally ordered set with a least and a greatest element, and is called strict if these elements are distinct. 𝔻_{1} is (a skeleton of) the category of finite strict intervals, so we can take its objects to be the intervals ⟨k⟩ = {0, …, k + 1} for k ≥ 0 and its morphisms to be the interval maps.

The cofaces of ⟨k⟩ are the surjections ⟨k⟩ \overline{}⟨k − 1⟩ (assuming k ≥ 1; if k = 0 then there are none). They are ϕ_{0}, …, ϕ_{k}, where ϕ_{i} identifies i and i + 1; of these, ϕ_{1}, …, ϕ_{k−1} are inner. The cohorn Λ_{ϕ_{i}}^{⟨k⟩} : 𝔻_{1} \overline{}Set sends ⟨l⟩ to

\{ψ : ⟨k⟩ \overline{}⟨l⟩ | ψ factors through ϕ_{i′} for some i′ ∈ \{0, …, i − 1, i + 1, …, k\}\}.

Now, let Δ be a skeleton of the category of nonempty finite totally ordered sets, with objects [k] = {0, …, k} (k ≥ 0). Then 𝔻_{1} ≅ Δ^{op}, with ⟨k⟩ corresponding to [k], the cofaces ϕ_{i} : ⟨k⟩ \overline{}⟨k − 1⟩ to the usual face maps [k − 1] \overline{}[k], and the inner cofaces to the inner faces (i.e. all but the first and last). Trivially, cohorns Λ_{ϕ_{i}}^{⟨k⟩} correspond to horns in the standard sense, and fillers to fillers. Hence a weak 1-category is a functor A : Δ^{op} \overline{}Set in which every inner horn has a unique filler—exactly the condition that A is the nerve of a category. So a weak 1-category is just a category.

n = 2. Again we use a duality. Given natural numbers l_{1}, …, l_{k}, let T_{l_{1},...,l_{k}} be the strict 2category generated by objects x_{0}, …, x_{k}, 1-cells p_{i}^{j} : x_{i−1} \overline{}x_{i} (1 ≤ i ≤ k, 0 ≤ j ≤ l_{i}), and 2-cells ξ_{i}^{j} : p_{i}^{j−1} \overline{}p_{i}^{j} (1 ≤ i ≤ k, 1 ≤ j ≤ l_{i}). (E.g. the lower half of Fig. 8(a) shows T_{1,0}.) Let Δ_{2} be the category whose objects are sequences (l_{1}, …, l_{k}) with k, l_{i} ≥ 0, and whose maps (l_{1}, …, l_{k}) \overline{}(l′_{1}, …, l′_{k′}) are the strict 2-functors T_{l_{1},...,l_{k}} \overline{}T_{l′_{1},...,l′_{k′}}. Then 𝔻_{2} ≅ Δ_{2}^{op}. On objects, this says that a finite 2-disk is just a finite sequence of numbers, e.g. (1, 0) in Fig. 8(a).

[figure: ∘ ∘ •^{w}∘ ∘ ∘ ∘; ∘; •u; •v; ∘; •; ∘ ∘ ∘^{w} ∘ ∘ ∘; ∘ •u  •v ∘; •; ∘ ∘^{w} ∘ ∘ ∘ ∘; ∘ •u  •v ∘; •; ∘ ∘ •^{w}∘ ∘; ∘; •u; ∘v; •; ∘ ∘ •^{w}∘ ∘; ∘; •_{u v}; ∘; •; •; •; •; •; •; •; •; •; •; •; •; •; •; (a)  (b)  (c)  (d)  (e)]

Figure 8: The duality. In the upper row, • denotes an interior element and ∘ an endpoint of a fibre, and the labels u, v, w show what the coface maps ‘ϕ’ do

Any bicategory B has a ‘nerve’ A : Δ_{2}^{op} \overline{}Set, where A(l_{1}, …, l_{k}) = \{weak functors T_{l_{1},...,l_{k}} \overline{}B strictly preserving identities\}. We can recover B from A, so weak 2-categories are the same as bicategories just as long as the definition gives the right conditions on functors 𝔻_{2} ≅ Δ_{2}^{op} \overline{}Set. I do not have a full proof that this is so, hence there are gaps in what follows.

Defining faces of an object of Δ_{2} as cofaces of the corresponding object of 𝔻_{2}, and similarly horns, a weak 2-category is a functor Δ_{2}^{op} \overline{}Set in which every 1- (respectively, 2-) dimensional horn has a filler (respectively, unique filler). Faces are certain subcategories: e.g. Fig. 8(b)–(e) shows the 4 cofaces of a disk and correspondingly the 4 faces of T_{1,0}, of which only (e) is inner.

For the converse of the nerve construction, we take a weak 2-category A and form a bicategory B. Its graph B_{2} [figure] B_{1} [figure] B_{0} is the image under A of the diagram • [figure] _{•} [figure: ⊃; ⊃] _{•} \overline{}_{•} [figure: ⊃; ⊃] ^{•} in Δ_{2}. A diagram _{a}_{•} ^{^{f} \overline{}}_{b}_{•} ^{^{g} \overline{}}_{c}_{•} in B is a horn in A for

the unique inner face of T_{0,0} = _{•}  \overline{}_{•}  \overline{}_{•}; choose a filler K_{f,g} and write g^{∘}f for its

third face. This gives 1-cell composition; vertical 2-cell composition works similarly but

without choice. Next, a diagram _{a}_{•} [figure: f; f′; α] _{b}_{•} ^{^{g} \overline{}}_{c}_{•} , with K_{f,g} and K_{f′,g}, forms a horn for the

unique inner face of T_{1,0} (Fig. 8), so has a unique filler K_{α,g}; write g^{∘}α : g^{∘}f \overline{}g^{∘}f′ for face (e) of K_{α,g}. Horizontal 2-cell composition is defined via this construction, its dual, and vertical composition. Next, _{a}_{•} ^{^{f} \overline{}}_{b}_{•} ^{^{g} \overline{}}_{c}_{•} ^{^{h} \overline{}}_{d}_{•} , together with K_{f,g}, K_{g,h}, K_{g^{∘}f,h}, gives an

inner horn for T_{0,0,0}. There’s a (unique?) filler, whose final face L_{f,g,h} is itself a filler of

a_{•} ^{^{f} \overline{}}_{b}_{•} \overline{}^{h^{∘}g} _{d}_{•} with third face h^{∘}(g^{∘}f). Considering _{a}_{•} [figure: f; f; 1] _{b}_{•} \overline{}^{h^{∘}g} _{d}_{•} with K_{f,h^{∘}g} and L_{f,g,h}

gives an invertible 2-cell (h^{∘}g)^{∘}f \overline{}h^{∘}(g^{∘}f).

Definition St

Simplicial Sets

The Simplicial Category. Let Δ be a skeleton of the category of nonempty finite totally ordered sets: that is, Δ has objects [m] = {0, …, m} for m ≥ 0, and maps are order-preserving functions (with respect to the usual ordering ≤). A simplicial set is a functor Δ^{op} \overline{}Set.

Maps in Δ. Let m ≥ 1: then there are injections δ_{0}, …, δ_{m} : [m − 1] \overline{}[m] in Δ, determined by saying that the image of δ_{i} is [m]\{i}.

Let A : Δ^{op} \overline{}Set and m ≥ 0. An element a ∈ A[m] is called degenerate if there exist a natural number m′ < m, a surjection σ : [m] \overline{}[m′], and an element a′ ∈ A[m′] such that a = (Aσ)a′.

Horns. Given 0 ≤ k ≤ m, we define the horn Λ_{m}^{k} : Δ^{op} \overline{}Set by

Λ_{m}^{k}[m′] = {ψ ∈ Δ([m′], [m]) | image(ψ) ⊉ [m]\{k}}.

That is, Λ_{m}^{k}[m′] is the set of all maps ψ : [m′] \overline{}[m] in Δ except for the surjections and the maps with image {0, …, k − 1, k + 1, …, m}. So for each m′ we have an inclusion Λ_{m}^{k}[m′]^{⊂} ^{}Δ([m′], [m]), and Λ_{m}^{k} is thus a subfunctor of Δ(−, [m]). Write i_{m}^{k} : Λ_{m}^{k} ^{⊂} ^{}Δ(−, [m]) for the inclusion.

Let A be a simplicial set. A horn in A is a natural transformation h : Λ_{m}^{k} \overline{}A, for some 0 ≤ k ≤ m. A filler for the horn h is a natural transformation \overline{h} : Δ(−, [m]) \overline{}A making the following diagram commute:

Λ_{m}^{k} ^{_{⊂} i_{m}^{k}^{}}Δ(−, [m])
[figure: h]
A.\overline{h}_{}

Orientation

Alternating Sets. A set of natural numbers is alternating if its elements, when written in ascending order, alternate in parity.

Let 0 ≤ k ≤ m, and write k^{±} = {k−1, k, k+1}∩[m]. A subset S ⊆ [m] is k-alternating if

• k^{±} ⊆ S

• the set k^{±} ∪ ([m]\S) is alternating.

Admissible Horns. A simplicial set with hollowness is a simplicial set A together with a subset H_{m} ⊆ A[m] for each m ≥ 1, whose elements are called the hollow elements of A[m] (and may also be thought of as ‘thin’ or ‘universal’).

Let (A, H) be a simplicial set with hollowness, and 0 ≤ k ≤ m. A horn h : Λ_{m}^{k} \overline{}A is admissible if for every m′ ≥ 1 and ψ ∈ Λ_{m}^{k}[m′],

image(ψ) is a k-alternating subset of [m] ⇒ h_{[m′]}(ψ) is hollow.

The Definition

Weak ω-Categories. A weak ω-category is a simplicial set with hollowness (A, H) such that

i. for m ≥ 1, H_{m} ⊇ {degenerate elements of A[m]}

ii. for m ≥ 1 and 0 ≤ k ≤ m, every admissible horn h : Λ_{m}^{k} \overline{}A has a filler \overline{h} satisfying \overline{h}_{[m]}(1_{[m]}) ∈ H_{m} (‘every admissible horn has a hollow filler’)

iii. for m ≥ 2 and 0 ≤ k ≤ m, if a ∈ H_{m} has the property that (Aδ_{i})a ∈ H_{m−1} for each i ∈ [m]\{k} then also (Aδ_{k})a ∈ H_{m−1}.

Weak n-Categories. Let n ≥ 0. A weak n-category is a weak ω-category (A, H) such that

i′. for m > n, H_{m} = A[m]

ii′. in condition (ii) above, when m > n there is a unique filler \overline{h} for h (which necessarily satisfies \overline{h}_{[m]}(1_{[m]}) ∈ H_{m}).

Definition St for n ≤ 2

Let m ≥ 0 and let S be a nonempty subset of [m]; then in Δ there is a unique injection ϕ into [m] with image S. Given a simplicial set A and an element a ∈ A[m], the S-face of a is the element (Aϕ)a of A[l], where l + 1 is the cardinality of S. Similarly, the S-face of a horn h : Λ_{m}^{k} \overline{}A is h_{[l]}(ϕ) ∈ A[l] (which makes sense as long as S ⊉ [m]\\{k\}).

To compare weak 1-(2-)categories with (bi)categories, we need to interpret elements a ∈ A[m] as arrows pointing in some direction. Our convention is: if S is an m-element subset of the (m + 1)-element set [m] and the missing element is odd, then we regard the S-face of a as a source; if even, a target. See Fig. 9.

Suppose (A, H) is a simplicial set with hollowness satisfying (i), and let h : Λ_{m}^{k} \overline{}A be a horn satisfying the defining condition for admissibility for just the injective ψ ∈ Λ_{m}^{k}[m′]. Then, in fact, h is admissible. So h is admissible if and only if: for every kalternating subset S of [m], the S-face of h is hollow. Table 2 shows the k-alternating subsets of [m] in the cases we need.

n = 0

A weak 0-category is a simplicial set A in which every horn has a unique filler—including those of shape Λ_{1}^{k}. It follows that the functor A : Δ^{op} \overline{}Set is constant, so a weak 0-category is just a set.

n = 1

A horn of shape Λ_{m}^{k} is called inner if 0 < k < m; a simplicial set is the nerve of a category if and only if every inner horn has a unique filler. If (A, H) is a simplicial set with hollowness satisfying (i′) for n = 1 then every inner horn is admissible, hence, if (ii) and (ii′) also hold, has a unique filler: so A is (the nerve of) a category. Working out the other conditions, we find that a weak 1-category is a category equipped with a set H_{1} of isomorphisms containing all the identity maps and closed under composition and inverses. So given a weak 1-category we obtain a category by forgetting H; conversely, given a category we can take H_{1} = {all isomorphisms} (or {all identities}) to obtain a weak 1-category.

k  k-alternating subsets of [m] of cardinality ≤ 3
0  {0, 1}, {0, 1, m}
1, …, m − 1  {k − 1, k, k + 1}
m  {m − 1, m}, {0, m − 1, m}

Table 2: k-alternating subsets of [m] of cardinality ≤ 3, for m ≥ 1

[figure: (a); •; •; a_{0}; a_{1}; f_{01}; (b); •; •; •; a_{0}; a_{1}; a_{2}; f_{01}; f_{12}; f_{02}; α_{012}; (c); •; •; •; a_{0}; a_{1}; a_{2}; f_{01}; f_{12}; (d); •; •; •; a_{0}; a_{1}; a_{2}; f_{01}; f_{02}; ∼; (e); •; •; •; •; ∼; a_{0}; a_{1}; a_{2}; a_{3}; •; •; •; •; ∼; ∼; a_{0}; a_{1}; a_{2}; a_{3}; (f); •; •; •; •; ∼; a_{0}; a_{1}; a_{2}; a_{3}; •; •; •; •; a_{0}; a_{1}; a_{2}; a_{3}]

Figure 9: (a) Element of A[1], with, for instance, the {0}-face labelled a_{0}; (b) element of A[2]; (c) (admissible) horn Λ_{2}^{1} \overline{}A; (d) admissible horn Λ_{2}^{0} \overline{}A, with ∼ indicating a hollow face; (e) admissible horn Λ_{3}^{0} \overline{}A, with labels f_{ij}, α_{ijk} omitted; (f) as (e), but for Λ_{3}^{1}.

n = 2

A weak 2-category is a simplicial set A equipped with subsets H_{1} ⊆ A[1] and H_{2} ⊆ A[2], satisfying certain axioms. It appears that this is the same as a bicategory equipped with a set H_{1} of 1-cells which are equivalences and a set H_{2} of 2-cells which are isomorphisms, satisfying closure conditions similar to those under ‘n = 1’ above.

So, let (A, H) be a weak 2-category. We construct a bicategory whose 0- and 1-cells are the elements of A[0] and A[1]; a 2-cell f \overline{}g is an element of A[2] of the form [figure: •; •; •; a; a; b; 1_{a}; g; f] , where 1_{a} indicates a degenerate 1-cell. Composition of 1-cells is defined

by making a random choice of hollow filler for each horn of shape Λ_{2}^{1}; composition of 2-cells is defined by filling in 3-dimensional horns Λ_{3}^{k}; identities are got from degeneracies. See Fig. 9. The associativity and unit isomorphisms are certain hollow cells, and the coherence axioms hold because of the uniqueness of certain fillers.

Conversely, let B be a bicategory, and construct a weak 2-category (A, H) as follows. A[0] and A[1] are the sets of 0- and 1-cells of B; an element of A[2] as in Fig 9(b) is a 2-cell α_{012} : f_{02} \overline{}f_{12}^{∘}f_{01} in B. (In general, an element of A[m] is a ‘strictly unitary colax morphism’ [m] \overline{}B, where [m] is regarded as a 2-category whose only 2-cells are identities.) H_{1} ⊆ A[1] is the set of 1-cells which are equivalences, and H_{2} is the set of 2-cells which are isomorphisms. Then all the axioms for a weak 2-category are satisfied.

Variant

We could add to conditions (i)–(iii) on (A, H) the further condition that H is maximal with respect to (i)–(iii): that is, if (A, H′) also satisfies (i)–(iii) and H′_{m} ⊇ H_{m} for all m then H′ = H. (Compare the issue of maximal atlases in the definition of smooth manifold.) With this addition, a weak 1-category is essentially just a category, and a weak 2-category a bicategory. This is in contrast to the original St (as analysed above), where the flexibility in the choice of H means that the correspondence between weak 1-(2-)categories and (bi)categories is inexact.

Definition X

This definition is not intended to be rigorous, although it can be made so.

Opetopic Sets

An opetopic set A is a commutative diagram of sets and functions

[figure: ⋯; \begin{matrix}s \\ t \end{matrix}; A′_{2}; \begin{matrix}s \\ t \end{matrix}; A′_{1}; \begin{matrix}s \\ t \end{matrix}; A′_{0} = A_{0}; ⋯; ⋯; \begin{matrix}s \\ t \end{matrix}; A_{2}; \begin{matrix}s \\ t \end{matrix}; A_{1}; \begin{matrix}s \\ t \end{matrix}; A_{0}]

where for each m ≥ 1, the set A′_{m} and the functions s : A′_{m}  \overline{}A′_{m−1} and t : A′_{m}  \overline{}

A_{m−1} are defined from the sets A_{m}, A′_{m−1}, A_{m−1}, …, A′_{1}, A_{1}, A_{0} and the functions s, t between them in the following way.

An element a ∈ A_{0} is regarded as a 0-cell, and drawn _{a}^{•}. An element f ∈ A_{1} is regarded as a 1-cell _{a}_{•} ^{^{f} \overline{}}_{b}_{•} , where a = s(f) and b = t(f). A′_{1} is the set of ‘1-pasting diagrams’

in A, that is, diagrams of 1-cells pasted together, that is, paths _{a_{0}}_{•} ^{^{f_{1}}\overline{}}_{a_{1}}_{•} ^{^{f_{2}}\overline{}}… ^{^{f_{k}}\overline{}}_{a_{k}}_{•}

(k ≥ 0) in A. An element α ∈ A_{2} has a source s(α) of this form and a target t(α) of the form _{a_{0}}_{•} ^{^{g} \overline{}}_{a_{k}}_{•} , and is drawn as

[figure: •; •; •; ⋯; a_{0}; a_{1}; a_{k}; f_{1}; f_{2}; f_{k}; g; α]  (∥)

Next, A′_{2} is the set of ‘2-pasting diagrams’, that is, diagrams of cells of the form (∥) pasted together, such as

[figure: •; •; •; •; •; •; •; •; •; •; a_{0}; a_{1}; a_{2}; a_{3}; a_{4}; a_{5}; a_{6}; a_{7}; a_{8}; a_{9}; f_{1}; f_{2}; f_{3}; f_{4}; f_{5}; f_{6}; f_{7}; f_{8}; f_{9}; f_{10}; f_{11}; f_{12}; f_{13}; α_{1}; α_{2}; α_{3}; α_{4}]  (∗∗)

Note that the arrows go in compatible directions: e.g. the target or ‘output’ edge f_{11} of α_{3} is a source or ‘input’ edge of α_{1}. The source of this element of A′_{2} is _{a_{0}}_{•} ^{^{f_{1}}\overline{}}⋯ ^{^{f_{9}}\overline{}}_{a_{9}}_{•} ∈ A′_{1}, and the target is f_{13} ∈ A_{1}. Next, if γ ∈ A_{3} and s(γ) is (∗∗) then t(γ) is of the form (∥) with k = 9 and g = f_{13}, and we picture γ as a 3-dimensional cell with a flat bottom face (labelled α) and four curved faces on top (labelled α_{1}, α_{2}, α_{3}, α_{4}). Carrying on, A′_{3} is the set of 3-pasting diagrams, A_{4} is the set of 4-cells, etc.

We need some terminology concerning cells. Let Φ ^{^{α}} g be an m-cell: that is, let α ∈ A_{m} with s(α) = Φ ∈ A′_{m−1} and t(α) = g ∈ A_{m−1}.

For any p-cell e, there is a p-pasting diagram ⟨e⟩ consisting of e alone. If ⟨g⟩ ^{^{β}} h then α and β can be pasted to obtain (Φ ^{β_{∗}(α)} h) ∈ A′_{m}.

The faces of Φ are the (m − 1)-cells which have been pasted together to form it, e.g. if α is as in (∥) then Φ has faces f_{1}, …, f_{k}, and the Φ of (∗∗) has faces α_{1}, α_{2}, α_{3}, α_{4}. If f is a face of Φ and e is a cell parallel to f (i.e. e ∈ A_{m−1} with s(e) = s(f), t(e) = t(f)) then we obtain a new pasting diagram Φ(e/f) ∈ A′_{m−1} by replacing f with e in Φ. (Read Φ(e/f) as ‘Φ with e replacing f’.) If also ⟨e⟩ ^{^{β}} f then α and β can be pasted to obtain

(Φ(e/f) ^{β^{∗}(α)} g) ∈ A′_{m}.

Universal Cells

Let A be an opetopic set and fix n ≥ 0. We define what it means for a cell Φ ^{^{ε}} g of A to be ‘universal’, and, when f is a face of Φ, what it means for f to be ‘liminal’ in the cell. The definitions depend on n, so I should really say ‘n-universal’ rather than just ‘universal’, and similarly ‘n-liminal’; but I will drop the ‘n’ since it is regarded as fixed.

The two definitions are given inductively in an interdependent way.

Universality. For m ≥ n+1, a cell (Φ ^{^{ε}} g) ∈ A_{m} is universal if whenever (Φ ^{ε′} g′) ∈ A_{m}, then ε′ = ε. Now let 1 ≤ m ≤ n. A cell (Φ ^{^{ε}} g) ∈ A_{m} is universal if both

i. for every α : Φ \overline{}h, there exist \overline{α} : ⟨g⟩ \overline{}h and a universal U : \overline{α}_{∗}(ε) \overline{}α,

ii. for every α : Φ \overline{}h, \overline{α} : ⟨g⟩ \overline{}h and universal U : \overline{α}_{∗}(ε) \overline{}α, \overline{α} is liminal in U.

Liminality. Let m ≥ 1, let (Φ ^{^{ε}} g) ∈ A_{m}, and let f be a face of Φ. Then f is liminal in ε if m ≥ n + 2 or

i. for every cell e parallel to f and β : Φ(e/f) \overline{}g, there exist \overline{β} : ⟨e⟩ \overline{}f and a universal cell U : \overline{β}^{∗}(ε) \overline{}β, and

ii. for every e parallel to f, β : Φ(e/f) \overline{}g, \overline{β} : ⟨e⟩ \overline{}f and universal U : \overline{β}^{∗}(ε) \overline{}β, \overline{β} is liminal in U.

The Definition

Let n ≥ 0. A weak n-category is an opetopic set A satisfying

existence of universal fillers: for every m ≥ 0 and Φ ∈ A′_{m}, there exists a universal cell of the form Φ ^{^{ε}} g

closure of universals under composition: if m ≥ 2, (Φ ^{^{ε}} g) ∈ A_{m}, each face of Φ is universal, and ε is universal, then g is universal.

Definition X for n ≤ 2

First consider high-dimensional cells in an n-category A. For every m ≥ n + 1 and Φ ∈ A′_{m−1}, there is a unique ε ∈ A_{m} whose source is Φ: in other words, s : A_{m} \overline{}A′_{m−1} is a bijection. This means that the entire opetopic set is determined by the part of dimension ≤ n and the map t : A_{n+1} \overline{}A_{n}. Identifying A_{n+1} with A′_{n}, this map t assigns to each n-pasting diagram Φ the target g of the unique (n + 1)-cell with source Φ, and we regard g as the composite of Φ. (In general, we may regard the target of a universal cell as a composite of its source; note that all cells of dimension > n are universal.) This composition of n-cells is strictly associative and unital. A weak n-category therefore consists of a commutative diagram

[figure: A′_{n}; \begin{matrix}s \\ t \end{matrix}; ⋯; \begin{matrix}s \\ t \end{matrix}; A′_{1}; \begin{matrix}s \\ t \end{matrix}; A′_{0} = A_{0}; A_{n}; ∘; \begin{matrix}s \\ t \end{matrix}; ⋯; \begin{matrix}s \\ t \end{matrix}; A_{1}; \begin{matrix}s \\ t \end{matrix}; A_{0}]  (††)

of sets and functions such that n-dimensional composition ^{∘} obeys strict laws and the defining conditions on existence and closure of universals hold in lower dimensions.

n = 0. A weak 0-category is a set A_{0} together with a function ^{∘} : A_{0} \overline{}A_{0} obeying strict laws—which say that ^{∘} is the identity. So a weak 0-category is just a set.

n = 1. When n = 1, diagram (††) is a directed graph A_{1} [figure: s; t] A_{0} together with a map

∘ : A′_{1} \overline{}A_{1} compatible with the source and target maps: in other words, assigning to each string of edges

a_{0} ^{f_{1}} a_{1} ^{f_{2}} ⋯ ^{f_{k}} a_{k}

in A a new edge a_{0} ^{(f_{k}^{∘}···^{∘}f_{1})} a_{k}. The axioms on ^{∘} say that

((f_{k}^{r_{k}}^{∘} ⋯^{∘}f_{k}^{1})^{∘} ⋯^{∘}(f_{1}^{r_{1}}^{∘} ⋯^{∘}f_{1}^{1})) = (f_{k}^{r_{k}}^{∘} ⋯^{∘}f_{1}^{1}), (f) = f

—in other words, that A forms a category. A weak 1-category is therefore a category satisfying the extra conditions that every object is the domain of some universal morphism and that the composite of universal morphisms is universal. I claim that the universal morphisms are the isomorphisms, so that these conditions hold automatically and a weak 1-category is just a category.

So: a morphism ε : a \overline{}b is universal if (i) every morphism α : a \overline{}c factors

as α = \overline{α}^{∘}ε, and (ii) such an \overline{α} is always liminal in the unique 2-cell [figure: •; •; •; a; b; c; ε; \overline{α}; α; U] .

Liminality of \overline{α} in U means, in turn, that if α̃ : b \overline{}c satisfies α̃^{∘}ε = α then α̃ = \overline{α}. (This is part (i) of the definition of liminality; part (ii) holds trivially.) So ε : a \overline{}b is universal if and only if every morphism out of a factors uniquely through ε, which holds if and only if ε is an isomorphism.

n = 2. A weak 2-category is essentially the same thing as a bicategory. More precisely, the category of bicategories and weak functors is equivalent to the category whose objects are weak 2-categories and whose morphisms are those maps of opetopic sets which send universal cells to universal cells. The equivalence works as follows.

Given a bicategory B, define a weak 2-category A by taking A_{0} and A_{1} to be the sets of 0- and 1-cells in B, and a 2-cell (∥) in A to be a 2-cell (f_{k}^{∘} ⋯^{∘}f_{1}) \overline{}g in B. Here (f_{k}^{∘} ⋯^{∘}f_{1}) is defined inductively as f_{k}^{∘}(f_{k−1}^{∘} ⋯^{∘}f_{1}) if k ≥ 1, or as 1 if k = 0; any other iterated composite would do just as well. Composition ^{∘} : A′_{2} \overline{}A_{2} is pasting of 2-cells in B. Then a 1-cell (respectively, 2-cell) in A turns out to be universal if and only if the corresponding 1-cell (2-cell) in B is an equivalence (isomorphism), and it follows that A is a weak 2-category.

Conversely, take a weak 2-category A and construct a bicategory B as follows. The 0- and 1-cells of B are just those of A, and a 2-cell of B is an element of A_{2} of the form

[figure: a; b; f; g; α; •; •] (i.e. α : ⟨f⟩ \overline{}g). For each diagram _{a}_{•} ^{^{f} \overline{}}_{b}_{•} ^{^{g} \overline{}}_{c}_{•} of 1-cells, choose at

random a universal filler [figure: •; •; •; a; b; c; f; g; g^{∘}f; ε_{f,g}] , where by definition g^{∘}f = t(ε_{f,g}); this defines

composition of 1-cells. Vertical composition of 2-cells comes from ^{∘} : A′_{2} \overline{}A_{2}. To

define the horizontal composite of _{a}_{•} [figure: f; g; α]_{a′}_{•} [figure: f′; g′; α′] _{a′′}_{•} , consider pasting ε_{g,g′} to α and α′, and

then use the universality of ε_{f,f′}. Next observe that given two universal fillers Φ ^{^{ε}} g, Φ ^{ε′} g′ for a 1-pasting diagram Φ = (_{a_{0}}_{•} ^{^{f_{1}}\overline{}}… ^{^{f_{k}}\overline{}}_{a_{k}}_{•} ), there is a unique 2-cell

⟨g⟩ ^{^{δ}} g′ such that the composite ^{∘}(δ_{∗}(ε)) is ε′. Applying this observation to a certain pair of universal fillers for (_{•} ^{^{f} \overline{}}_{•} ^{^{g} \overline{}}_{•} ^{^{h} \overline{}}_{•}) gives the associativity isomorphism, and the word ‘unique’ in the observation gives the pentagon axiom. Identities work similarly, where this time we choose a random universal filler for each degenerate 1-pasting diagram a^{•}.

Further Reading

This section contains the references and historical notes missing in the main text. It is not meant to be a survey of the literature. Where I have omitted relevant references it is almost certainly a result of my own ignorance, and I hope that the authors will forgive me.

First are some references to introductory and general material, and a very brief account of the history of higher-dimensional category theory. Then there are references for each of the sections in turn: ‘Background’, followed by the ten definitions. Finally there are references to some proposed definitions of n-category which I didn’t include, and a very few references to areas of mathematics related to n-categories.

Citations such as math.CT/9810058 and alg-geom/9708010 refer to the electronic mathematics archive at http://arXiv.org. Readers unfamiliar with the archive may find it easiest to go straight to the address of the form http://arXiv.org/abs/math.

CT/9810058 .

Introductory Texts

Introductions to n-categories come slanted towards various different audiences. One for theoretical computer scientists and logicians is

[1] A. J. Power, Why tricategories?, Information and Computation 120 (1995), no. 2, 251–262; also LFCS report ECS-LFCS-94-289, April 1994, available via http://

     www.lfcs.informatics.ed.ac.uk

and another with a logical slant, but this time with foundational concerns, is

[2] M. Makkai, Towards a categorical foundation of mathematics, in Logic Colloquium ’95 (Haifa), Lecture Notes in Logic 11, Springer, 1998, pp. 153–190.

Moving to introductions for those more interested in topology, geometry and physics, one which starts at a very basic level (the definition of category) is

[3] John C. Baez, A tale of n-categories, available via http://math.ucr.edu/home/

     baez/week73.html, 1996–97.

With similar themes but at a more advanced level, there are

[4] John C. Baez, An introduction to n-categories, in Category theory and computer science (Santa Margherita Ligure, 1997), Lecture Notes in Computer Science 1290, Springer, 1997, pp. 1–33; also e-print q-alg/9705009, 1997

(especially sections 1–3) and

[5] Tom Leinster, Topology and higher-dimensional category theory: the rough idea, e-print math.CT/0106240, 2001, 15 pages.

The ambitious might, if they can find a copy, like to look at the highly discursive 600-page letter of Grothendieck to Quillen,

[6] A. Grothendieck, Pursuing stacks, manuscript, 1983,

in which (amongst many other things) the idea is put that tame topology is really the study of weak ω-groupoids. A more accessible discussion of what higher-dimensional algebra might ‘do’, especially in the context of topology, is

[7] Ronald Brown, Higher dimensional group theory, available via http://www.bangor.

     ac.uk/∼mas010.

General Comments and History

The easiest way to begin a history of n-categories is as follows.

0-categories—sets or classes—came into the mathematical consciousness around the end of the 19th century. 1-categories—categories—arrived in the middle of the 20th century. Strict 2-categories and, implicitly, strict n-categories, made their presence felt around the late 1950s and early 1960s, with the work of Ehresmann [19]. Weak 2-categories were first introduced by Bénabou [23] in 1967, under the name of bicategories, and thereafter the question was in the air: ‘what might a weak n-category be?’ The first precise proposal for a definition was given by Street [66] in 1987. This was followed by three more proposals around 1995: Baez and Dolan’s [73], Batanin’s [31], and Tamsamani’s [46]. A constant stream of further proposed definitions has issued forth since then, and will doubtless continue for a while. Work on low values of n was also going on at the same time: an axiomatic definition of weak 3-category was proposed in

[8] R. Gordon, A. J. Power, Ross Street, Coherence for Tricategories, Memoirs of the American Mathematical Society 117, no. 558, 1995,

and a proposal in similar vein for n = 4 was made in

[9] Todd Trimble, The definition of tetracategory, manuscript, 1995.

Crucially, it was shown in [8] that not every tricategory is equivalent to a strict 3-category (in contrast to the situation for n = 2), from which it follows that the theory of weak n-categories is genuinely different from that of strict ones.

But this is far too simplistic. A realistic history must take account of categorical structures other than n-categories per se: for instance, the various kinds of monoidal category (plain, symmetric, braided, tortile/ribbon, …), and of monoidal 2-categories and monoidal bicategories. The direct importance of these is that a monoidal category is a bicategory with only one 0-cell, and similarly a braided monoidal category is a tricategory with only one 0-cell and one 1-cell. The basic reference for braided monoidal categories is

[10] André Joyal, Ross Street, Braided tensor categories, Advances in Mathematics 102 (1993), no. 1, 20–78,

and they can also be found in the new edition of Mac Lane’s book [18].

Moreover, around the same time as the theory of n-categories was starting to develop, another theory was emerging with which it was later to converge: the theory of multicategories and operads. Multicategories first appeared in

[11] Joachim Lambek, Deductive systems and categories II: standard constructions and closed categories, in Category Theory, Homology Theory and their Applications, I (Battelle Institute Conference, Seattle, 1968, Vol. One), ed. Hilton, Lecture Notes in Mathematics 86, Springer, 1969, pp. 76–122.

(The definition is on page 103.) A multicategory is like a category, but each arrow has as its source or input a sequence of objects (and, as usual, as its target or output a single object). An operad is basically just a multicategory with only one object. For this reason, multicategories are sometimes called ‘coloured operads’, and the objects are then named after colours (black, white, etc.). The development of operads is generally attributed to Boardman, Vogt and May:

[12] J. M. Boardman, R. M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics 347, Springer, 1973,

[13] J. P. May, The Geometry of Iterated Loop Spaces, Lectures Notes in Mathematics 271, Springer, 1972,

although I am told that essentially the same idea was the subject of

[14] Michel Lazard, Lois de groupes et analyseurs, Annales Scientifiques de l’École Normale Supérieure (3) 72 (1955), 299–400

(where operads go by the name of ‘analyseurs’).

It seems to have taken a long time before it was realized that operads and multicategories were so closely related; I do not know of any pre-1995 text which mentions both Lambek and Boardman-Vogt or May in its bibliography. This can perhaps be explained by the different fields in which they were being studied: multicategories were introduced in the context of logic and found application in linguistics, whereas operads were used for the theory of loop spaces. Moreover, if one uses the terms in their original senses then it is not strictly true that an operad is the same thing as a one-object multicategory; operads are also equipped with a symmetric structure, and the ‘hom-sets’ (sets of operations) are topological spaces rather than just sets. (It is also very natural to consider multicategories with both these pieces of extra structure, but historically this is beside the point.)

Many short introductions to operads have appeared as section 1 of papers by topologists and quantum algebraists. The interested reader may also find useful the following texts dedicated to the subject:

[15] J. P. May, Definitions: operads, algebras and modules, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemporary Mathematics 202, AMS, 1997, pp. 1–7; also available via http://www.math.uchicago.edu/

     ∼may,

[16] J. P. May, Operads, algebras and modules, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemporary Mathematics 202, AMS,

     1997, pp. 15–31; also available via http://www.math.uchicago.edu/∼may,

[17] Martin Markl, Steve Shnider, Jim Stasheff, Operads in Algebra, Topology and Physics, book in preparation.

A glimpse of the role of operads and multicategories in higher-dimensional category theory can be seen in the definitions of weak n-category above. Often the ‘operads’ and ‘multicategories’ used are not the original kinds, but more general kinds adapted for the different shapes and dimensions which occur in the subject; for references, see ‘Definitions B and L’ below.

Background

Category Theory. Almost any book on the subject will provide the necessary background. The second edition of the classic book by Mac Lane,

[18] Saunders Mac Lane, Categories for the Working Mathematician, second edition, Graduate Texts in Mathematics 5, Springer, 1998,

is especially useful, containing as it does two new chapters on such topics as bicategories and nerves of categories. Strict n-Categories. I do not know of any good text introducing strict n-categories. Ehresmann’s original book

[19] Charles Ehresmann, Catégories et Structures, Dunod, Paris, 1965

could be consulted, but is generally regarded as a very demanding read. Probably more useful is

[20] G. M. Kelly, Ross Street, Review of the elements of 2-categories, in Category Seminar (Sydney, 1972/1973), Lecture Notes in Mathematics 420, Springer, 1974, pp. 75–103,

which only covers strict 2-categories (traditionally just called ‘2-categories’) but should give a good idea of strict n-categories for general n. This could usefully be supplemented by

[21] Samuel Eilenberg, G. Max Kelly, Closed categories, in Proceedings of Conference on Categorical Algebra (La Jolla, California, 1965), Springer, 1966, pp. 421–562

(see e.g. page 552), which also covers enrichment. For another reference on enriched categories, see chapter 6 of

[22] Francis Borceux, Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 51, Cambridge University Press,

Bicategories. Bicategories were first explained by Bénabou:

[23] Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, ed. Bénabou et al, Lecture Notes in Mathematics 47, Springer, 1967, pp. 1–77,

and further important work on them is in

[24] John W. Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Mathematics 391, Springer, 1974.

(At least) two texts contain summaries of the ‘basic theory’ of bicategories: that is, the definitions of bicategory and of weak functor (homomorphism), transformation and modification between bicategories, together with the result that any bicategory is in a suitable sense equivalent to a strict 2-category. These are section 9 of

[25] Ross Street, Categorical structures, in Handbook of Algebra 1, ed. M. Hazewinkel, North-Holland, 1996, pp. 529–577

and the whole of

[26] Tom Leinster, Basic bicategories, e-print math.CT/9810017, 1998, 11 pages.

Definition Tr

The definition was given in a talk,

[27] Todd Trimble, What are ‘fundamental n-groupoids’?, seminar at DPMMS, Cambridge, 24 August 1999,

and has not been written up previously. Trimble used the term ‘flabby n-category’ rather than ‘weak n-category’.

As the title of the talk suggests, the idea was not to develop the weakest possible notion of n-category, but to provide (in his words) ‘a sensible niche for discussing fundamental n-groupoids’. In a world where all the definitions have been settled, it may be that fundamental n-groupoids of topological spaces have certain special features (other than the invertibility of their cells) not shared by all weak n-categories. Thus it may be that the word ‘weak’ is less appropriate for definition Tr than the other definitions.

Evidence that this is the case comes from two directions. Firstly, the maps γ_{a_{0},...,a_{k}} describing composition of hom-(n − 1)-categories in an n-category are strict (n − 1)- functors. This corresponds to having strict interchange laws. It therefore seems likely that a precise analysis of n = 3 would show that every weak 3-category gives rise to a tricategory (in a similar manner to n = 2) but that not every tricategory is triequivalent to one arising from a weak 3-category.

Secondly, forget Tr for the moment and consider in naive terms what the fundamental ω-groupoid A of a space X might look like. In the notation of the Background section on strict n-categories, we would probably imagine taking

A_{0} = underlying set of X,
A_{1} = \{maps f : [0, 1] \overline{}X\},
A_{2} = \{maps α : [0, 1]^{2}  \overline{}X such that α(0, −) and α(1, −) are constant\},

with the obvious source and target functions. There are then composition functions

∘_{0} : A_{1} ×_{A_{0}} A_{1}  \overline{}A_{1}, ^{∘}_{0} : A_{2} ×_{A_{0}} A_{2}  \overline{}A_{2}, ^{∘}_{1} : A_{2} ×_{A_{1}} A_{2}  \overline{}A_{2}

defined by the standard method of travelling at double speed whenever necessary, satisfying the usual source and target axioms. They are not strictly associative, nor are there strict identities; however, they do obey the strict interchange law,

(β′ ^{∘}_{1} α′)^{∘}_{0} (β^{∘}_{1} α) = (β′ ^{∘}_{0} β)^{∘}_{1} (α′ ^{∘}_{0} α)

for any α, α′, β, β′ ∈ A_{2} such that these composites make sense. More generally, one would expect the fundamental n-groupoid of a space to obey strict interchange in all dimensions. This provides the kind of ‘special feature’ of fundamental n-groupoids referred to above.

Other ideas on fundamental n-groupoids, n-categories, and how they tie together can be found in Grothendieck’s letter [6]. More practical material on fundamental 1- and 2-groupoids is in

[28] K. H. Kamps, T. Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific Publishing Co., 1997.

For more on operads, see the references under ‘General Comments and History’ above. The name of the operad E was chosen not only to stand for ‘endpoint-preserving’, but also because it comes after D for ‘disk’—the idea being that E is something like the onedimensional little disks operad D (crucial in the theory of loop spaces). A touch more precisely, E seems to play the same kind of role for paths as D does for closed loops.

Prospects for comparing Tr with B and L look bright: it seems very likely that weak n-categories according to Tr are just the algebras for a certain globular n-operad (in the sense of B or L), namely, a truncated version of the globular operad E^{op}(D) of endomorphisms of the coglobular space of disks. E^{op}(D) is discussed in section 9 of Batanin’s paper [31], and can be regarded as a higher-dimensional version of the operad E used in Tr.

More on how the theory of bicategories comes from the operad of trees can be found in Appendix A (and Chapter 1) of my thesis, [34], and in my [90].

Definition P

Definition P of weak ω-category is in

[29] Jacques Penon, Approche polygraphique des ∞-categories non strictes, Cahiers de Topologie et Géométrie Différentielle 40 (1999), no. 1, 31–80.

His chosen term for weak ω-category is ‘prolixe’, whose closest English translation is perhaps ‘waffle’. As far as I can see there is no actual definition of weak n-category or n-dimensional prolixe in the paper, although he clearly has one in mind on page 48:

Les prolixes de dimension ≤ 2 s’identifient exactement aux bicatégories […] la preuve de ce résultat sera montré dans un article ultérieur

(‘waffles of dimension ≤ 2 correspond exactly to bicategories […] the proof of this result will be given in a forthcoming paper’).

Other translations: my category [ℝ, Set] of reflexive globular sets is his category ∞-𝔾rr of reflexive ∞-graphs; my s and t are his s and b; my strict ω-categories are his ∞-categories; my category Q is called by him 𝔼tC, the category of étirements catégoriques (‘categorical stretchings’); my contractions γ are written [−, −] (with the arguments reversed: γ_{m}(f_{0}, f_{1}) = [f_{1}, f_{0}]); and my adjunction F ⊣ U is called Ê⊣ V̂.

The word ‘magma’ is borrowed from Bourbaki, who used it to mean a set equipped with a binary operation. It is a slightly inaccurate borrowing, in that ω-magmas are equipped with (nominal) identities as well as binary compositions; put another way, it would have been more suitable if Bourbaki had used the word to mean a set equipped with a binary operation and a distinguished basepoint.

It seems plausible that Penon’s construction can be generalized to provide weak versions of structures other than ω-categories (e.g. up-to-homotopy topological monoids). Batanin has done something precise along the lines of generalizing Penon’s definition and comparing it to his own:

[30] M. A. Batanin, On the Penon method of weakening algebraic structures, to appear in Journal of Pure and Applied Algebra; also available via http://www.math.mq.

     edu.au/∼mbatanin/papers.html, 2001, 25 pages.

Definitions B and L

Batanin gave his definition, together with an examination of n = 2, in

[31] M. A. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Advances in Mathematics 136 (1998), no. 1, 39–103; also

     available via http://www.math.mq.edu.au/∼mbatanin/papers.html.

Another account of it is

[32] Ross Street, The role of Michael Batanin’s monoidal globular categories, in Higher category theory (Evanston, IL, 1997), Contemporary Mathematics 230, AMS, 1998,

     pp. 99–116; also available via http://www.math.mq.edu.au/∼street.

The definition of weak n-category which appears as 8.7 in [31] is (I believe) what is here called definition B1. More precisely, let O be the category whose objects are (globular) operads on which there exist a contraction and a system of compositions, and whose maps are just maps of operads. What Batanin does is to construct an operad K which is weakly initial in O. ‘Weakly initial’ means that there is at least one map from K to any other object of O, so this does not determine K up to isomorphism; one needs some further information. But in Remark 2 just before Definition 8.6, Batanin suggests that, once given the appropriate extra structure, K is initial in the category OCS of operads equipped with a contraction and a system of compositions, which does determine K. This is the approach taken in B1.

A weak n-category according to B2 is (I believe) almost exactly what Batanin calls a ‘weak n-categorical object in Span’ in his Definition 8.6. The only difference is my extra condition that the operad C is (in his terminology) normalized: C(0) ≅ 1. Now C(0) is the set of operations in the operad which take a 0-cell of an algebra and turn it into another 0-cell, so normality means that there are no such operations except, trivially, the identity. This seems reasonable in the context of n-categories, since one expects to have operations for composing m-cells only when m ≥ 1. The lack of normality in Batanin’s version ought to be harmless, since the contraction means that all the operations on 0-cells are in some sense equivalent to the identity operation, but it does make the analysis of n ≤ 2 a good deal messier. (Note that the operad K is normalized, so any weak ω-/n-category in the sense of B1 is also one in the sense of B2; the same goes for L1 and L2.)

My modification L1 of Batanin’s definition first appeared in

[33] Tom Leinster, Structures in higher-dimensional category theory, e-print math.CT/ 0109021, 1998, 81 pages,

but a more comprehensive and, I think, comprehensible account is in

[34] Tom Leinster, Operads in higher-dimensional category theory, Ph.D. thesis, University of Cambridge, 2000; also e-print math.CT/0011106, 2000, viii + 127 pages.

(There, C ⊗ C′ is called C^{∘}C′, C · − is T_{C}, and P_{C}(π) is P_{π}(C).) [34] also contains a precise analysis of L1 for n ≤ 2, including proofs of (a) the equivalence of the two different categories of weak n-categories (for finite n) mentioned at the start of the analysis of n ≤ 2 above, and (b) the equivalence of the category of unbiased bicategories and weak functors with that of (classical) bicategories and weak functors. Definition L2 has not appeared before, and has just been added here for symmetry.

The globular operads in B and L arise as a special case of two different, more general theories of operads. Firstly, they are a special case of generalized operads, a family of higher-dimensional categorical structures which are perhaps as interesting and applicable as n-categories themselves. Briefly, the theory goes as follows. Given a monad T on a category E, satisfying some natural conditions, one can define a category of T -multicategories. For example, when T is the identity monad on the category E of sets, a T -multicategory is just a category, and when T is the free-monoid monad on Set, a T -multicategory is just an ordinary multicategory (see ‘General Comments and History’ above). A T -operad is a one-object T -multicategory, so in the first of these examples it is a monoid and in the second it is an operad in the original sense (but without symmetric or topological structure). Now take T to be the free strict ω-category monad on the category E of globular sets, as in B and L: a T -operad is then exactly a globular operad. Algebras for T -multicategories can be defined in the general context, and again this notion specializes to the one in B and L.

Generalized (operads and) multicategories were first put forward in

[35] Albert Burroni, T -catégories (catégories dans un triple), Cahiers de Topologie et Géométrie Différentielle 12 (1971), 215–321

and were twice rediscovered independently:

[36] Claudio Hermida, Representable multicategories, Advances in Mathematics 151 (2000), no. 2, 164–225; also available via http://www.cs.math.ist.utl.pt/s84.

     www/cs/claudio.html,

[37] Tom Leinster, General operads and multicategories, e-print math.CT/9810053, 1997, 35 pages.

As far as I know, the notion of algebra for a T -multicategory only appears in the third of these. ([37] also appears, more or less, as Chapter I of [33] and Chapter 2 of [34].)

Secondly, globular operads arise as the principal special case of Batanin’s theory of operads in a monoidal globular category (see [31] and [32]). A strict monoidal globular category is a strict ω-category in Cat; a monoidal globular category is a slightly relaxed version of this. There is a particular monoidal globular category Span, and an operad in Span is exactly what we call a globular operad. This general theory is capable of handling coglobular structures and coalgebras for operads, as, for instance, in the discussion above of Definition Tr, where the coglobular space (functor 𝔾^{op} \overline{}Top) of disks was mentioned.

The difference between definitions L1 and B1 can be summarized by saying that L1 takes B1, dispenses with the notions of system of compositions and B-style contraction, and merges them into a single more powerful notion of contraction. A few more words on the difference are in section 4.5 of [34]. The operad L canonically carries a B-style contraction and a system of compositions, so there is a canonical map K \overline{}L of operads, and this induces a functor in the opposite direction on the categories of algebras. Hence every weak ω-/n-category in the sense of L1 gives rise canonically to one in the sense of B1.

Definition L′

This is the first time in print for definition L′. Once we have the language of generalized multicategories (described in the previous section) and the theory of strict ω-categories, it is very quickly stated. My papers [34] and [33] (and to some extent [37]) cover generalized multicategories and globular operads, but not specifically globular multicategories. The 1dimensional case, 1-globular multicategories, are the ‘fc-multicategories’ described briefly in

[38] Tom Leinster, fc-multicategories, e-print math.CT/9903004, 1999, 8 pages,

at a little more length in

[39] Tom Leinster, Generalized enrichment of categories, to appear in Journal of Pure and Applied Algebra,

and in detail in

[40] Tom Leinster, Generalized enrichment for categories and multicategories, e-print

     math.CT/9901139, 1999, 79 pages.

Logicians might like to view L′ through proof-theoretic spectacles, substituting the word ‘proof’ for ‘reason’. They (and others) might also be interested to read

[41] M. Makkai, Avoiding the axiom of choice in general category theory, Journal of Pure and Applied Algebra 108 (1996), no. 2, 109–173; also available via http://

     www.math.mcgill.ca/makkai

in which Makkai defines anafunctors and anabicategories and discusses the philosophical viewpoint which led him to them. In the same vein, see also Makkai’s [2] and the remarks on ‘a composite’ vs. ‘the composite’ towards the end of the Introduction to the present paper.

A weak ω-/n-category in the sense of L2 (and so L1 too) gives rise to one in the sense of L′. For just as ‘algebras’ for a category C (functors C \overline{}Set) correspond one-to-one with discrete opfibrations over C, via the so-called Grothendieck construction, so the same is true in a suitable sense for globular multicategories. This generalization is explained in section 4.2 of [37], section I.3 of [33], and section 3.4 of [34] (any one of which will do, but they are listed in increasing order of clarity). What this means is that an algebra for a globular operad gives rise to a globular multicategory (the domain of the opfibration), and if the operad admits a contraction in the sense of L then the resulting multicategory is a weak ω-category in the sense of L′.

Midway between L′ and J is another possible definition of weak ω-category, which for various reasons I have not included here. It was presented in a talk,

[42] Tom Leinster, Not quite Joyal’s definition of n-category (a.k.a. ‘algebraic nerves’), seminar at DPMMS, Cambridge, 22 February 2001,

notes from which, in the (2 + 2)-page format of this paper, are available on request. The idea behind it can be traced back to Segal’s formalization of the notion of up-to-homotopy topological commutative monoid, special Γ-spaces (and their non-commutative counterparts, special Δ-spaces). The analogy is that just as Segal took the theory of honest topological commutative monoids and did something to it to obtain an up-to-homotopy version, so we take the theory of strict n-categories and do something similar to obtain a weak version. Segal’s original paper is

[43] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.

A different generalization of his idea defines up-to-homotopy algebras for any (classical) operad. This is done at length in

[44] Tom Leinster, Homotopy algebras for operads, e-print math.QA/0002180, 2000, 101 pages,

or a much briefer explanation of the idea is

[45] Tom Leinster, Up-to-homotopy monoids, e-print math.QA/9912084, 1999, 8 pages.

Definitions Si and Ta

Tamsamani’s original definition appeared in

[46] Zouhair Tamsamani, Sur des notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux, K-Theory 16 (1999), no. 1, 51–99; also e-print

     alg-geom/9512006, 1995.

What I have called truncatability of a functor (Δ^{r})^{op} \overline{}Set is called ‘r-troncabilité’ by Tamsamani. It is not immediately obvious that the two conditions are equivalent, but a thoroughly mundane induction shows that they are. Other translations: my 1^{p} is his I_{p}, my s and t are his s and b, my Q^{(m)} is his T^{m}, my π^{(m)} is his t^{m}, my internal equivalence of cells x_{1}, x_{2} (as in the text of Ta) is his (r − p)-équivalence intérieure, and my external equivalence of functors (Δ^{r})^{op} \overline{}Set is his r-équivalence extérieure. His term for a weak n-category is ‘n-nerf’ or ‘n-catégorie large’. (‘Large’ has nothing to do with large and small categories: it means broad or generous, and can perhaps be translated here as ‘lax’; compare the English word ‘largesse’.)

Tamsamani also offers a proof that his weak 2-categories are essentially the same as bicategories, but I believe that it is slightly flawed, in that he has omitted a necessary axiom for the 2-nerve of a bicategory (the last bulleted item in ‘Definition Ta for n ≤ 2’, starting ‘ι_{uwx}^{z}’). Without this, the constructed functor (Δ^{2})^{op} \overline{}Set will not necessarily be a weak 2-category in the sense of Ta. (In this context, my a’s are his x’s, my α’s are his λ’s, and my ι’s are his ε’s.)

Working with him in Toulouse, Simpson produced a simplified version of Tamsamani’s definition, which first appeared in

[47] Carlos Simpson, A closed model structure for n-categories, internal \underline{Hom}, n-stacks and generalized Seifert-Van Kampen, e-print alg-geom/9704006, 1997, 69 pages.

He used the term ‘easy n-category’ for his weak n-categories, and ‘easy equivalence’ for what is called a contractible map in Si.

The simplification lies in the treatment of equivalences. Weak 1-categories according to either Ta or Si are just categories, but whereas a Ta-style equivalence of weak 1categories is a functor which is full, faithful and essentially surjective on objects (that is, an ordinary equivalence of categories), an easy equivalence is a functor which is full, faithful and genuinely surjective on objects. The latter property of functors is expressible at a significantly more primitive conceptual level than the former, since it is purely in terms of the underlying directed graphs and has nothing to do with the actual category structure. For this reason, Si is much shorter than Ta. (But to develop the theory of weak n-categories we still need Tamsamani’s more general notion of equivalence; this is, for instance, the missing piece of vocabulary referred to at the very end of ‘Definition Si for n ≤ 2’.)

As one would expect from this description, any easy equivalence (contractible map) is an equivalence in the sense of Ta. So as long as it is true that any weak n-category (Δ^{n})^{op} \overline{}Set in the sense of Si is truncatable (which I cannot claim to have proved), it follows that any weak n-category in the sense of Si is also one in the sense of Ta.

Following on from his definition, Tamsamani investigated homotopy n-groupoids of spaces:

[48] Zouhair Tamsamani, Equivalence de la théorie homotopique des n-groupoïdes et celle des espaces topologiques n-tronqués, e-print alg-geom/9607010, 1996, 24 pages.

Numerous papers by Simpson, using a mixture of his definition and Tamsamani’s and largely in the language of Quillen model categories, push the theory of weak n-categories further along:

[49] Carlos Simpson, Limits in n-categories, e-print alg-geom/9708010, 1997, 92 pages,

[50] Carlos Simpson, Homotopy types of strict 3-groupoids, e-print math.CT/9810059, 1998, 29 pages,

[51] Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak n-categories, e-print math.CT/9810058, 1998, 36 pages,

[52] Carlos Simpson, Calculating maps between n-categories, e-print math.CT/0009107, 2000, 13 pages,

[53] Carlos Simpson, Some properties of the theory of n-categories, e-print math.CT/ 0110273, 2001, 11 pages.

Toen has also applied Tamsamani’s definition, as in

[54] B. Toen, Dualité de Tannaka supérieure I: structures monoidales, Max-Planck-Institut preprint MPI-2000-57, available via http://www.mpim-bonn.mpg.de, 2000, 71 pages

and

[55] B. Toen, Notes on higher categorical structures in topological quantum field theory,

     available via http://guests.mpim-bonn.mpg.de/rosellen/etqft00.html, 2000,

14 pages,

and the theory finds its way into some very grown-up mathematics in

[56] Carlos Simpson, Algebraic aspects of higher nonabelian Hodge theory, e-print math. AG/9902067, 1999, 186 pages.

The connection between categories and their nerves is covered briefly in one of the new chapters of Mac Lane’s book [18]; two early sources are

[57] Graeme Segal, Classifying spaces and spectral sequences, Institut des Hautes Études Scientifiques Publications Mathématiques 34 (1968), 105–112,

[58] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Springer, 1967.

Presumably the ‘Segal maps’ are so named because of the prominent role they play in Segal’s paper [43] on loop spaces and homotopy-algebraic structures.

The basic method by which a Simpson or Tamsamani weak 2-category gives rise to a bicategory seems implicit in Segal’s [43], is made explicit in section 3 of my [45], and is done in even more detail in section 3.3 of my [44]. (Actually, these last two papers only describe the method for monoidal categories rather than bicategories in general, but there is no substantial difference.) There is also a discussion of the converse process in section 4.4 of [44], and the idea behind this is once more implicit in the work of Segal.

Definition J

Joyal gave his definition in an unpublished note,

[59] A. Joyal, Disks, duality and Θ-categories, preprint, 1997, 6 pages.

There he defined a notion of weak ω-category, which he called ‘θ-category’. He also wrote a few informal words about structures called θ^{n}-categories, and how one could derive from them a definition of weak n-category; but I was unable to interpret his meaning, and consequently definition J of weak n-category might not be what he envisaged.

The term ‘disk’ comes from the case where, in the notation of J, D_{m} is the closed mdimensional unit disk (= ball) in ℝ^{m}, p_{m} is projection onto the first (m − 1) coordinates, and the order on the fibres is given by the usual order on the real numbers. The second bulleted condition in the paragraph headed ‘Disks’ holds at a point d of D_{m} if and only if d is on the boundary of D_{m}. From another point of view, this condition can be regarded as a form of exactness.

The handling of faces in J is not necessarily equivalent to that in [59]; again, I had trouble understanding the intended meaning and made my own path. In fact, Joyal works the duality discussed under n ≤ 2 into the definition itself, putting Θ = 𝔻^{op} and calling Θ the category of ‘Batanin cells’ (for reasons suggested by Figures 3 and 8). So he does not speak of cofaces and cohorns in 𝔻, but rather of faces and horns in Θ.

Of the analyses of n ≤ 2 for the ten definitions, that for J is probably the furthest from complete. It appears to be the case that in a weak n-category A : 𝔻_{n} \overline{}Set, any cohorn Λ_{ϕ}^{D} \overline{}A where D has volume > n has a unique filler. (We know that this is true when the dimension of D is n.) If this conjecture holds then we can complete the proof (sketched in ‘n ≤ 2’) that any weak 2-category gives rise to a bicategory; for instance, applied to T_{0,0,0} it tells us that there is a canonical choice of associativity isomorphism, and applied to T_{0,0,0,0} it gives us the pentagon axiom. However, I have not been able to find a proof (or counterexample).

Introductory material on simplicial sets and horns can be found in, for instance, Kamps and Porter’s book [28].

The duality between the skeletal category Δ of nonempty finite totally ordered sets and the skeletal category 𝕀 of finite strict intervals has been well-known for a long time. Nevertheless, I have been unable to trace the original reference, or even a text where it is explained directly—except for Joyal’s preprint [59], which the reader may have trouble obtaining. A slightly indirect explanation is in section VIII.7 of

[60] Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic, Springer, 1992,

as part of a discussion of the related fact that the category of simplicial sets is a classifying topos for the theory of intervals. The duality is also a restricted version of the duality between finite distributive lattices and finite posets. Put briefly, it comes from mapping into the 2-element ordered set: if k is a natural number then the set Δ([k], [1]) naturally has the structure of an interval (isomorphic to ⟨k⟩) and the set 𝕀(⟨k⟩, ⟨0⟩) naturally has the structure of a totally ordered set (isomorphic to [k]). This provides functors Δ(−, [1]) : Δ^{op} \overline{}𝕀 and 𝕀(−, ⟨0⟩) : 𝕀^{op} \overline{}Δ which are mutually inverse, so Δ^{op} ≅ 𝕀.

Some of the fundamental facts on the higher duality were established in

[61] Clemens Berger, A cellular nerve for higher categories, to appear in Advances in Mathematics; Université de Nice—Sophia Antipolis Prépublication 602 (2000), 50

     pages; also available via http://math.unice.fr/∼cberger

(where a closed model category structure on [𝔻, Set] is also discussed), and more on this and on the relationship between definitions J and B is in

[62] Michael Batanin, Ross Street, The universal property of the multitude of trees, Journal of Pure and Applied Algebra 154 (2000), no. 1-3, 3–13; also available via

     http://www.math.mq.edu.au/∼mbatanin/papers.html.

The higher duality has also been the subject of detailed investigation by Makkai and Zawadowski:

[63] Mihaly Makkai, Marek Zawadowski, Duality for simple ω-categories and disks, Theory and Applications of Categories 8 (2001), 114–243, [64] Marek Zawadowski, Duality between disks and simple categories, talk at 70th Peri-

patetic Seminar on Sheaves and Logic, Cambridge, 1999,

[65] Marek Zawadowski, A duality theorem on disks and simple ω-categories, with applications to weak higher-dimensional categories, talk at CT2000, Como, Italy, 2000.

(Slides and notes from Zawadowski’s talks have the virtue of containing some pictures absent in the published version.)

As mentioned above, there is another way to define weak n-category which has strong connections to both J and L′: [42].

Definition St

Street proposed his definition of weak ω-category in a very tentative manner, in the final sentence of

[66] Ross Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra 49 (1987), no. 3, 283–335.

He did not explicitly formulate a notion of weak n-category for finite n; this small addition is mine, as is the Variant at the end of the section on n ≤ 2.

There is one minor but material difference, and a small number of cosmetic differences, between Street’s definition and St. The material difference is that in a weak ω-category as proposed in [66], the only hollow 1-cells are the degenerate ones. One terminological difference is that a pair (A, H) is called a ‘simplicial set with hollowness’ in [66] only when (i) and the aforementioned condition on 1-cells hold: so the term has a narrower meaning there than here. (Street informs me that he and Verity have used the term ‘stratified simplicial set’ for the same purpose, either with the two conditions or without.) Another is that he uses ‘ω-category’ in a wider sense: his potentially have infinite-dimensional cells, and the category of strict ω-categories in the sense of the present paper is denoted ω-Cath. Further translations: I say that a subset S ⊆ [m] is k-alternating where Street says that the set [m]\S is ‘k-divided’, and he calls a map [l] \overline{}[m] ‘k-monic’ if it is monic and its image is a k-alternating subset of [m].

The focus of [66] is actually on strict n- and ω-categories. To this end he considers the condition on 1-cells mentioned above, and conditions (i)–(iii) of St with ‘unique’ inserted before the word ‘filler’ in (ii). Having spent much of the paper constructing the nerve of a strict ω-category (this being a simplicial set with hollowness), he conjectures that a given simplicial set with hollowness is the nerve of some strict ω-category if and only if all the conditions just mentioned hold. (The conjecture was a result of joint work with John Roberts.) The necessity of these conditions was proved soon afterwards in

[67] Ross Street, Fillers for nerves, in Categorical algebra and its applications (Louvain-La-Neuve, 1987), Lecture Notes in Mathematics 1348, Springer, 1988, pp. 337–341.

A proof of their sufficiency was supplied by Dominic Verity; this has not appeared in print, but was presented at various seminars in Berkeley, Bangor and Sydney around 1993.

It is entirely possible that most of the detailed work for n ≤ 2 has already been done by Duskin. A short account of his work on this was presented as

[68] John W. Duskin, A simplicial-matrix approach to higher dimensional category theory, talk at CT2000, Como, Italy, 2000,

and a full-length version is in preparation:

[69] John W. Duskin, A simplicial-matrix approach to higher dimensional category theory I: nerves of bicategories, preprint, 2001, 82 pages,

[70] John W. Duskin, A simplicial-matrix approach to higher dimensional category theory II: bicategory morphisms and simplicial maps, preprint, 2001, 53 pages.

What Duskin does is to construct the nerve of any bicategory (this being a simplicial set) and to give exact conditions saying which simplicial sets arise in this way. He moreover shows how to recover a bicategory from its nerve. Duskin does not deal explicitly with Street’s conditions or his notion of hollowness (although he does mention them); indeed, the results just mentioned suggest that for n = 2, the hollow structure on the nerve of a bicategory is superfluous. The same point is addressed in a recent note,

[71] Ross Street, Weak omega-categories, to appear in Contemporary Mathematics; also

     available via http://www.math.mq.edu.au/∼street, 2001, 7 pages,

where Street modifies definition St so that a weak ω-category is now a simplicial set with properties rather than structure.

The word ‘thin’ has been used for the same purpose as ‘hollow’, hence the name Tcomplex, as discussed in III.2.26 and onwards in Kamps and Porter’s book [28] (which also contains basic information on simplicial sets and horn-filling). The original definition of T-complex was given by M. K. Dakin in his 1975 Ph.D. thesis, published as

[72] M. K. Dakin, Kan complexes and multiple groupoid structures, Mathematical sketches (Esquisses Mathématiques) 32 (1983), xi+92 pages, University of Amiens.

T-complexes are simplicial sets with hollowness satisfying conditions (i)–(iii), but with ‘admissible’ dropped and ‘filler’ changed to ‘unique filler’ in (ii). The dropping of ‘admissible’ means that the delicate orientation considerations of Street’s paper are ignored and any direction is as good as any other—everything can be run backwards. Thus, T-complexes are meant to be like strict ω-groupoids rather than strict ω-categories.

Definition X

The story of X is complicated. Essentially it is a combination of the ideas of Baez, Dolan, Hermida, Makkai and Power. Baez and Dolan proposed a definition of weak n-category, drawing on that of Street, in

66  LEINSTER

[73] John C. Baez, James Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, Advances in Mathematics 135 (1998), no. 2, 145–206; also

e-print q-alg/9702014, 1997.

An informal account is in section 4 of Baez’s [4], and another account, which also discusses relations between homotopy theory and n-category theory, is in

[74] John C. Baez, James Dolan, Categorification, in Higher category theory (Evanston, IL, 1997), Contemporary Mathematics 230, AMS, 1998, pp. 1–36; also e-print

math.QA/9802029, 1998.

In turn, Hermida, Makkai and Power drew on the work of Baez and Dolan, producing a modified version of Baez and Dolan’s opetopic sets, which they called multitopic sets. (My use in X of the former term rather than the latter should not be interpreted as significant.) Their original preprint still seems to be available somewhere on the web:

[75] Claudio Hermida, Michael Makkai, John Power, On weak higher dimensional cate-

gories, available via http://fcs.math.sci.hokudai.ac.jp/doc/info/ncat.html,

1997, 104 pages

and is currently enjoying a journal serialization:

[76] Claudio Hermida, Michael Makkai, John Power, On weak higher dimensional categories I: Part 1, Journal of Pure and Applied Algebra 154 (2000), no. 1-3, 221–246,

[77] Claudio Hermida, Michael Makkai, John Power, On weak higher-dimensional categories I—2, Journal of Pure and Applied Algebra 157 (2001), no. 2-3, 247–277,

[78] Claudio Hermida, Michael Makkai, John Power, On weak higher-dimensional categories I—3, Journal of Pure and Applied Algebra 166 (2002), no. 1-2, 83–104.

A related paper with a somewhat different slant and in a much more elementary style is

[79] Claudio Hermida, Michael Makkai, John Power, Higher-dimensional multigraphs, in Thirteenth Annual IEEE Symposium on Logic in Computer Science (Indianapolis, IN, 1998), IEEE Computer Society, Los Alamitos, CA, 1998, pp. 199–206.

Hermida, Makkai and Power’s original work did not go as far as an alternative definition of weak n-category, although see the description below of [84].

I learned something near to definition X from

[80] Martin Hyland, Definition of lax n-category, seminar at DPMMS, Cambridge, based on a conversation with John Power, 18 June 1997.

Whether this is closer to the approach of Baez and Dolan or of Hermida, Makkai and Power is hard to say. The Baez-Dolan definition falls into two parts: the definition of opetopic set, then the definition of universality. Certainly the universality in X is Baez and Dolan’s, but the sketch of the definition of opetopic set is very elementary, in contrast to the highly involved definitions of opetopic/multitopic set given by both these groups of authors.

Opetopic sets are, it is claimed in [73], just presheaves on a certain category, the category of opetopes. (The situation can be compared with that of simplicial sets, which are just presheaves on the category Δ.) Multitopic sets are shown in [75] to be presheaves on a category of multitopes. A third notion of opetope, going (perhaps reprehensibly) by the same name, is given briefly in section 4.1 of my [37], and is laid out in more detail in Chapter IV of my [33]. Roughly speaking, it is shown that all three notions are equivalent in

[81] Eugenia Cheng, The relationship between the opetopic and multitopic approaches to weak n-categories, available via http://www.dpmms.cam.ac.uk/∼elgc2, 2000, 36 pages

(which compares Baez-Dolan’s notion with Hermida-Makkai-Power’s) and

[82] Eugenia Cheng, Equivalence between approaches to the theory of opetopes, available

     via http://www.dpmms.cam.ac.uk/∼elgc2, 2000, 36 pages

(which adds in my own). More accurately, Cheng begins [81] by modifying Baez and Dolan’s notion of operad; the effect of this is that the symmetries present in Baez and Dolan’s account are now handled much more cleanly and naturally, especially when it comes to the crucial process of ‘slicing’. So this means that the Baez-Dolan opetopes are not necessarily the same as the three equivalent kinds of opetope involved in Cheng’s result, and it remains to be seen whether they fit in.

Let us now turn from opetopic sets to universality. The notion of liminality does not appear in Baez and Dolan’s paper, and is in some sense a substitute for their notion of ‘balanced punctured niche’. I made this change in order to shorten the inductive definitions; it is just a rephrasing and has no effect on the definition of universal cell. The price to be paid is that in isolation, liminality is probably a less meaningful concept than that of balanced punctured niche.

More on the formulation of universality can be found in

[83] Eugenia Cheng, A notion of universality in the opetopic theory of n-categories,

     available via http://www.dpmms.cam.ac.uk/∼elgc2, 2001, 12 pages.

Makkai appears to have hit upon a notion of ‘ω-dimensional universal properties’, and thereby developed the definition of multitopic set into a new definition of weak ω-category:

[84] M. Makkai, The multitopic ω-category of all multitopic ω-categories, available via

     http://mystic.biomed.mcgill.ca/M Makkai, 1999, 67 pages.

I do not, unfortunately, know enough about this to include an account here. Nor have I included the definition of

[85] Tom Leinster, Batanin meets Baez and Dolan: yet more ways to define weak ncategory, seminar at DPMMS, Cambridge, 6 February 2001,

which uses opetopic shapes but an algebraic approach like that of L. This definition can be repeated for various other shapes, such as globular (giving exactly L) and computads (which are like opetopes but with many outputs as well as many inputs), and perhaps simplicial and even cubical.

Finally, the analysis of n ≤ 2 has been done in a very precise way, in

[86] Eugenia Cheng, Equivalence between the opetopic and classical approaches to bicategories, available via http://www.dpmms.cam.ac.uk/∼elgc2, 2000, 68 pages.

This uses the notion of opetopic/multitopic set given by Cheng’s modification of Baez and Dolan, or by my opetopes, or by Hermida, Makkai and Power (for by her equivalence result, all three notions are the same), together with the Baez-Dolan notion of universality. I am fairly confident that this gives the same definition of weak 2-category as is described in X above.

Other Definitions of n-Category

I have already mentioned several proposed definitions of weak n-category which are not presented here. My own [42] and [85] are missing, as is Street’s recent modification [71] of St. The opetopic definitions—those related to definition X—are under-represented, as I have not given any such definition in precise terms; in particular, there is no exact presentation of Baez-Dolan’s definition [73], of Cheng’s modification of Baez-Dolan’s definition ([81], [82]), or of Makkai’s definition [84].

In the final stages of writing this I received a preprint,

[87] J. P. May, Operadic categories, A_{∞}-categories and n-categories, available via http://

     www.math.uchicago.edu/∼may, 2001, 10 pages,

containing another definition of weak n-category. I have not had time to assimilate this; nor have I yet digested the approach to weak n-categories in

[88] Hiroyuki Miyoshi, Toru Tsujishita, Weak ω-categories as ω-hypergraphs, e-print

     math.CT/0003137, 2000, 26 pages,

[89] Akira Higuchi, Hiroyuki Miyoshi, Toru Tsujishita, Higher dimensional hypercategories, e-print math.CT/9907150, 1999, 25 pages.

Comparing Definitions

It seems that not a great deal of rigorous work has been done on comparing the proposed definitions, although there are plenty of informal ideas floating about. The papers that I know of are listed above under the appropriate definitions.

The ‘n ≤ 2’ sections show that there are many reasonable notions even of weak 2category. This is not diminished by restricting to one-object weak 2-categories, that is, monoidal categories. So by examining and trying to compare various possible notions of monoidal category, one can hope to get some idea of what things will be like for weak ncategories in general. A proof of the equivalence of various ‘algebraic’ or ‘definite’ notions of monoidal category is in

[90] Tom Leinster, What’s a monoidal category?, poster at CT2000, Como, Italy, 2000,

and a similar but less general result is in Chapter 1 of my [34] (actually stated for bicategories). Hermida compares the indefinite with the definite in his paper [36] on representable multicategories, and a different definite/indefinite comparison is in section 3 of my [45] or section 3.3 of my [44]. (I use the terms ‘definite’ and ‘algebraic’ in the sense of the Introduction.)

No-one who has seen the definition of tricategory given by Gordon, Power and Street in [8] will take lightly the prospect of analysing the case n = 3. However, it is worth pointing out an aspect of this definition less well-known than its complexity: that it is not quite algebraic. In precise terms, what I mean by this is that the category whose objects are tricategories and whose maps are strict maps of tricategories is not monadic over the category of 3-globular sets. (3-globular sets are globular sets as in the ‘Strict n-Categories’ section of ‘Background’, but with m only running from 0 up to 3. So the graph structure of a tricategory is a 3-globular set.) For whereas most of the definition of tricategory consists of some data subject to some equations, a small part does not: in items (TD5) and (TD6), it is stipulated that certain transformations of bicategories are equivalences. This is not an algebraic axiom; to make it into one, we would have to add in as data a pseudo-inverse for each of these equivalences, together with two invertible modifications witnessing the fact that it is a pseudo-inverse, and then we would want to add more coherence axioms (saying, amongst other things, that this data forms an adjoint equivalence). So there is little chance that the category of weak 3-categories and strict maps according to P, B1, L1, or any other algebraic definition is equivalent to the category of tricategories and strict maps, in contrast to the situation for n = 2.

Related Areas

I will be extremely brief here; as stated above, this is not meant to be a survey of the literature. However, there are two areas I feel it would be inappropriate to omit. Most of the references that follow are meant to function as ‘meta-references’, and are chosen for their comprehensive bibliographies.

The first area is the Australian school of 2-dimensional algebra, a representative of which is

[91] R. Blackwell, G. M. Kelly, A. J. Power, Two-dimensional monad theory, Journal of Pure and Applied Algebra 59 (1989), no. 1, 1–41.

The issues arising there merge into questions of coherence, one starting point for which is the paper ‘On braidings, syllepses and symmetries’ by Sjoerd Crans:

[92] Sjoed Crans, On braidings, syllapses and symmetries, Cahiers de Topologie et Géométrie Différentielle 41 (2000), no. 1, 2–74; also available via http://math.

     unice.fr/∼crans,

[93] S. Crans, Erratum: ‘On braidings, syllapses and symmetries’, Cahiers de Topologie et Géométrie Différentielle 41 (2000), no. 2, 156.

More references for work in this area are to be found in Street’s [25].

The second area is from algebraic topology: where higher-dimensional category theorists want to take strict algebraic structures and weaken them, stable homotopy theorists like to take strict topological-algebraic structures and do them up to homotopy (in a more sensitive way than one might at first imagine). The two have much in common. Various systematic ways of doing the latter have been proposed, and some of these are listed on the last page of text in my [44]. Missing from that list is the method of

[94] Mikhail A. Batanin, Homotopy coherent category theory and A_{∞}-structures in monoidal categories, Journal of Pure and Applied Algebra 123 (1998), no. 1-3, 67–103;

     also available via http://www.math.mq.edu.au/∼mbatanin/papers.html.

Another connection with homotopy theory and loop spaces is in

[95] C. Balteanu, Z. Fiedorowicz, R. Schwänzl, R. Vogt, Iterated monoidal categories, e-print math.AT/9808082, 1998, 55 pages.

Further references for these two areas and more can be found in the ‘Introductory Texts’ listed above.

Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences, Wilberforce Road Cambridge CB3 0WB, UK

Email: leinster@dpmms.cam.ac.uk

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Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca