Theory and Applications of Categories, Vol. 28, No. 25, 2013, pp. 804–856.

MULTITENSOR LIFTING AND STRICTLY UNITAL HIGHER CATEGORY THEORY

MICHAEL BATANIN, DENIS-CHARLES CISINSKI AND MARK WEBER

Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak n-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units.

1. Introduction

This paper continues the developments of [Batanin-Weber, 2011] and [Weber, 2013] on the interplay between monads and multitensors in the globular approach to higher category theory, and expands considerably on an earlier preprint [Batanin-Cisinski-Weber, 2009]. To take an important example, according to [Weber, 2013] there are two related combinatorial objects which can be used to describe the notion of Gray category. One has the monad A on the category G^{3}(Set) of 3-globular sets whose algebras are Gray categories, which was first described in [Batanin,1998]. On the other hand there is a multitensor (ie a lax monoidal structure) on the category G^{2}(Set) of 2-globular sets, such that categories enriched in E are exactly Gray categories. The theory described in [Weber, 2013] explains how A and E are related as part of a general theory which applies to all operads of the sort defined originally in [Batanin,1998].

However there is a third object which is missing from this picture, namely, the Gray tensor product of 2-categories. It is a simpler object than A and E, and categories enriched in 2-Cat for the Gray tensor product are exactly Gray categories. The purpose of this paper is to exhibit the Gray tensor product as part of our emerging framework. This is done by means of the lifting theorem for multitensors – theorem(3.5) of this article.

MULTITENSOR LIFTING AND STRICTLY UNITAL HIGHER CATEGORY THEORY 805

Strict n-categories can be defined by iterated enrichment, and the lifting theorem leads one to hope that such an inductive definition can be found for a wider class of higher categorical structures. The second main result of this article – theorem(6.2) – says that one has a similar process of iterated enrichment to capture weak n-categories with strict units. In this result the appropriate (lax) tensor product L_{≤n} of weak n-categories with strict units is identified, so that categories enriched in weak n-categories with strict units using this tensor product are exactly weak (n+1)-categories with strict units.

Received by the editors 2011-11-20 and, in revised form, 2013-09-12.

Transmitted by Stephen Lack. Published on 2013-09-23.

2010 Mathematics Subject Classification: 18A05; 18D20; 18D50; 55P48.

Key words and phrases: multitensors, strictly unital higher categories, higher operads.

\begin{matrix}c \\ ◯ \end{matrix} Michael Batanin, Denis-Charles Cisinski and Mark Weber, 2013. Permission to copy for private use granted.

In order to formulate this second result, we identify a new class of higher operad, these being the reduced T_{≤n}-operads of section(5.3). Moreover we describe the notion of contractibility for such operads which enables one to formalise the idea that a given higher categorical structure should have strict units. Then weak n-categories with strict units are defined as algebras of the universal contractible reduced T_{≤n}-operad, analogous to how weak n-categories were defined operadically in [Batanin,1998]. It turns out that the lifting theorem is particularly compatible with these new notions, and it is this fact which is chiefly responsible for theorem(6.2).

Let us turn now to a more detailed introduction to this article. Recall [Batanin-Weber, 2011, Weber, 2013] that a multitensor (E, u, σ) on a category V consists of n-ary tensor product functors E_{n} : V^{n} → V, whose values on objects are denoted in any of the following ways

E(X_{1}, ..., X_{n}) E_{n}(X_{1}, ..., X_{n}) E_{1≤i≤n} X_{i}  E_{i} X_{i}

depending on what is most convenient, together with unit and substitution maps

u_{X} : Z → E_{1}X σ_{X_{ij}} : E_{i} E_{j} X_{ij} → E_{ij} X_{ij}

for all X, X_{ij} from V which are natural in their arguments and satisfy the obvious unit and associativity axioms. It is also useful to think of (E, u, σ) more abstractly as a lax algebra structure on V for the monoid monad^{1} M on CAT, and so to denote E as a functor E : MV → V. The basic example to keep in mind is that of a monoidal structure on V, for in this case E is given by the n-ary tensor products, u is the identity and the components of σ are given by coherence isomorphisms for the monoidal structure.

A category enriched in E consists of a V -enriched graph X together with composition maps

κ_{x_{i}} : E_{i} X(x_{i−1}, x_{i}) → X(x_{0}, x_{n})

for all n ∈ ℕ and sequences (x_{0}, ..., x_{n}) of objects of X, satisfying the evident unit and associativity axioms. With the evident notion of E-functor (see [Batanin-Weber, 2011]), one has a category E-Cat of E-categories and E-functors together with a forgetful functor

U^{E} : E-Cat → GV.

When E is a distributive multitensor, that is when E_{n} commutes with coproducts in each variable, one can construct a monad ΓE on GV over Set. The object map of the underlying endofunctor is given by the formula

1Recall that for a category V, an object of MV is a finite sequence (X_{1}, ..., X_{n}) of objects of V, that one only has morphisms in MV between sequences of the same length, and that such a morphism (X_{1}, ..., X_{n}) → (Y_{1}, ..., Y_{n}) consists of morphisms f_{i} : X_{i} → Y_{i} for 1 ≤ i ≤ n.

ΓEX(a, b) = ∐_{a=x_{0},...,x_{n}=b} E_{i} X(x_{i−1}, x_{i}),

the unit u is used to provide the unit of the monad and σ is used to provide the multiplication. The identification of the algebras of ΓE and categories enriched in E is witnessed by a canonical isomorphism E-Cat ≅ G(V)^{ΓE} over GV. In section(2) we recall the relevant aspects of the theory of multitensors and monads from [Weber, 2013].

If one restricts attention to unary operations, then E_{1}, u and the components σ_{X} : E_{1}^{2}X → E_{1}X provide the underlying endofunctor, unit, and multiplication for a monad on V. This monad is called the unary part of E. When the unary part of E is the identity monad, the multitensor is a functor operad. This coincides with existing terminology, see [McClure-Smith, 2004] for instance, except that we don’t in this paper consider any symmetric group actions. Since units for functor operads are identities, we denote any such as a pair (E, σ), where as for general multitensors E denotes the functor part and σ the substitution.

By definition then, a functor operad is a multitensor. On the other hand, as observed in [Batanin-Weber, 2011] lemma(2.7), the unary part of a multitensor E acts on E, in the sense that as a functor E factors as

[figure: MV; V^{E_{1}}; V; U^{E_{1}}]

and in addition, the substitution maps are morphisms of E_{1}-algebras. Moreover an Ecategory structure on a V -enriched graph X includes in particular an E_{1}-algebra structure on each hom X(a, b) of X with respect to which the composition maps are morphisms of E_{1}-algebras. These observations lead to

1.1. Question. Given a multitensor (E, u, σ) on a category V can one find a functor operad (E′, σ′) on V^{E_{1}} such that E′-categories are exactly E-categories?

The first main result of this paper, theorem(3.5), says that question(1.1) has a nice answer: when E is distributive and accessible and V is cocomplete, one can indeed find a unique distributive accessible such E′.

One case of our lifting theorem in the literature is in the work of Ginzburg and Kapranov on Koszul duality [Ginzburg-Kapranov, 1994]. Formula (1.2.13) of that paper, in the case of a K-collection E coming from an operad, implicitly involves the lifting of the multitensor corresponding (as in [Batanin-Weber, 2011] example(2.6)) to the given operad. For instance, our lifting theorem gives a general explanation for why one must tensor over K in that formula. As we shall see in section(3.11), another existing source of examples comes from Day convolution [Day, 1970].

Taking E to be the multitensor on G^{n}(Set) such that ΓE is the monad whose algebras are weak (n+1)-categories, one might hope that the statement that “E′-categories are exactly E-categories” in this case expresses a sense in which weak (n+1)-categories are categories enriched in weak n-categories for an appropriate tensor product E′. However as we explain in section(5.2), the presence of weak identity arrows as part of the structure of weak (n+1)-category prevents such a pleasant interpretation directly, because composition with weak identity arrows gives the homs of an E-category extra structure, as can already be seen for the case n = 1 of bicategories.

Having identified this issue, it is natural to ask

1.2. Question. Does the lifting theorem produce the tensor products enabling weak ncategories with strict units to be obtained by iterated enrichment?

For this question to be well-posed it is necessary to define weak n-categories with strict units.

To do this it is worth first meditating a little on the operadic definition of weak ncategory. Indeed since such a definition first appeared in [Batanin,1998], as the algebras of a weakly initial higher operad of a certain type, there have been a number of works that have refined our understanding. In [Leinster, 2003] this topic was given a lovely exposition, the focus was shifted to considering operads that were strictly initial with the appropriate properties rather than weakly so, and an alternative notion of contractibility was given which, for expository purposes, is a little simpler. Thus in this article we use Leinster contractibility throughout, however as we point out in remark(4.9), one can give a completely analogous development using the original notions.

On a parallel track, Grandis and Tholen initiated an algebraic study of the weak factorisation systems arising in abstract homotopy theory in [Grandis-Tholen, 2006], this was refined by Garner in [Garner, 2009(2)] and then applied to higher category theory in [Garner, 2009] and [Garner, 2010]. We devote a substantial part of section(4) to a discussion of these developments, both by way of an updated review of the definition of weak n-category, and because we make essential use of Garner’s theory of weak factorisation systems later in the article.

Assimilating these developments one can present an operadic definition of weak ncategories in the following way. One has a presheaf category T_{≤n}-Coll_{0} of T_{≤n}-collections over Set (called “normalised collections” in [Batanin,1998]) and there are two finitary monads on T_{≤n}-Coll_{0} the first of whose algebras are T_{≤n}-operads over Set. To give an algebra structure for the second of these monads is to exhibit a given T_{≤n}-collection over Set as contractible, and one obtains this monad from an algebraic weak factorisation system, in the sense of [Garner, 2009(2)], on T_{≤n}-Coll_{0}. In general the monad coproduct of a pair of finitary monads on a locally finitely presentable category is itself finitary, and so its category of algebras will also be locally finitely presentable. Applied to the two monads in question, the algebras of their monad coproduct are contractible T_{≤n}-operads over Set, the initial such being the operad for weak n-categories.

We now adapt this to give an operadic definition weak n-category with strict units in the following way. First, one considers only reduced T_{≤n}-operads, which are those that contain a unique unit operation of each type. Second, one strengthens the notion of contractibility, by giving an algebraic weak factorisation system on the category PtRd-T_{≤n}-Coll of pointed reduced T_{≤n}-collections. This notion of underlying collection includes the substitution of unique unit operations as part of the structure. The purpose of this refined contractibility notion is to encode the idea that the unique unit operations of the operads under consideration really do behave as strict identities. So as before one has a pair of finitary monads, but this time on the category PtRd-T_{≤n}-Coll, one whose algebras are reduced operads, and the other whose algebras are pointed reduced collections that are exhibited as contractible in this stronger sense. One then imitates the scheme laid out in the previous paragraph, defining the operad for weak n-categories with strict units as the initial object of the category of algebras of the coproduct of these two monads on PtRd-T_{≤n}-Coll.

A novelty of this definition is that it uses the extra generality afforded by the theory of algebraic weak factorisation system’s of [Garner, 2009(2)]. Namely, while in the standard theory of weak factorisation systems one considers those that are generated, via the small object argument of Quillen, by a set of generating cofibrations. In Garner’s algebraic version of the small object argument, one can consider a category of generating cofibrations. The algebraic weak factorisation system we consider on PtRd-T_{≤n}-Coll has a category of generating cofibrations, and it is the morphisms of this category by means of which the strictness of units is expressed.

Having obtained a reasonable definition of weak n-category with strict units, we then proceed to answer question(1.2) in the affirmative, this being our second main result theorem(6.2). For any T_{≤n+1}-operad B over Set, there is a T_{≤n}-operad h(B), whose algebras are by definition the structure possessed by the homs of a B-algebra. This is the object map of a functor h, and if one restricts attention to reduced B, and then contractible reduced B, h in either of these contexts becomes a left adjoint, which is the formal fact that enables theorem(6.2) to go through.

This paper is organised in the following way. In section(2) we review the theory of multitensors from [Weber, 2013]. Then in section(3) we discuss the lifting theorem. The theorem itself is formulated and proved in section(3.2), applications are presented in sections(3.6) and (3.11), and the 2-functorial aspects of the lifting theorem are presented in section(3.13). We give a review of the definition of weak n-category and the theory of algebraic weak factorisation systems in section(4). In section(5) we present our definition of weak n-category with strict units, and then the tensor products which exhibit this notion as arising by iterated enrichment are produced in section(6).

Notation and terminology. Efforts have been made to keep the notation and terminology of this article consistent with [Weber, 2013]. The various other standard conventions and abuses that we adopt include regarding the Yoneda embedding ℂ → ℂ̂ as an inclusion, and so by the Yoneda lemma regarding x ∈ XC for X ∈ ℂ̂ as an arrow x : C → X in ℂ̂. To any such X, and more generally any pseudo-functor X : ℂ^{op} → Cat, the Grothendieck construction gives the associated fibration into ℂ, the domain of which we denote as el(X) and call the “category of elements of X”. The objects of the topologists’ simplicial category Δ are as usual written as ordinals [n] = {0 < ... < n} for n ∈ ℕ, and regarded as a full subcategory of Cat. Thus in particular [1] is the category consisting of one non-identity arrow, and E^{[1]} denotes the arrow category of a category E. We use a different notation for the algebraists’ simplicial category Δ_{+}, objects here being regarded as ordinals n = {1 < ... < n} for n ∈ ℕ. In section(5.3), we adopt the abuse of identifying the left adjoint reflection of the inclusion of a full subcategory with the idempotent monad generated by the adjunction.

2. Review of the theory of multitensors

Given a category V, GV is the category of graphs enriched in V. Thus G1 = Set and G^{n}(Set) is equivalent to the category of n-globular sets. As a category GV is at least as good as V. Two results from [Weber, 2013] which express this formally, are proposition(2.11) which explains how colimits in GV are constructed from those in V, and theorem(2.15) from which it follows that if V is locally presentable (resp. a Grothendieck topos, resp. a presheaf topos) then so is GV.

Certain colimits in GV are very easy to understand directly, namely those connected colimits in which the arrows of the diagram are identities on objects. For in this case, as pointed out in remark(2.13) of [Weber, 2013], one can just take the object set of the colimit to be that of any of the V -graphs appearing in the diagram, and compute the colimit one hom at a time in the expected way. These colimits will play a basic role in the developments of section(3.2).

Formal properties of the endofunctor G : CAT → CAT were discussed in section(2) of [Weber, 2013]. In particular G preserves Eilenberg Moore objects, so that given a monad T on a category V, one has an isomorphism G(V)^{G(T)} ≅ G(V^{T}) expressing that the algebras of the monad GT on GV are really just graphs enriched in the category V^{T} of algebras of T.

As explained in the introduction to this article, given a distributive multitensor E on a category V with coproducts, the formula

ΓEX(a, b) = ∐_{a=x_{0},...,x_{n}=b} E_{i} X(x_{i−1}, x_{i})

describes the object map of the underlying endofunctor of the monad ΓE on the category GV of graphs enriched in V, whose algebras are E-categories. The assignment (V, E) ↦ (GV, ΓE) is itself the object map of a locally fully faithful 2-functor

Γ : DISTMULT ⟶ MND(CAT/Set)

from the full sub-2-category DISTMULT of the 2-category of lax monoidal categories, lax monoidal functors and monoidal natural transformations, consisting of those (V, E) where V has coproducts and E is distributive, to the 2-category of monads, as defined in [Street, 1972], in the 2-category CAT/Set.

From this perspective then, the category GV is regarded as being “over Set”, that is to say, it is taken together with the functor (−)_{0} : GV → Set which sends a V -graph to its set of objects. A monad in CAT/Set is called a monad over Set, and the monad ΓE is over Set because: (1) for any V -graph X, the objects of ΓEX are those of X; (2) for any morphism of V -graphs f : X → Y, the object map of ΓEf is that of f; and (3) the components of the unit and multiplication of ΓE are identity-on-objects morphisms of V -graphs. From the point of view of structure, ΓE being over Set says that the structure of an E-category gives nothing at the level of objects.

Section(3), and especially theorem(3.7) of [Weber, 2013], contains a careful analysis of how properties of E correspond to properties of the induced monad ΓE. The particular fact following from that discussion that we shall use below is recorded here in lemma(2.1). Recall that a multitensor E is λ-accessible for some regular cardinal λ, when the underlying functor E : MV → V preserves λ-filtered colimits, and that this condition is equivalent to each of the E_{n} : V^{n} → V preserving λ-filtered colimits in each variable (see [Weber, 2013] section(3) for more discussion).

2.1. Lemma. Suppose that V is a cocomplete category, λ is a regular cardinal and E is distributive multitensor on V. Then E is λ-accessible iff ΓE is λ-accessible.

Monads T on GV over Set of the form ΓE for some distributive multitensor E on V a category with coproducts, are those that are path-like and distributive in the sense that we now recall. These properties concern only the functor part of the given monad, and so for the sake of the proof of theorem(3.5) below, we shall formulate these definitions more generally than in [Weber, 2013], for functors over Set between categories of enriched graphs.

When the category V has an initial object ∅, sequences (X_{1}, ..., X_{n}) of objects of V can be regarded as V -graphs. The object set for the V -graph (X_{1}, ..., X_{n}) is {0, ..., n}, for 1 ≤ i ≤ n the hom from (i−1) to i is X_{i}, and the other homs are ∅. An evocative informal picture of this V -graph is

[figure: 0; 1; ...; n−1; n; X_{1}; X_{2}; X_{n−1}; X_{n}]

because it encourages one to imagine X_{i} as being the object of ways of moving from (i−1) to i in this simple V -graph.

Given a functor T : GV → GW over Set, one can define a functor \overline{T} : MV → W whose object map is given by

\overline{T}_{1≤i≤n} X_{i} = T (X_{1}, ..., X_{n})(0, n).

Note that since T is over Set, the W -graph T (X_{1}, ..., X_{n}) also has object set {0, ..., n}, and so the expression on the right hand side of the above equation makes sense as a hom of this W -graph. By definition \overline{T} amounts to functors \overline{T}_{n} : V^{n} → W for each n ∈ ℕ.

2.2. Definition. Let V and W be categories with coproducts. A functor T : GV → GW over Set is distributive when for each n ∈ ℕ, \overline{T}_{n} preserves coproducts in each variable. A monad T on GV over Set is distributive when its underlying endofunctor is distributive.

As in [Weber, 2013] section(4), this definition can be reexpressed in more elementary terms without mention of \overline{T}.

Intuitively, path-likeness says that the homs of the W -graphs T X are in some sense some kind of abstract path object. Formally for a functor T : GV → GW over Set, given a V -graph X and sequence x = (x_{0}, ..., x_{n}) of objects of X, one can define the morphism

\overline{x} : (X(x_{0}, x_{1}), X(x_{1}, x_{2}), ..., X(x_{n−1}, x_{n})) → X

whose object map is i ↦ x_{i}, and whose hom map between (i − 1) and i is the identity. For all such sequences x one has

T (\overline{x})_{0,n} : \overline{T}_{i} X(x_{i−1}, x_{i}) → T X(x_{0}, x_{n})

and so taking all sequences x starting at a and finishing at b one induces the canonical map

π_{T,X,a,b} : ∐_{a=x_{0},...,x_{n}=b} \overline{T}_{i} X(x_{i−1}, x_{i}) → T X(a, b)

in W.

2.3. Definition. Let V and W be categories with coproducts. A functor T : GV → GW over Set is path-like when for all X ∈ GV and a, b ∈ X_{0}, the maps π_{T,X,a,b} are isomorphisms. A monad T on GV over Set is path-like when its underlying endofunctor is path-like.

A basic result concerning these notions that will be useful below is

2.4. Lemma. Let V, W and Y be categories with coproducts and R : V → W, T : GV → GW and S : GW → GY be functors.

1. If R preserves coproducts then GR is distributive and path-like.

2. If S and T are distributive and path-like, then so is ST.

Proof. (1): Since R preserves the initial object one has GR(Z_{1}, ..., Z_{n}) = (RZ_{1}, ..., RZ_{n}) and so \overline{GR} : MV → W sends sequences of length n ≠ 1 to ∅, and its unary part is just R. Thus GR is distributive since R preserves coproducts, and coproducts of copies of ∅ are initial. The summands of the domain of π_{GR,X,a,b} are initial unless (x_{0}, ..., x_{n}) is the sequence (a, b), thus π_{GR,X,a,b} is clearly an isomorphism, and so GR is path-like.

(2): Since S and T are path-like and distributive one has

ST (Z_{1}, ..., Z_{n})(0, n) ≅ ∐_{0=r_{0}≤...≤r_{m}=n} \overline{S}_{1≤i≤m} \overline{T}_{r_{i−1}<j≤r_{i}} Z_{j}

and so ST is path-like and distributive since S and T are, and since a coproduct of coproducts is a coproduct.

The aforementioned characterisation of monads of the form ΓE is then given by the following result.

2.5. Theorem. ([Weber, 2013] theorem(4.9)). For a category V with coproducts, a monad T on GV over Set is of the form ΓE for a distributive multitensor E iff T is path-like and distributive, in which case one can take E = \overline{T}.

So far we have discussed two important constructions, Γ which produces a monad from a multitensor, and \overline{(−)} which produces a multitensor from a monad. We now recall a third construction (−)^{×} which also produces a multitensor from a monad. If V has finite products and T is any monad on V, then one can define a multitensor T^{×} on V as follows

T^{×}_{1≤i≤n} X_{i} = ∏_{i=1}^{n} T X_{i}.

A category enriched in T^{×} is exactly a category enriched in V^{T} using the cartesian product. As explained in section(5) of [Weber, 2013], this is an instance of a general phenomenon of a distributive law of a multitensor over a monad, in this case witnessed by the fact that any monad T will be opmonoidal with respect to the cartesian product on V. If moreover V has coproducts and V ’s products distribute over them, so that cartesian product ∏ on V is a distributive multitensor, and if T preserves coproducts, then T^{×} is a distributive multitensor. The 2-functoriality of Γ and the formal theory of monads [Street, 1972] then ensures that at the level of monads one has a distributive law G(T)Γ(∏) → Γ(∏)G(T) between the monads Γ(∏) and G(T) on GV.

Fundamental to the globular approach to higher category are the monads T_{≤n}, defined for n ∈ ℕ, on the category G^{n}(Set) of n-globular sets, whose algebras are strict ncategories. Higher categorical structures in dimension n in this approach are by definition algebras of T_{≤n}-operads, and a T_{≤n}-operad is by definition a monad A on G^{n}(Set) equipped with a cartesian monad morphism^{2} α : A → T_{≤n}. In terms of the theory described so far, the monads T_{≤n} have a simple inductive description

• T_{≤0} is the identity monad on Set.

• Given the monad T_{≤n} on G^{n}Set, define the monad T_{≤n+1} = ΓT_{≤n}^{×} on G^{n+1}Set.

The fact that T_{≤n} algebras are strict n-categories is immediate from this definition, our understanding of what the constructions Γ and (−)^{×} correspond to at the level of algebras and enriched categories, and the definition of strict n-category via iterated enrichment using cartesian product. Moreover, an inductive definition of the T_{≤n} via an iterative system of distributive laws [Cheng, 2011], is also an immediate consequence of this point of view.

That one has this good notion of T_{≤n}-operad expressible so generally in terms of cartesian monad morphisms, so that it fits within the framework of Burroni [Burroni, 1971], Hermida [Hermida, 2000] and Leinster [Leinster, 2003], is due to the fact that T_{≤n} is a cartesian monad. That is, its functor part preserves pullbacks and the naturality squares for its unit and multiplication are pullback squares. The underlying endofunctor T_{≤n} satisfies a stronger condition than pullback preservation, namely, it is a local right adjoint, that is to say, for any X ∈ G^{n}(Set), the functor

2That is, a natural transformation between underlying endofunctors, whose naturality squares are pullbacks, and which is compatible with the monad structures.

(T_{≤n})_{X} : G^{n}(Set)/X ⟶ G^{n}(Set)/T_{≤n}X

obtained by applying T_{≤n} to arrows into X, is a right adjoint. A cartesian monad whose functor part satisfies this stronger property is called a local right adjoint monad^{3} in this article, and this notion is important because for such monads one can define the nerve of an algebra in a useful way. See [Berger, 2002], [Weber, 2007], [Melliès, 2010] and [Berger-Melliès-Weber, 2012] for further discussion.

All these pleasant properties enjoyed by the monads T_{≤n} can be understood in terms the compatibility of these properties with the constructions Γ and (−)^{×}. See [Weber, 2013], especially theorem(3.7) and example(5.7), for more details.

A T_{≤n}-operad over Set is a T_{≤n}-operad α : A → T_{≤n} such that the monad A is over Set and α’s components are identities on objects. We denote by T_{≤n}-Op_{0} the category of T_{≤n}-operads over Set. There is also a notion of E-multitensor for a given cartesian multitensor E on a category V, which consists of another multitensor F on V equipped with a cartesian morphism of multitensors ϕ : F → E. We denote by E-Mult the category of E-multitensors. Thanks to the functoriality of Γ and its compatibility with the categorical properties participating in these definitions one has

2.6. Theorem. The constructions Γ and \overline{(−)} give an equivalence of categories

T_{≤n}^{×}-Mult ≃ T_{≤n+1}-Op_{0}.

This result first appeared as corollary(7.10) of [Batanin-Weber, 2011], and was subsequently generalised in [Weber, 2013] theorem(6.2) with T_{≤n}^{×} replaced by a general cartesian multitensor E on a lextensive category V.

3. Lifting theorem

3.1. Overview. In [Crans, 1999] the so-called “Crans tensor product” of Gray categories was constructed by hand, and categories enriched in this tensor product were called “4teisi” and were unpacked in [Crans, 2000], these being Crans’ candidate notion of semistrict 4-category. From the perspective of these papers it is not at all clear that one can proceed in the reverse order, first constructing a T_{≤4}-operad for 4-teisi, and then obtaining the tensor product from this. By ordering the discussion in this way one makes a direct formal connection between the combinatorial work of Crans and the theory of higher operads. Even given an intuition that one can begin with the operad and then obtain the tensor product, it is very far from clear from the way that things are described in [Crans, 1999] and [Crans, 2000], how one would express such an intuition formally without drowning in the complexity of the combinatorial notions involved.

3In [Berger-Melliès-Weber, 2012] the terminology strongly cartesian monad is used.

The lifting theorem applied to this example gives a formal expression of this intuition in a way independent of any of these combinatorial details. It does not however say anything about how the T_{≤4}-operad for 4-teisi is constructed. Put simply, the T_{≤4}-operad for 4-teisi is the input, and the Crans tensor product is the output for this application of the lifting theorem.

This section is organised as follows. In section(3.2) we formulate and prove the lifting theorem as an answer to question(1.1). Then in section(3.6) we discuss how this result brings the Gray and Crans tensor products, of 2-categories and Gray categories respectively, into our framework. Section(3.11) exhibits Day convolution as arising via an application of the lifting theorem. Finally in section(3.13) we exhibit the process of lifting a multitensor as a right 2-adjoint to an appropriate inclusion of functor operads among multitensors.

3.2. The theorem and its proof. The idea which enables us to answer question(1.1) is the following. Given a distributive multitensor E on V one can consider also the multitensor Ẽ_{1} whose unary part is also E_{1}, but whose non-unary parts are all constant at the initial object. This is clearly a sub-multitensor of E, also distributive, and moreover as we shall see one has Ẽ_{1}-Cat ≅ G(V^{E_{1}}) over GV. Thus from the inclusion Ẽ_{1} ↪ E one induces the forgetful functor U fitting in the commutative triangle

[figure: G(V^{E_{1}}); E-Cat; GV; U; U^{E}; G(U^{E_{1}})]

For sufficiently nice V and E this forgetful functor has a left adjoint. The category of algebras of the induced monad T will be E-Cat since U is monadic. Thus problem is reduced to that of establishing that this monad T arises from a multitensor on V^{E_{1}}. By theorem(2.5) this amounts to showing that T is path-like and distributive.

In order to implement this strategy, we must understand something about the explicit description of the left adjoint G(V^{E_{1}}) → E-Cat, and this understanding is a basic part of monad theory that we now briefly recall. Suppose that (M, η^{M}, μ^{M}) and (S, η^{S}, μ^{S}) are monads on a category E, and ϕ : M→S is a morphism of monads. Then ϕ induces the forgetful functor ϕ^{∗} : E^{S} → E^{M} and we are interested in computing the left adjoint ϕ_{!} to ϕ^{∗}. By the Dubuc adjoint triangle theorem [Dubuc, 1970], one may compute the value of ϕ_{!} at an M-algebra (X, x : MX→X) as a reflexive coequaliser

[figure: (SMX, μ_{MX}^{S}); (SX, μ_{X}^{S}); ϕ_{!}(X, x); μ_{X}^{S}S(ϕ_{X}); Sη_{X}^{M}; Sx; q_{(X,x)}]  (1)

in E^{S}, when this coequaliser exists. Writing T for the monad induced by ϕ_{!} ⊣ ϕ^{∗}, the components of the unit η^{T} of T are given by

X ⟶^{η_{X}^{S}} SX ⟶^{q_{(X,x)}} ϕ_{!}(X, x).

The coequaliser (1) is taken in E^{S}, and we will need to have some understanding of how this coequaliser is computed in terms of colimits in E. So we suppose that E has filtered colimits and coequalisers, and that S is λ-accessible for some regular cardinal λ. Then the underlying object U^{S}ϕ_{!}(X, x) in E of ϕ_{!}(X, x) is constructed in the following way. We shall construct morphisms

v_{n,X,x} : SQ_{n}(X, x) → Q_{n+1}(X, x)  q_{n,X,x} : Q_{n}(X, x) → Q_{n+1}(X, x)
q_{<n,X,x} : SX → Q_{n}(X, x)

starting with Q_{0}(X, x) = SX by transfinite induction on n.

Initial step. Define q_{<0} to be the identity, q_{0} to be the coequaliser of μ^{S}(Sϕ) and Sx, q_{<1} = q_{0} and v_{0} = q_{0}b. Note also that q_{0} = v_{0}η^{S}.

Inductive step. Assuming that v_{n}, q_{n} and q_{<n+1} are given, we define v_{n+1} to be the coequaliser of S(q_{n})(μ^{S}Q_{n}) and Sv_{n}, q_{n+1} = v_{n+1}(η^{S}Q_{n+1}) and q_{<n+2} = q_{n+1}q_{<n+1}.

Limit step. Define Q_{n}(X, x) as the colimit of the sequence given by the objects Q_{m}(X, x) and morphisms q_{m} for m < n, and q_{<n} for the component of the universal cocone at m = 0.

[figure: colim_{m<n} S^{2}Q_{m}; colim_{m<n} SQ_{m}; colim_{m<n} Q_{m}; Q_{n}; SQ_{n}; S^{2}Q_{n}; μ_{<n}; v_{<n}; (Sv)_{<n}; η_{<n}; μ; η; o_{n,2}; o_{n,1}]

We write o_{n,1} and o_{n,2} for the obstruction maps measuring the extent to which S and S^{2} preserve the colimit defining Q_{n}(X, x). We write μ_{<n}^{S}, (Sv)_{<n}, v_{<n} and η_{<n}^{S} for the maps induced by the μ^{S}Q_{m}, Sv_{m}, v_{m} and η^{S}Q_{m} for m < n respectively. Define v_{n} as the coequaliser of o_{n,1}μ_{<n} and o_{n,1}(Sv)_{<n}, q_{n} = v_{n}(η^{S}Q_{n}) and q_{<n+1} = q_{n}q_{<n}.

Then since S preserves λ-filtered colimits, this sequence stabilises in the sense that for any ordinal n such that |n| ≥ λ, q_{n,X,x} is an isomorphism. Thus for any such n one may take

ϕ_{!}(X, x) = (Q_{n}(X, x), q_{n}^{−1}v_{n})  q_{<n} : (SX, μ_{X}) → (Q_{n}(X, x), q_{n}^{−1}v_{n})

as an explicit definition of ϕ_{!}(X, x) and the associated coequalising map in V^{S}. The proof of this is essentially standard – see theorem(3.9) of [Barr-Wells, 2005] for example, and so we omit the details.

In fact all that we require of the above details is that the transfinite construction involves only connected colimits. Moreover in the context that we shall soon consider, these will be connected colimits of diagrams of V -graphs which live wholly within a single fibre of (−)_{0} : GV → Set. As was explained in section(2) and in remark(2.13) of [Weber, 2013], such colimits in GV are straight forward.

3.3. Lemma. Let V be a category with coproducts, W be a cocomplete category, J be a small connected category and

F : J → [GV, GW]

be a functor. Suppose that F sends objects and arrows of J to functors and natural transformations over Set.

(1) Then the colimit K : GV →GW of F may be chosen to be over Set.

Given such a choice of K:

(2) If F j is path-like for all j ∈ J, then K is also path-like.

(3) If F j is distributive for all j ∈ J, then K is also distributive.

Proof. Colimits in [GV, GW] are computed componentwise from colimits in GW and so for X ∈ GV we must describe a universal cocone with components

κ_{X,j} : F j(X) → KX.

By remark(2.13) of [Weber, 2013] we may demand that the κ_{X,j} are identities on objects, and then compute the hom of the colimit between a, b ∈ X_{0} by taking a colimit cocone

{κ_{X,j}}_{a,b} : F j(X)(a, b) → KX(a, b)

in W. This establishes (1). Since the properties of path-likeness and distributivity involve only colimits at the level of the homs as does the construction of K just given, (2) and (3) follow immediately since colimits commute with colimits in general.

Recall the structure-semantics result of Lawvere, which says that for any category E, the canonical functor

Mnd(E)^{op} → CAT/E  T ↦ U : E^{T} → E

with object map indicated is fully faithful (see [Street, 1972] for a proof). An important consequence of this is that for monads S and T on E, an isomorphism E^{T} ≅ E^{S} over E is induced by a unique isomorphism S ≅ T of monads. We now have all the pieces we need to implement our strategy. First, in the following lemma, we give the result we need to recognise the induced monad on G(V^{E_{1}}) as arising from a multitensor.

3.4. Lemma. Let λ be a regular cardinal. Suppose that V is a cocomplete category, R is a coproduct preserving monad on V, S is a λ-accessible monad on GV over Set, and ϕ : GR→S is a monad morphism over Set. Denote by T the monad on G(V^{R}) induced by ϕ_{!} ⊣ ϕ^{∗}.

(1) One may choose ϕ_{!} so that T is over Set.

Given such a choice of ϕ_{!}:

(2) If S is distributive and path-like then so is T.

(3) If R is λ-accessible then so is T.

Proof. Let us denote by ρ : RU^{R} → U^{R} the 2-cell datum of the Eilenberg-Moore object for R, and note that since G preserves Eilenberg Moore objects, one may identify U^{GR} = G(U^{R}) and Gρ as the 2-cell datum for GR’s Eilenberg-Moore object. Now T is over Set iff G(U^{R})T is. Moreover since R preserves coproducts U^{R} creates them, and so T is path-like and distributive iff G(U^{R})T is. Since G(U^{R})T = U^{S}ϕ_{!}, it follows that T is over Set, path-like and distributive iff U^{S}ϕ_{!} is. Since the monads S and GR are over Set, as are ρ and ϕ, it follows by a transfinite induction using lemma(3.3) that all successive stages of this construction give functors and natural transformations over Set, whence U^{S}ϕ_{!} is itself over Set. Lemma(2.4) ensures that the functors GR and G(RU^{R}) are distributive and path-like, since R preserves coproducts and U^{R} creates them. When S is also distributive and path-like, then by the same sort of transfinite induction using lemmas(2.4) and (3.3), all successive stages of this construction give functors that are distributive and path-like, whence U^{S}ϕ_{!} is itself distributive and path-like.

Supposing R to be λ-accessible, note that GR is also λ-accessible. One way to see this is to consider the distributive multitensor R̃ on V whose unary part is R and non-unary parts are constant at the initial object. Thus R̃ will be λ-accessible since R is. To give an R̃-category structure on X ∈ GV amounts to giving R-algebra structures to the homs of X, and similarly on morphisms, whence one has a canonical isomorphism R̃-Cat ≅ G(V^{R}) over GV, and thus by structure-semantics one obtains ΓR̃ ≅ GR. Hence by lemma(2.1), GR is indeed λ-accessible. But then it follows that U^{GR} = G(U^{R}) creates λ-filtered colimits, and so T is λ-accessible iff G(U^{R})T = U^{S}ϕ_{!} is. In the transfinite construction of U^{S}ϕ_{!}, it is now clear that the functors involved at every stage are λ-accessible by yet another transfinite induction, and so U^{S}ϕ_{!} is λ-accessible as required.

To finish the proof we must check that T ’s monad structure is over Set. Since μ^{T} is a retraction of η^{T} T it suffices to verify that η^{T} is over Set, which is equivalent to asking that the components of G(U^{R})η^{T} are identities on objects. Returning to the more general setting of a morphism of monads ϕ : M → S on E and (X, x) ∈ E^{M} discussed above, the outside of the diagram on the left

[figure: MX; X; U^{S}ϕ_{!}(X, x); SX; SMX; SX; S^{2}X; x; U^{M} η_{(X,x)}^{T}; U^{S}q_{(X,x)}; ϕ_{X}; Sx; μ_{X}^{S}; Sϕ_{X}; η_{MX}^{S}; η_{X}^{S}; U^{S}q_{(X,x)}; η_{SX}^{S}; 1; G(RU^{R}); GU^{R}; U^{S}ϕ_{!}; SG(U^{R}); Gρ; G(U^{R})η^{T}; q; ϕG(U^{R})]

clearly commutes, thus one has a commutative square as on the right in the previous display, and so the result follows.

3.5. Theorem. (Multitensor lifting theorem) Let λ be a regular cardinal and let E be a λ-accessible distributive multitensor on a cocomplete category V. Then there is, to within isomorphism, a unique functor operad (E′, σ′) on V^{E_{1}} such that

1. (E′, σ′) is distributive.

2. E′-Cat ≅ E-Cat over GV.

Moreover E′ is also λ-accessible.

Proof. Write ψ : Ẽ_{1} ↪ E for the multitensor inclusion of the unary part of E, and then apply lemma(3.4) with S = ΓE, R = E_{1} and ϕ = Γψ to produce a λ-accessible distributive and path-like monad T on G(V^{E_{1}}) over Set. Thus by theorem(2.5) \overline{T} is a distributive multitensor on V^{E_{1}} with \overline{T} -Cat ≅ E-Cat. Moreover since T ≅ Γ\overline{T} it follows by lemma(2.1) that \overline{T} is λ-accessible. As for uniqueness suppose that (E′, σ′) is given as in the statement. Then by theorem(2.5) Γ(E′) is a distributive monad on G(V^{E_{1}}) and one has

G(V^{E_{1}})^{Γ(E′)} ≅ E-Cat

over G(V^{E_{1}}). By structure-semantics one has an isomorphism Γ(E′)≅T of monads. Since Γ is locally fully faithful, one thus has an isomorphism E′≅\overline{T} of multitensors as required.

From the above proofs in the explicit construction of E′ one first obtains ΓE′. In particular one has a coequaliser

[figure: ΓE(E_{1}X_{1}, ..., E_{1}X_{n}); ΓE(X_{1}, ..., X_{n}); ΓE′((X_{1}, x_{1}), ..., (X_{n}, x_{n}))]  (2)

in E-Cat, and then one takes

E′_{i=1}^{n}(X_{i}, x_{i}) = ΓE′((X_{1}, x_{1}), ..., (X_{n}, x_{n}))(0, n).

The set of objects for each of the E-categories appearing in (2) is {0, ..., n}, and the morphisms are all identities on objects. The explicit construction of (2) at the level of V -graphs proceeds, as we have seen, by a transfinite construction. From this and the definition of V -graphs of the form ΓE(X_{1}, ..., X_{n}), it is clear that for all the E-categories appearing in (2), the hom between a and b for a > b is initial. Thus to understand (2) completely it suffices to understand the homs between a and b for a ≤ b.

By the explicit description of the monad ΓE, the hom of each stage of the transfinite construction in GV of the coequaliser (2), depends on the homs between c and d – where a ≤ c ≤ d ≤ b – of the earlier stages of the construction. Thus the hom between a and b of (2) is

[figure: E_{a<i≤b}E_{1}X_{i}; E_{a<i≤b}X_{i}; E′_{a<i≤b}(X_{i}, x_{i}); σ; E_{i} x_{i}]  (3)

and moreover, by virtue of its dependence on the intermediate homs (ie between c and d as above), this will not simply be the process of taking (3) to be the coequaliser in V^{E_{1}}. For instance when E′ is the Gray tensor product as in example(3.9) below, what we have here is a description of the Gray tensor product of 2-categories in terms of certain coequalisers in Gray-Cat.

However note that when one applies E′ to sequences of free E_{1}-algebras, as in

E′_{i=1}^{n}(E_{1}X_{i}, σ_{(X_{i})}) = ΓE′((E_{1}X_{1}, σ_{(X_{1})}), ..., (E_{1}X_{n}, σ_{(X_{n})}))(0, n)
= ΓE′GE_{1}(X_{1}, ..., X_{n})(0, n)
= ΓE(X_{1}, ..., X_{n})(0, n) = E_{i=1}^{n} X_{i}

one simply recovers E.

3.6. Gray and Crans tensor products. In the examples that we present in this section we shall use the following notation. We denote by A the appropriate T_{≤n+1}- operad over Set and by E the T_{≤n}^{×}-multitensor associated to it by theorem(2.6), so that one has

A = ΓE  E = \overline{A}

and G^{n+1}(Set)^{A} ≅ E-Cat over G^{n+1}(Set). The monad E_{1} on G^{n}(Set) has as algebras the structure borne by the homs of an A-algebra. Theorem(3.5) produces the functor operad E′ on G^{n}(Set)^{E_{1}} such that

G^{n+1}(Set)^{A} ≅ E′-Cat ≅ E-Cat

over G^{n}(Set)^{E_{1}}. Moreover E′ is the unique such functor operad which is distributive. The first of our examples is the most basic.

3.7. Example. When A is the terminal T_{≤n+1}-operad, E is the terminal T_{≤n}^{×}-multitensor, and so E_{1} = T_{≤n}. Since strict (n+1)-categories are categories enriched in n-Cat using cartesian products, and these commute with coproducts (in fact all colimits), it follows by the uniqueness part of theorem(3.5) that E′ is just the cartesian product of n-categories.

The general context in which this example can be generalised is that of a distributive law of a multitensor over a monad, as described in section(5) of [Weber, 2013]. Recall that in theorem(5.4) of [Weber, 2013] such a distributive law was identified with structure on the monad making it opmonoidal with respect to the multitensor, and that analogously to the usual theory of distributive laws between monads, one has a lifting of the multitensor to the category of algebras of the monad. As the following example explains, the two senses of the word “lifting” – coming from the theory of distributive laws, and from the lifting theorem – are in fact compatible.

3.8. Example. Let E be a multitensor on V and T be an opmonoidal monad on (V, E). Then one has by theorem(5.4) of [Weber, 2013] a lifted multitensor E′ on V^{T}. On the other hand if moreover V is cocomplete, E is a distributive and accessible functor operad, and T is coproduct preserving and accessible, then E′ may also be obtained by applying theorem(3.5) to the composite multitensor EM(T). When E is given by cartesian product EM(T) is just another name for the multitensor T^{×}, making E′ the cartesian product of T -algebras by the uniqueness part of theorem(3.5) and proposition(2.8) of [Batanin-Weber, 2011]. Specialising further to the case T = T_{≤n}, we recover example(3.7).

In the above examples we used the uniqueness part of theorem(3.5) to enable us to identify the lifted multitensor E′ as the cartesian product. In each case we had the cartesian product on the appropriate category of algebras as a candidate, and the aforementioned uniqueness told us that this candidate was indeed our E′ because the resulting enriched categories matched up. In the absence of this uniqueness, in order to identify E′ one would have to unpack its construction, and as we saw in the proof of lemma(3.4), this involves a transfinite colimit construction in the appropriate category of enriched graphs. The importance of this observation becomes greater as the operads we are considering become more complex. We now come to our leading example.

3.9. Example. Take A to be the T_{≤3}-operad for Gray categories constructed in [Batanin,1998] (example(4) after corollary(8.1.1)). Since E_{1} is the monad on G^{2}(Set) for 2-categories, in this case E′ is a functor operad for 2-categories. However the Gray tensor product of 2-categories [Gray, 1974] is part of a symmetric monoidal closed structure. Thus it is distributive as a functor operad, and since Gray categories are categories enriched in the Gray tensor product by definition, it follows that E′ is the Gray tensor product.

Lemma(2.5) of [Crans, 2000] unpacks the notion of 4-tas (“tas” being the singular form, “teisi” being the plural) in detail. This explicit description can be interpretted as an explicit description of the T_{≤4}-operad for 4-teisi. On the other hand from [Crans, 1999], one can verify that ⊗_{Crans} is distributive in the following way. First we note that to say that ⊗_{Crans} is distributive is to say that

id ⊗ c_{i} : A ⊗_{Crans} B_{i} ⟶ A ⊗_{Crans} B

is a coproduct cocone, for all Gray categories A and coproduct cocones (B_{i} ⟶^{c_{i}} B : i ∈ I) of Gray categories, since ⊗_{Crans} is symmetric. Since the forgetful functor Gray-Cat → G(2-Cat) creates coproducts, and using the explicit description of coproducts of enriched graphs, a discrete cocone (C_{i} ⟶^{k_{i}} C : i ∈ I) in Gray-Cat is universal iff it is universal at the level of objects and each of the k_{i}’s is fully faithful (in the sense that the hom maps are isomorphisms of 2-categories). From the explicit description of ⊗_{Crans} given in section(4) of [Crans, 1999], one may witness that

A ⊗_{Crans} (−) : Gray-Cat ⟶ Gray-Cat

preserves fully faithful Gray-functors. Thus the distributivity of ⊗_{Crans} follows since at the level of objects ⊗_{Crans} is the cartesian product.

3.10. Example. Take A to be the T_{≤4}-operad for 4-teisi. The associated multitensor E has E_{1} equal to the T_{≤3}-operad for Gray categories. Thus theorem(3.5) constructs a functor operad E′ of Gray categories whose enriched categories are 4-teisi. As we explained above ⊗_{Crans} is distributive, and so the uniqueness part of theorem(3.5) ensures that E′ = ⊗_{Crans}, since teisi are categories enriched in the Crans tensor product by definition.

3.11. Day convolution. While this article is directed primarily at an improved understanding of the examples discussed above, within a framework that one could hope will lead to an understanding of the higher dimensional analogues of the Gray tensor product, it is interesting to note that Day convolution can be seen as an instance of the multitensor lifting theorem.

The set of multimaps (X_{1}, ..., X_{n}) → Y in a given multicategory ℂ shall be denoted as ℂ(X_{1}, ..., X_{n}; Y). Recall that a linear map in ℂ is a multimap whose domain is a sequence of length 1. The objects of ℂ and linear maps between them form a category, which we denote as ℂ_{l}, and we call this the linear part of ℂ. The set of objects of ℂ is denoted as ℂ_{0}. Given objects

A_{11}, ..., A_{1n_{1}}, ......, A_{k1}, ..., A_{kn_{k}} B_{1}, ..., B_{k} C

of ℂ, we denote by

σ_{A,B,C} : ℂ(B_{1}, ..., B_{k}; C) × ∏_{i} ℂ(A_{i1}, ..., A_{in_{i}}; B_{i}) → ℂ(A_{11}, ..., A_{kn_{k}}; C)

the substitution functions of the multicategory ℂ. One thus induces a function

σ_{A,C} : ∫ ^{B_{1},...,B_{k}} ℂ(B_{1}, ..., B_{k}; C) × ∏_{i} ℂ(A_{i1}, ..., A_{in_{i}}; B_{i}) → ℂ(A_{11}, ..., A_{kn_{k}}; C)

in which for the purposes of making sense of this coend, the objects B_{1}, ..., B_{k} are regarded as objects of the category ℂ_{l}. A promonoidal category in the sense of Day [Day, 1970], in the unenriched context, can be defined as a multicategory ℂ such that these induced functions σ_{A,C} are all bijective. A promonoidal structure on a category 𝔻 is a promonoidal category ℂ such that ℂ_{l} = 𝔻^{op}.

A lax monoidal category (V, E) is cocomplete when V is cocomplete as a category and E_{n} : V^{n} → V preserves colimits in each variable for all n ∈ ℕ. In this situation the multitensor E is also said to be cocomplete. When ℂ is small it defines a functor operad on the functor category [ℂ_{l}, Set] whose tensor product F is given by the coend

F_{i} X_{i} = ∫ ^{C_{1},...,C_{n}} ℂ(C_{1}, ..., C_{n}; −) × ∏_{i} X_{i}C_{i}

and substitution is defined in the evident way from that of ℂ. By proposition(2.1) of [Day-Street, 2003] F is a cocomplete functor operad and is called the standard convolution structure of ℂ on [ℂ_{l}, Set]. By proposition(2.2) of [Day-Street, 2003], for each fixed category 𝔻, standard convolution gives an equivalence between multicategories on ℂ such that ℂ_{l} = 𝔻 and cocomplete functor operads on [𝔻, Set], which restricts to the well-known [Day, 1970] equivalence between promonoidal structures on 𝔻^{op} and closed monoidal structures on [𝔻, Set].

We have recalled these facts in a very special case compared with the generality at which this theory is developed in [Day-Street, 2003]. In that work all structures are considered as enriched over some nice symmetric monoidal closed base V, and moreover rather than 𝔻 = ℂ_{l} as above, one has instead an identity on objects functor 𝔻 → ℂ_{l}. The resulting combined setting is then what are called V-substitudes in [Day-Street, 2003], and in the V = Set case the extra generality of the functor 𝔻 → ℂ_{l}, corresponds at the level of multitensors, to the consideration of general closed multitensors on [𝔻, Set] instead of mere functor operads. We shall now recover standard convolution, for the special case that we have described above, from the lifting theorem.

Given a multicategory ℂ we define the multitensor E on [ℂ_{0}, Set] via the formula

( E_{1≤i≤n} X_{i}) (C) = ∐_{C_{1},...,C_{n}} (ℂ(C_{1}, ..., C_{n}; C) × ∏_{1≤i≤n} X_{i}(C_{i}))

using the unit and compositions for ℂ in the evident way to give the unit u and substitution σ for E. When ℂ_{0} has only one element, this is the multitensor on Set coming from the operad P described in [Batanin-Weber, 2011] and [Weber, 2013], whose tensor product is given by the formula

E_{1≤i≤n} X_{i} = P_{n} × X_{1} × ... × X_{n}.

An E-category with one object is exactly an algebra of the coloured operad P in the usual sense. A general E-category amounts to a set X_{0}, sets X(x_{1}, x_{2})(C) for all x_{1}, x_{2} ∈ X_{0} and C ∈ ℂ_{0}, and functions

ℂ(C_{1}, ..., C_{n}; C) × ∏_{i} X(x_{i−1}, x_{i})(C_{i}) → X(x_{0}, x_{n})(C)  (4)

compatible in the evident way with the multicategory structure of ℂ. On the other hand an F -category amounts to a set X_{0}, sets X(x_{1}, x_{2})(C) natural in C, and maps as in (4) but which are natural in C_{1}, ..., C_{n}, C, and compatible with ℂ’s multicategory structure. However this added naturality enjoyed by an F -category isn’t really an additional condition, because it follows from the compatibility with the linear maps of ℂ. Thus E and F -categories coincide, and one may easily extend this to functors and so give E-Cat ≅ F -Cat over G[ℂ_{0}, Set].

The unary part of E is given on objects by

E_{1}(X)(C) = ∐_{D} ℂ_{l}(D, C) × X(D)

which should be familiar – E_{1} is the monad on [ℂ_{0}, Set] whose algebras are functors ℂ_{l} → Set, and may be recovered from left Kan extension and restriction along the inclusion of objects ℂ_{0} ↪ ℂ_{l}. Thus the category of algebras of E_{1} may be identified with the functor category [ℂ_{l}, Set]. Since the multitensor E is clearly cocomplete, it satisfies the hypotheses of theorem(3.5), and so one has a unique finitary distributive multitensor E′ on [ℂ_{l}, Set] such that E-Cat ≅ E′-Cat over G[ℂ_{0}, Set]. By uniqueness we have

3.12. Proposition. Let ℂ be a multicategory, F be the standard convolution structure on [ℂ_{l}, Set] and E be the multitensor on [ℂ_{0}, Set] defined above. Then one has an isomorphism F ≅ E′ of multitensors.

In particular when ℂ is a promonoidal category proposition(3.12) expresses classical unenriched Day convolution as a lift in the sense of theorem(3.5).

3.13. The 2-functoriality of multitensor lifting. We now express the lifting theorem as a coreflection to the inclusion of functor operads within a 2-category of multitensors which are sufficiently nice that theorem(3.5) can be applied to them.

Recall [Street, 1972] that when a 2-category K has Eilenberg-Moore objects, one has a 2-functor

sem_{K} : MND(K) ⟶ K^{[1]}  (V, T) ↦ U^{T} : V^{T} → V

which on objects sends a monad T to the forgetful arrow U^{T} which forms part of the Eilenberg-Moore object of T, and that a straight forward consequence of the universal property of Eilenberg-Moore objects is that sem_{K} (“sem” being short for “semantics”) is 2-fully-faithful. In the case K = CAT if one restricts attention to the sub-2-category of MND(CAT) consisting of the 1-cells of the form (1_{E}, −), then one refinds the structuresemantics result of Lawvere referred to earlier.

The 2-fully-faithfulness of sem_{K} says that the one and 2-cells of the 2-category MND(K) admit an alternative “semantic” description. Given monads (V, T) and (W, S) in K, to give a monad functor (H, ψ) : (V, T) → (W, S), is to give H̃ : V^{T} → W^{S} such that U^{S}H̃ = HU^{T}; and to give a monad 2-cell ϕ : (H_{1}, ψ_{1}) → (H_{2}, ψ_{2}) is to give ϕ : H_{1}→H_{2} and ϕ̃ : H̃_{1}→H̃_{2} commuting with U^{T} and U^{S}. Note that Eilenberg-Moore objects in CAT/Set are computed as in CAT, and we shall soon apply these observations to the case K = CAT/Set.

In view of theorem(2.5) of this article and proposition(4.4.2) of [Weber, 2013], the 2-functor Γ restricts to a 2-equivalence

Acc-DISTMULT ≃ Acc-G-MND.  (5)

Here Acc-DISTMULT is the full sub-2-category of DISTMULT consisting of the (V, E) such that V is cocomplete and the multitensor E is accessible. The 2-category Acc-G-MND is defined as follows:

• objects are pairs (V, T), where V is a cocomplete category and T is a monad on GV which is distributive, path-like and accessible.

• an arrow (V, T) → (W, S) is a pair (H, ψ), where H : V → W is a functor, and ψ is a natural transformation making (GH, ψ) : (GV, T) → (GW, S) a morphism of MND(CAT/Set).

• a 2-cell (H, ψ) → (K, κ) is a monad 2-cell (GH, ψ) → (GK, κ).

Since by [Weber, 2013] proposition(2.1.6), the effect G_{1} : CAT ⟶ CAT/Set of G on arrows into 1 is locally fully faithful, the data of a 2-cell of Acc-G-MND will be of the form Gϕ for a unique natural transformation ϕ : H → K. Moreover from the recollections of the previous paragraph, the one and 2-cells of Acc-G-MND can be reinterpreted in semantic terms.

Let us denote by Acc-FOp the full sub-2-category of Acc-DISTMULT consisting of the (V, E) such that E is a functor-operad, and by

I : Acc-FOp ⟶ Acc-DISTMULT

the inclusion.

3.14. Proposition. The assignment (V, E) ↦ (V^{E_{1}}, E′), with E′ defined by theorem(3.5), is the object map of a right adjoint

C : Acc-DISTMULT ⟶ Acc-FOp

to the inclusion I.

Proof. We shall define the coreflection C and a 2-natural transformation ε : IC → 1 such that Cε = id and εI = id, so that ε is the counit of an adjunction I ⊣ C whose unit is an identity. Let (H, ψ) : (W, F) → (V, E) be a lax monoidal functor. Then we have the following serially commutative diagram of forgetful functors

[figure: F -Cat; G(W^{F_{1}}); GW; GV; G(V^{E_{1}}); E-Cat; U^{F′}; G(U^{F_{1}}); GH; G(U^{E_{1}}); U^{E′}; ψ^{∗}; G(ψ_{1}^{∗}); U^{F}; U^{E}]

The left-most square is the semantic interpretation of a morphism of monads

(G(ψ_{1}^{∗}), ψ̃) : (G(W^{F_{1}}), ΓF′) ⟶ (G(V^{E_{1}}), ΓE′)

and so by (5), there is a unique natural transformation ψ′ giving the coherence data of a lax monoidal functor

(ψ_{1}^{∗}, ψ′) : (W^{F_{1}}, F′) ⟶ (V^{E_{1}}, E′)

such that Γ(ψ_{1}^{∗}, ψ′) = (G(ψ_{1}^{∗}), ψ̃). This enables us to define the 1-cell map of C as C(H, ψ) = (ψ_{1}^{∗}, ψ′), and similarly using the semantic interpretation of the 2-cells of Acc-G-MND and (5), one defines the 2-cell map of C. The 2-functoriality of C follows from that of Γ and sem_{CAT/Set}.

For (V, E) ∈ Acc-DISTMULT one has

[figure: E-Cat; G(V^{E_{1}}); GV; E-Cat; U^{E′}; G(U^{E_{1}}); U^{E}; 1]

which is the semantic interpretation of

(U^{E_{1}}, \overline{ε_{E}}) : (V^{E_{1}}, E′) ⟶ (V, E) ∈ Acc-DISTMULT

and we take this to be the component ε_{(V,E)} of ε. To verify naturality with respect to (H, ψ) as above, note that sem_{CAT/Set}(ε_{(V,E)}(ψ_{1}^{∗}, ψ′)) is the composite morphism U^{F′} → U^{E_{1}} depicted on the left in

[figure: F -Cat; G(W^{F_{1}}); G(V^{E_{1}}); GV; E-Cat; E-Cat; U^{F′}; G(ψ_{1}^{∗}); G(U^{E_{1}}); U^{E}; 1; ψ^{∗}; U^{E′}; F -Cat; G(W^{F_{1}}); GW; GV; E-Cat; F -Cat; U^{F′}; G(U^{F_{1}}); GH; U^{E}; ψ^{∗}; 1; U^{F}]

whereas sem_{CAT/Set}((H, ψ)ε_{(W,F)}) is the composite depicted on the right. Since U^{E_{1}}ψ_{1}^{∗} = HU^{F_{1}} these are equal and so ε is natural with respect to (H, ψ) by the fully faithfulness of sem_{CAT/Set}. The 2-naturality of ε follows similarly using the 2-fully faithfulness of sem_{CAT/Set}. When E is itself a functor operad one has that E_{1} is the identity monad, and so U^{E_{1}} and thus ε_{(V,E)} are identities, whence εI = id. On the other hand for any (V, E) ∈ Acc-DISTMULT, \overline{ε_{E}}_{1}^{∗} = id by definition, and so Cε_{(V,E)} = id.

As pointed out in section(6.1) of [Weber, 2013], the data ϕ : E → F of a morphism of multitensors on some category V can also be regarded as the coherences making the identity functor 1_{V} into a lax monoidal functor (1_{V} , ϕ) : (V, F) → (V, E), or alternatively as the coherences making 1_{V} into an oplax monoidal functor (1_{V} , ϕ) : (V, E) → (V, F).

3.15. Remark. Suppose that ψ : E → T_{≤n}^{×} is a T_{≤n}^{×}-multitensor. Then the datum ψ can also be regarded as the coherence data for a lax monoidal functor

(1_{G^{n}(Set)}, ψ) : (G^{n}(Set), T_{≤n}^{×}) ⟶ (G^{n}(Set), E),

applying C to it gives a lax monoidal functor

(ψ_{1}^{∗}, ψ′) : (G^{n}(Set)^{T_{≤n}}, ∏) ⟶ (G^{n}(Set)^{E_{1}}, E′)

and the components of ψ′ are morphisms

ψ′_{(X_{1},x_{1}),...,(X_{n},x_{n})} : E_{i}′ψ_{1}^{∗}(X_{i}, x_{i}) ⟶ ∏_{i=1}^{n} ψ_{1}^{∗}(X_{i}, x_{i})

of E_{1}-algebras defined for each sequence ((X_{1}, x_{1}), ..., (X_{n}, x_{n})) of strict n-categories. The codomain of these morphisms can be written as above since ψ_{1}^{∗} as a right adjoint preserves products. This gives a general comparison map between the functor operad E′ produced by theorem(3.5) and cartesian products, defined for sequences of E_{1}-algebras that underlie strict n-categories. In particular when E is as in example(3.9), then ψ_{1}^{∗} is the identity, and one has the comparison maps between the Gray tensor product and the cartesian product of 2-categories.

One of the expected features of the higher dimensional analogues of the Gray tensor product are such comparisons with cartesian product, which moreover are expected to be equivalences in the appropriate higher categorical sense. We shall see in remark(6.5) below, that in the context of reduced T_{≤n}-operads, one always gets such comparison maps with the cartesian product, and these are defined for all E_{1}-algebras.

4. Weak n-categories via algebraic weak factorisation systems

4.1. Overview. In this section we review the operadic definition of weak n-category (more precisely Leinster’s variant [Leinster, 2003] of Batanin’s [Batanin,1998] original definition) in a way that allows adaptation to the strictly unital case in section(5). The overall scheme of this definition is as follows. One has the category T_{≤n}-Coll_{0} of T_{≤n}-collections over Set, and this is a presheaf category. There are two finitary monads on T_{≤n}-Coll_{0}, Opd_{≤n} and Cont_{≤n}, whose algebras are T_{≤n}-operads over Set and T_{≤n}-collections over Set with chosen contractions (in the sense to be recalled below), respectively. The coproduct of these monads Opd_{≤n} ∐ Cont_{≤n} exists and is finitary, and its algebras are T_{≤n}-operads over Set equipped with chosen contractions. As the category of algebras of a finitary monad on a presheaf category, the category of T_{≤n}-operads over Set with chosen contractions is locally finitely presentable, and so has an initial object K_{≤n}. A weak n-category is by definition an algebra for K_{≤n}.

The monad Opd_{≤n} can be thought about in a few ways. Most generally, one has a finite limit sketch S_{1} whose Set-valued models are T_{≤n}-operads over Set, another such sketch S_{2} (this time without any distinguished limit cones) for T_{≤n}-collections over Set, and an inclusion S_{2} ↪ S_{1} of finite limit sketches which induces the finitary monad Opd_{≤n}. Another viewpoint is that T_{≤n}-Op_{0} is the category of monoids for the substitution tensor product on T_{≤n}-Coll_{0}, this tensor product being filtered colimit preserving in one variable and cocontinuous in the other, and so by general well-known results such as those of [Kelly, 1980], the monoid monad exists and is finitary. Most particularly, an explicit description of Opd_{≤n} is given in [Leinster, 2003] Appendix D, from which its finitariness and many other desirable properties may be witnessed.

As for Cont_{≤n}, our preferred approach will be that of [Garner, 2009], which is to use the theory of cofibrantly-generated algebraic weak factorisation systems from [Garner, 2009(2)] to give a conceptual understanding of this monad. We shall spend some time reviewing the relevant aspects of [Garner, 2009(2)] as we shall use this machinery in section(5).

4.2. Algebraic weak factorisation systems. Composition in any category E can be regarded as a morphism comp_{E} of spans of categories

[figure: E; E^{[2]}; E; E^{[1]}; t_{E}; s_{E}; ^{comp^{E}}]

where s_{E} (resp. t_{E}) takes the source (resp. target) of a given morphism of E. A functorial factorisation is a morphism of spans that is a section of comp_{E}. On objects a given functorial factorisation F : E^{[1]} → E^{[2]} takes an arrow f : X → Y of E to a composable pair

X ⟶^{Lf} Kf ⟶^{Rf} Y

of morphisms whose composite is f. With the evident morphisms one has a category FF(E) of functorial factorisations for E. For a given F the assignment f ↦ Lf defines a copointed endofunctor ε : L → 1 on E^{[1]} such that s_{E}ε = id_{s_{E}}, where the f-component of ε is given as on the left in

[figure: X; X; Y; Kf; 1_{X}; f; Rf; Lf; X; Kf; Y; Y; Lf; Rf; 1_{Y}; f]

and similarly, the assignment f ↦ Rf defines a pointed endofunctor η : 1 → R on E^{[1]} such that t_{E}η = id_{t_{E}}, with the f-component of η given as on the right in the previous display. Clearly these give three equivalent viewpoints on the notion, that is to say, to give a functorial factorisation F, is to give (L, ε) such that s_{E}ε = id_{s_{E}}, which in turn is equivalent to giving (R, η) such that t_{E}η = id_{t_{E}}. Thus (following [Garner, 2009(2)]) we shall also denote F as the pair (L, R). The assignment f ↦ Kf is the object map of a functor K : E^{[1]} → E that we call the image of a given functorial factorisation.

Composition of the associated pointed endofunctors gives FF(E) a strict monoidal structure, with tensor product denoted as ∘_{r}, and whose unit is initial. Composition of the associated copointed endofunctors gives FF(E) another strict monoidal structure, with terminal unit and tensor product denoted as ∘_{l}. A key observation of [Garner, 2009(2)] is that one has natural maps

λ_{F_{1},F_{2},F_{3},F_{4}} : (F_{1} ∘_{l} F_{2}) ∘_{r} (F_{3} ∘_{l} F_{4}) ⟶ (F_{1} ∘_{r} F_{3}) ∘_{l} (F_{2} ∘_{r} F_{4})

providing the interchange maps of a duoidal structure. Recall that a duoidal structure [Aguiar-Mahajan, 2010] [Batanin-Markl, 2012] on a category V is a pair of monoidal structures (I_{1}, ⊗_{1}, α_{1}, λ_{1}, ρ_{1}) and (I_{2}, ⊗_{2}, α_{2}, λ_{2}, ρ_{2}), together with morphisms η : I_{1} → I_{2}, μ : I_{2} ⊗ I_{2} → I_{2} and δ : I_{1} → I_{1} ⊗_{2} I_{1}, and natural morphisms

λ_{A,B,C,D} : (A ⊗_{2} B) ⊗_{1} (C ⊗_{2} D) ⟶ (A ⊗_{1} C) ⊗_{2} (B ⊗_{1} D)

called interchange maps, such that

• (η, μ) makes I_{2} a ⊗_{1}-monoid.

• (η, δ) makes I_{1} a ⊗_{2}-comonoid.

• (ι, δ) are the coherences making ⊗_{2} : V × V → V lax monoidal for ⊗_{1}.

• (ι, μ) are the coherences making ⊗_{1} : V × V → V oplax monoidal for ⊗_{2}.

The key observation alluded to above is precisely that one has a duoidal structure on FF(E) for which I_{1} = 0, I_{2} = 1, ⊗_{1} = ∘_{r} and ⊗_{2} = ∘_{l}.

A duoidal category is a natural environment for a notion of bialgebra. A bialgebra structure on X in a general duoidal category V consists of morphisms i : I_{1} → X, m : X ⊗_{1} X → X, c : X → I_{2} and d : X → X ⊗_{2} X, such that (i, m) is a ⊗_{1}-monoid structure, (c, d) is a ⊗_{2}-comonoid structure, and these are compatible in the following equivalent ways:

• i and m are morphisms of ⊗_{2}-comonoids.

• c and d are morphisms of ⊗_{1}-monoids.

An algebraic weak factorisation system on E (originally called a natural weak factorisation system in [Garner, 2009(2)]) is by definition a bialgebra in the duoidal category FF(E).

In particular for an algebraic weak factorisation system F, the copointed endofunctor (L, ε) has in addition a comultiplication making it into a comonad, and the pointed endofunctor (R, η) has a multiplication making it into a monad. Thus the given factorisation of f : X → Y as

X ⟶^{Lf} Kf ⟶^{Rf} Y

is into a map Lf which has the structure of an L-coalgebra, followed by the map which has the structure of an R-algebra. In general L-coalgebras and R-algebras admit lifting conditions with respect to each other, that is to say, given a commutative square as on the left in

[figure: A; C; D; B; u; r; v; l; A; C; Kr; D; B; Kl; u; Lr; Rr; v; Rl; Ll; λ; K(u,v); ρ]

together with an L-coalgebra structure λ on l and an R-algebra structure ρ on r, one constructs the diagonal filler as on the right in the previous display. Thus everything works as with the usual theory of weak factorisation systems, except that here instead of classes of left and right maps one has the categories of L-coalgebras and R-algebras, and the liftings are constructed as shown above using the algebraic data. In fact as explained in [Garner, 2009(2)], every algebraic weak factorisation system has an underlying weak factorisation system.

For any X ∈ E, L restricts to a comonad L_{X} on the coslice X/E whose coalgebras are L-coalgebras with domain X. Similarly R restricts to a monad R_{X} on the slice E/X whose algebras are R-algebras with codomain X. In particular if E has an initial object 0, then L_{0} is a comonad on E whose coalgebras are objects X of E together with the structure of an L-coalgebra on the unique map 0 → X. This is the algebraic analogue of cofibrant replacement, when the underlying weak factorisation system is the factorisation of morphisms of a Quillen model category into cofibrations followed by trivial fibrations. Dually when E has a terminal object 1, the monad R_{1} on E is an algebraic analogue of fibrant replacement, when the underlying weak factorisation system is the factorisation of morphisms of a Quillen model category into trivial cofibrations followed by fibrations.

Given a functorial factorisation F = (L, R) with image denoted as K, since colimits in E^{[1]} and E^{[2]} are computed componentwise as in E, the following statements are clearly equivalent for a given category C:

• F : E^{[1]} → E^{[2]} preserves limits (resp. colimits) of functors from C.

• L : E^{[1]} → E^{[1]} preserves limits (resp. colimits) of functors from C.

• R : E^{[1]} → E^{[1]} preserves limits (resp. colimits) of functors from C.

• K : E^{[1]} → E preserves limits (resp. colimits) of functors from C.

When any of these conditions is satisfied, we say that the functorial factorisation preserves the given limit or colimit. In particular given a regular cardinal λ an algebraic weak factorisation system is said to be λ-ary when its underlying functorial factorisation preserves λ-filtered colimits, and finitary in the case λ = ℵ_{0} of filtered colimits. Note that for a finitary algebraic weak factorisation system (L, R), the comonad L_{0} and the right monad R_{1} are both finitary. Below we shall exhibit Cont_{≤n} as the monad R_{1} for a finitary algebraic weak factorisation system (L, R) on T_{≤n}-Coll_{0}.

Since 0 ∈ FF(E) has underlying monad the identity, it preserves all limits and colimits as a functorial factorisation, and similarly for 1 ∈ FF(E) since its underlying comonad is the identity. Thus if we write FF(E)′ for the full subcategory of FF(E) consisting of the functorial factorisations preserving some class of limits and colimits, then the duoidal structure of FF(E) restricts to FF(E)′. So the algebraic weak factorisation system’s which preserve the limits and colimits in the given class could equally well be regarded as bialgebras in FF(E)′.

We conclude this section with an observation that will be of use in section(4.4).

4.3. Lemma. Let P : A → B be an opfibration, and let F = (L, R) be the functorial factorisation on A of a map into a cocartesian arrow for P followed by a vertical arrow (ie one sent to an identity by P). Then F preserves any colimit that P preserves.

Proof. For any functor P : A → B one has the functor i_{P} : A → P/B given on objects by P A = (A, 1_{P A}, P A), and the structure of an opfibration amounts to a left adjoint π : P/B → A to i_{P} over B (see [Street, 1974] for instance). One also has the canonical functor

P′ : A^{[1]} → P/B  A ⟶^{α} A′ ↦ (A, P α, P A′)

and the image for the cocartesian-vertical factorisation for P is the composite πP′. Now the forgetful functor P/B → A × B creates all colimits that are preserved by P. Thus P′ preserves all colimits that P does since colimits in E^{[1]} are formed componentwise as in E, and so the result follows since π as a left adjoint preserves all colimits.

4.4. Garner’s small object argument. The algebraic weak factorisation system’s relevant for us will all be cofibrantly generated, in the sense that they all arise from Garner’s small object argument, which we now recall in a special case of interest to us (for the general theory see [Garner, 2009(2)]). Suppose that E is locally finitely presentable and that a functor J : I → E^{[1]} is given, where I is a small category. Then Garner’s small object argument produces an algebraic weak factorisation system on E in three steps. Before we begin to recall these, we invite the reader to keep in mind the more familiar special case when I is discrete, and so J may be regarded as a set of maps

J = {s_{i} ⟶^{ϕ_{i}} d_{i} : i ∈ I}

that one might call the “generating cofibrations” of the resulting algebraic weak factorisation system.

The first step is to take the left Kan extension of J along itself to produce L_{1} : E^{[1]} → E^{[1]}, which comes with a canonical comonad structure (comonads arising this way are commonly known as “density” comonads). The underlying copointed endofunctor (L_{1}, ε_{1}) of this comonad does not satisfy the condition s_{E}ε_{1} = id_{s_{E}} that would make it part of a functorial factorisation, and the point of the second step is to force this condition. Observe that s_{E} : E^{[1]} → E is an opfibration, in which the cocartesian arrows are those whose underlying square in E is a pushout^{4}. Then the second step is carried out by factoring the counit components ε_{1,f} : L_{1}f → f into

L_{1}f ⟶^{c_{f}} L_{2}f ⟶^{ε_{2,f}} f

a cocartesian arrow followed by a vertical arrow for s_{E}. In this way (L_{2}, ε_{2}) satisfies the condition s_{E}ε_{2} = id_{s_{E}} and so we have a functorial factorisation. Moreover L_{2} inherits a comultiplication from that of L_{1}, making it into a comonad. What is still missing is the compatible monad structure on R, and the third step of the construction will supply this.

Before proceeding to describe this third step, let us note that the factorisation provided by the first two steps yields something very familiar in the case where I is discrete. For suppose that f : X → Y is in E, then L_{1}f and L_{2}f are as indicated on the left in

[figure: ∐_{i,α,β}s_{i}; X; Y; ∐_{i,α,β}d_{i}; (α); f; L_{1}f; (β); P_{f}; L_{2}f; R_{2}f; s_{i}; X; Y; d_{i}; α; f; β; ϕ_{i}; =]

the summands are indexed over the set of triples (i, α, β) giving rise to a commutative square as indicated on the right in the previous display. So we have here what is usually regarded as the first step of Quillen’s small object argument. In general, that is for I not necessarily discrete, the above coproducts will be replaced by more general colimits.

On completing the first two steps we have produced a functorial factorisation (L_{2}, R_{2}) in which L_{2} is a comonad, in other words we have a ∘_{l}-comonoid in FF(E). The third step of Garner’s small object argument is to take the free bialgebra on this comonoid, and this encapsulates the algebraic analogue of the transfinite part of Quillen’s small object argument.

Let us see why this free bialgebra exists in our setting with E locally finitely presentable. First we note that since I is small and every object of E has rank, the functor J factors through the inclusion E_{λ} ↪ E of the λ-presentable objects where λ is some regular cardinal. Thus L_{1} is the left extension of its restriction to E_{λ}, and so it preserves λ-filtered colimits. Since the opfibration s_{E} : E^{[1]} → E preserves all colimits and thus λ-filtered ones in particular, by lemma(4.3) L_{2} preserves λ-filtered colimits. So in fact (L_{2}, R_{2}) is a ∘_{l}-comonoid in FF_{λ}(E), the duoidal category of λ-ary functorial factorisations on E.

4Note that for any E, s_{E} is a fibration, but what is important for us is that pushouts in E make s_{E} into an opfibration.

In FF_{λ}(E) the tensor products ∘_{l} and ∘_{r} are cocontinuous in one variable and λfiltered colimit preserving in the other, since in each case these are just obtained by the composition of some kind of λ-ary functors. Because of the compatibility of these tensor products within the duoidal structure, the tensor product ∘_{r} lifts to a tensor product on the category ∘_{l}-CoMon_{λ}(E) of λ-ary ∘_{l}-comonoids, and since the forgetful functor U_{l} : ∘_{l}-CoMon_{λ}(E) → FF_{λ}(E) is strict monoidal and creates colimits, ∘_{r} as a tensor product on ∘_{l}-CoMon_{λ}(E) is also cocontinuous in one variable and λ-filtered colimit preserving in the other. Since FF_{λ}(E) and thus ∘_{l}-CoMon_{λ}(E) is evidently cocomplete, one can apply the construction of the free monoid on a pointed object from [Kelly, 1980] to (L_{2}, R_{2}) ∈ ∘_{l}-CoMon_{λ}(E), to obtain the free bialgebra.

One reason we have discussed these details is to make the following observation.

4.5. Lemma. Suppose that J : I → E^{[1]} where E is locally finitely presentable, I is small and for all i ∈ I, the domain and codomain of J(i) is finitely presentable. Then the algebraic weak factorisation system obtained from J via Garner’s small object argument is finitary.

Proof. In this case the λ of the above discussion is ℵ_{0} and so the third part of the construction stays within the category of finitary functorial factorisations.

Note that in the general construction of the free bialgebra, one can think of it as taking place just within FF_{λ}(E), ie one can forget about the ∘_{l}-comonoid structures, since U_{l} is strict monoidal and creates colimits, so this extra structure is automatically compatible with everything that is going on. It is also worth noting that the passage from (L, R) ∈ FF(E) to R_{X}, for all X ∈ E, preserves colimits and sends ∘_{r} to composition of pointed endofunctors of E/X, and so for the (L, R) obtained from Garner’s small object argument, R_{X} is the free monad on the pointed endofunctor (R_{2})_{X}.

The algebraic weak factorisation system obtained from J : I → E^{[1]} via Garner’s small object argument is said to be cofibrantly generated by J. For such algebraic weak factorisation systems one has a direct description of what an R-algebra structure on a given f : X → Y amounts to. In order to describe this let us denote by i ↦ s(i) ⟶^{ϕ_{i}} d(i) the object map of J. Then to give an R-algebra structure on f is to give a choice of diagonal fill

[figure: si; X; Y; di; α; f; β; ϕ_{i}; γ(i,α,β)]

for every (i, α, β) such that fα = βϕ_{i}; and these are compatible in the sense that given δ : j → i in I and α and β as above, one has

[figure: sj; si; X; Y; di; dj; sδ; α; f; β; dδ; ϕ_{j}; ϕ_{i}; γ(i,α,β); γ(j,αs(δ),βd(δ))]

γ(i, α, β)d(δ) = γ(j, αs(δ), βd(δ)). This compatibility condition is one of the novel features of the theory of algebraic weak factorisation systems, which owes its existence to the fact that one has a category of “generating cofibrations” as opposed to a mere set. We shall use this feature below in section(5) to express what it means for unital operations to be “strict” in our operadic definition of “weak n-category with strict units”.

4.6. T_{≤n}-collections. Up to this point in this article we have been using exclusively the general definitions of T_{≤n}-collections (resp. operads) as cartesian natural transformations (resp. cartesian monad morphisms) into T_{≤n}. The advantage of this is that all the combinatorial aspects of pasting diagrams are neatly packaged into a single very wellbehaved object – the monad T_{≤n} on G^{n}(Set) for strict n-categories. We now, for both the convenience of the reader and to set our notation and terminology, recall some of these combinatorial aspects.

We denote also by Tr_{≤n} the n-globular set T_{≤n}(1). This is the free strict n-category on the terminal n-globular set, the set of k-cells of which are denoted as Tr_{k}, whose elements can be regarded as k-stage trees in the sense of [Batanin,1998], a k-stage tree p being a sequence

[figure: p^{(k)}; p^{(k−1)}; ...; p^{(1)}; p^{(0)}; ξ_{k}; ξ_{1}]

in Δ_{+} such that p^{(0)} = 1. An element x ∈ p^{(r)} is a node of p whose height is r, the element y = ξ_{r}...ξ_{s+1}(x) is x’s ancestor of height s, x is said to be a descendant of y, and a node x is said to be a leaf when it doesn’t have any descendants. The leaves of a given tree p have a linear order determined by the order on the p^{(r)}’s and the ξ_{r}’s. For k > 0 the source and target functions s, t : Tr_{k+1} → Tr_{k} are equal, they correspond to forgetting the nodes at height (k + 1), that is to say “truncation to height k”, and so we write s = tr = t. For k < n any k-stage tree p can be regarded as a (k+1)-stage tree zp such that tr(zp) = p and (zp)^{(k+1)} = 0, and one has an inclusion z : Tr_{k} ↪ Tr_{k+1}. The k-stage tree with exactly one leaf at height k is denoted as U_{k}, and so the k-stage trees with exactly one leaf are those of the form z^{s}U_{k−s}. Such trees are said to be linear.

Trees in the sense just described define the n-globular sets to be regarded as globular pasting schemes. In [Batanin,1998] the notation p^{∗} was used for the globular set determined by the tree p. From a general perspective the assignation of a globular set from a tree is the object map of a functor el(Tr_{≤n}) → G^{n}(Set) (el(Tr_{≤n}) is the category of elements of the n-globular set Tr_{≤n}) which comes from the fact that T_{≤n} is a local right adjoint monad on a presheaf category – so the trees/globular pasting diagrams are the canonical arities for T_{≤n}, see [Weber, 2007] [Berger-Melliès-Weber, 2012]. For an explicit description of this operation, see [Batanin,1998] or [Weber, 2004]. We shall omit the (−)^{∗} and just refer to the n-globular set p. In particular the n-globular sets p and zp coincide.

The category T_{≤n}-Coll of T_{≤n}-collections admits two useful reformulations because of the equivalences of categories

T_{≤n}-Coll ≃ G^{n}(Set)/Tr_{≤n} ≃ [el(Tr_{≤n})^{op}, Set]

the first of which is given by evaluating at 1, and the second is the standard description of a slice of a presheaf category as a presheaf category. Given p ∈ Tr_{k} where k ≤ n, the set A_{p} is just the fibre over p ∈ Tr_{k} of (α_{1})_{k} : A(1)_{k} → Tr_{k}. When A underlies an operad, a ∈ A_{p} is an operation which in any A-algebra composes such a pasting diagram to a k-cell. By the Yoneda lemma a k-stage tree p may also be regarded as a morphism p : k → Tr_{≤n} of n-globular sets, and these are exactly the representables when regarding T_{≤n}-Coll as a presheaf category. For k > 0 one has cosource and cotarget morphisms σ, τ : (k−1) → k of representable globular sets, giving cosource and cotarget maps σ_{p}, τ_{p} : tr(p) → p in T_{≤n}-Coll, since in G^{n}(Set)/Tr_{≤n} one has tr(p) = pσ = pτ.

To say that a T_{≤n}-collection is over Set is to say that for the corresponding α : A → Tr_{≤n}, the fibre over U_{0} is singleton, that is that A_{U_{0}} is singleton. The category T_{≤n}-Coll_{0} is also a presheaf category

T_{≤n}-Coll_{0} ≃ [N_{≤n}^{op}, Set]

where N_{≤n} is the full subcategory of el(Tr_{≤n}) consisting of those trees p ≠ U_{0}. Right Kan extension along the inclusion i : N_{≤n} ↪ el(Tr_{≤n}) corresponds to the inclusion T_{≤n}-Coll_{0} ↪ T_{≤n}-Coll, and restriction along i applied to A amounts to replacing A_{U_{0}} with a singleton. So the trees p ≠ U_{0} are also representables in T_{≤n}-Coll_{0}, but viewed in G^{n}(Set)/Tr_{≤n} this will be different from p : k → Tr_{≤n} in that the two 0-cells of the n-globular set k will have been identified. Despite this difference we shall, when there is no risk of confusion, denote by p this representable object of T_{≤n}-Coll_{0}. Thus for a T_{≤n}-operad A over Set, one may regard an element of a ∈ A_{p}, that is to say an operation a of A of arity p, as a morphism a : p → A in T_{≤n}-Coll_{0} by the Yoneda lemma. Similarly, for k-stage trees p where k > 1, one has σ_{p}, τ_{p} : tr(p) → p in T_{≤n}-Coll_{0}.

4.7. T_{≤n}-operads with chosen contractions. The (Leinster-)contractibility of a T_{≤n}-operad is a condition on the underlying collection. A T_{≤n}-collection is contractible iff the corresponding morphism α : A → Tr_{≤n} of n-globular sets satisfies the right lifting property (right lifting property) with respect to the set

I_{≤n} = {∂(k) ⟶^{ϕ_{k}} k : 0 ≤ k ≤ n} ∪ {∂(n+1) ⟶^{ϕ′_{n}} n}.

To equip the T_{≤n}-collection with chosen contractions is by definition to give choices of liftings that witness α’s right lifting property.

As such, possessing chosen contractions is exactly the structure of an R-algebra on α, where (L, R) is the algebraic weak factorisation system on G^{n}(Set) cofibrantly generated by I_{≤n}. As observed in [Garner, 2009], possessing chosen contractions may be similarly identified from a cofibrantly generated algebraic weak factorisation system on T_{≤n}-Coll_{0}. One has

[figure: T_{≤n}-Coll_{0}; T_{≤n}-Coll; G^{n}(Set),; ^{ran^{i}}; dom; ^{res^{i}}; ⊥]

one defines I′_{≤n} to be the set of morphisms ϕ of T_{≤n}-Coll such that dom(ϕ) ∈ I_{≤n}, and then I′′_{≤n} is the set of morphisms res_{i}(ϕ) of T_{≤n}-Coll_{0} such that ϕ ∈ I′_{≤n}. Denote by (L′, R′) and (L′′, R′′) the algebraic weak factorisation systems cofibrantly generated by I_{≤n} and I′′_{≤n} respectively. Let B ∈ T_{≤n}-Coll_{0} such that ran_{i}(B) corresponds to α. Then by the definition of I′_{≤n}, an R-algebra structure on α amounts to an R′-algebra structure on the unique map ran_{i}(B) → 1, which in turn amounts to an R′′-algebra structure on B → 1 by the definition of I′′_{≤n}.

Moreover note that the boundaries ∂k of representables are clearly finite colimits of representables and so are finitely presentable objects of G^{n}(Set). Thus (L, R) is a finitary algebraic weak factorisation system. The objects of a slice E/X whose domains are finitely presentable are finitely presentable in E/X, thus the domains and codomains of ϕ ∈ I′_{≤n} are finitely presentable, whence (L′, R′) is also finitary. The functor ran_{i} is clearly finitary – this is most easily seen by regarding it as the inclusion, thus its left adjoint preserves finitely presentable objects, whence (L′′, R′′) is also a finitary algebraic weak factorisation system. We denote by Cont_{≤n} the monad R′′_{1}, and we have proved

4.8. Proposition. The monad Cont_{≤n} on T_{≤n}-Coll_{0} just described is finitary and its algebras are T_{≤n}-collections over Set equipped with chosen contractions.

In view of section(4.1), this concludes our recollection of the definition of weak n-category.

4.9. Remark. We have in this work used the definition of Leinster [Leinster, 2003] for ease of exposition, and because its compatibility with the technology of algebraic weak factorisation systems has been well discussed in the literature [Garner, 2009]. However one can, in a similar way, recast the original definition of [Batanin,1998] in similar terms, though working with strictly rather than weakly initial T_{≤n}-operads of the appropriate type. In this variant instead of T_{≤n}-Coll_{0} one works with pointed collections, that is, with the coslice 1_{G^{n}(Set)}/T_{≤n}-Coll_{0}. On this coslice there are three finitary monads whose algebras are T_{≤n}-operads, pointed T_{≤n}-collections equipped with a system of compositions, and pointed T_{≤n}-collections that are contractible in the sense of [Batanin,1998], with the last of these obtained via a finitary algebraic weak factorisation system. The algebras of the coproduct of these monads are, in the language of [Batanin,1998], exactly “contractible normalised n-operads equipped with a system of compositions”, and the algebras of the initial such are weak n-categories in the sense of [Batanin,1998].

4.10. Further remarks on generating cofibrations. We analyse the sets I_{≤n}, I′_{≤n} and I′′_{≤n} of generating cofibrations a little more, and develop some further notation and terminology for the sake of section(5). The role of the “extra” arrow ϕ′_{n} in the definition of I_{≤n} is that it ensures that for the resulting algebraic weak factorisation system (L, R), the structure of R-algebra involves unique lifting against the map ϕ_{n} : ∂(n) → n. This is part of a general phenomenon first observed by Bousfield in section(4) of [Bousfield, 1977]. We are grateful to Richard Garner for pointing us to this reference.

4.11. Lemma. (Bousfield [Bousfield, 1977]) Let E be a category with pushouts, ϕ : S → D in E, and form ϕ′ as follows:

[figure: S; D; S′; D; ϕ; τ; σ; ϕ; po; D; 1_{D}; 1_{D}; ϕ′]

Then f : X → Y satisfies the right lifting property with respect to ϕ and ϕ′ iff it satisfies the unique right lifting property with respect to ϕ.

Given a cocomplete category E satisfying either (∗) or (†) of [Garner, 2009(2)] section(4)^{5}, and that J : I → E^{[1]} with I small, and denote by (L, R) the algebraic weak factorisation system cofibrantly generated by J. Suppose that a subset S of the objects of I are given, and regard S as a discrete subcategory of I. Define I_{S} = I ∐ S and J_{S} : I_{S} → E^{[1]} for the functor which agrees with J on I, and for s ∈ S is given by J_{S}(s) = J(s)′ defined as in lemma(4.11). Denote by (L_{S}, R_{S}) the algebraic weak factorisation system cofibrantly generated by J_{S}.

4.12. Corollary. For E, J : I → E^{[1]} and S as just defined, an R_{S}-algebra structure on f : X → Y in E is exactly an R-algebra structure together with the property that f has the unique right lifting property with respect to the morphisms of S.

4.13. Remark. In corollary(4.12) S was just a subset of I regarded as a discrete subcategory. There is an evident way to extend (−)′ of lemma(4.11) to morphisms of generating cofibrations, and then one could take S to be a full subcategory of I. However this gives nothing different, because the uniqueness of the lifting conditions would imply that the compatibilities expressed by these extra morphisms of generating cofibrations would be automatically satisfied by any R_{S}-algebra. In fact, if one wished to be completely minimalistic, one could also remove any morphisms of I ⊆ I_{S} whose domains are in S, because for the same reason the resulting cofibrantly generated algebraic weak factorisation system would be the same.

Returning to I_{≤n}, the above discussion shows that instead of defining I_{≤n} one can instead define

Ĩ_{≤n} = {∂(k) ⟶^{ϕ_{k}} k : 0 ≤ k ≤ n}

and distinguish S = {ϕ_{n}} so that (Ĩ_{≤n})_{S} = I_{≤n}. Thus we introduce 4.14. Terminology. Let J : I → E^{[1]} and S be as in corollary(4.12). Then the elements of S are said to be strict, and the morphisms J(s) for s ∈ S are said to be generating strict cofibrations. The algebraic weak factorisation system cofibrantly generated by (S, J) is defined to be the algebraic weak factorisation system cofibrantly generated by J_{S}.

5These are general conditions under which Garner’s small object argument works, and in particular are satisfied when E is locally finitely presentable.

So as an alternative to defining I_{≤n}, we define instead Ĩ_{≤n} in which ϕ_{n} is distinguished as being strict, and consider the algebraic weak factorisation system cofibrantly generated therefrom.

Similarly instead of considering I′_{≤n}, we define instead the set

Ĩ′_{≤n} = {∂(p) ⟶^{ϕ_{p}} p : p ∈ Tr_{k}, 0 ≤ k ≤ n}

of morphisms of T_{≤n}-Coll, where p is regarded as a morphism p : k → Tr_{≤n} of n-globular sets by Yoneda, and thus as a collection, in fact the collections of this form are exactly the representables – remembering that T_{≤n}-Coll is a presheaf category. The T_{≤n}-collection ∂(p) is by definition the composite

∂(k) ⟶^{ϕ_{k}} k ⟶^{p} Tr_{≤n}

and so ϕ_{k} underlies the morphism ϕ_{p} : ∂(p) → p of collections. Then we distinguish the ϕ_{p} for p ∈ Tr_{n} as being strict. In the same way, as an alternative to I′′_{≤n} we define

Ĩ′′_{≤n} = {∂(p) ⟶^{ϕ_{p}} p : p ∈ Tr_{k}, 0 < k ≤ n}

of morphisms of T_{≤n}-Coll_{0}, again distinguishing ϕ_{p} for p ∈ Tr_{n} as strict. As before the p here are representables in the presheaf category T_{≤n}-Coll_{0}, noting that res_{i} sends U_{0} to the initial object and all the other representables to the representables of the same name in T_{≤n}-Coll_{0}, and so we define ϕ_{p} in T_{≤n}-Coll_{0} for p ≠ U_{0} to be res_{i}(ϕ_{p}). There is no harm in excluding the case res_{i}(ϕ_{U_{0}}) since this is the identity on the initial object, against which every map satisfies the unique right lifting property, and so its inclusion or exclusion has no effect on the resulting algebraic weak factorisation system.

Finally we note that Ĩ_{≤n}, Ĩ′_{≤n} and Ĩ′′_{≤n} admit alternative inductive descriptions, intrinsic to G^{n}(Set), T_{≤n}-Coll and T_{≤n}-Coll_{0}, an analogue of which will be important for section(5.5). For Ĩ_{≤n} the initial step is that ∂(0) is initial and ϕ_{0} is the unique map ∂(0) → 0. For 0 ≤ k < n, ∂(k + 1) and ϕ_{k+1} are defined by the pushout

[figure: ∂k; k; ∂(k + 1); k; ϕ_{k}; ϕ_{k}; po; k + 1; σ; τ; ϕ_{k+1}]

where σ and τ are the cosource and cotarget maps from 𝔾_{≤n} regarded as morphisms between representable n-globular sets. For Ĩ′_{≤n} one defines ϕ_{p} : ∂(p) → p for p ∈ Tr_{k} (regarded as a representable) by induction on k. The base case is that ∂(U_{0}) is initial and ϕ_{U_{0}} is determined uniquely and the inductive step is indicated in

[figure: ∂tr(p); tr(p); ∂(p); tr(p); ϕ_{tr(p)}; τ′_{p}; σ′_{p}; ϕ_{tr(p)}; po; p; σ_{p}; τ_{p}; ϕ_{p}]

where the cosource and target maps σ_{p} and τ_{p} are obtained in the evident way from those for G^{n}(Set) (ie using T_{≤n}-Coll ≃ G^{n}(Set)/Tr_{≤n} and Yoneda). The inductive definition of Ĩ′′_{≤n} within T_{≤n}-Coll_{0} is described identically, except that the base case is that ∂(p) is initial for all p ∈ Tr_{1}.

5. Weak n-categories with strict units

5.1. Overview. It is natural to ask whether the lifting theorem gives us an inductive formulation of the notion weak n-category, as recalled above in section(4), via iterated enrichment. That is, one can ask whether there is a functor operad L_{≤n} on the category G^{n}(Set)^{K_{≤n}} of K_{≤n}-algebras, such that

L_{≤n}-Cat ≅ G^{n+1}(Set)^{K_{≤n+1}}  (6)

over G^{n+1}(Set). Note that given a T_{≤n+1}-operad B over Set, we can consider the unary part of its corresponding T_{≤n}^{×}-multitensor, which is a T_{≤n}-operad, and denote this by h(B). This is the object map of a functor

h : T_{≤n+1}-Op_{0} ⟶ T_{≤n}-Op.

The operad h(B) describes the structure that the homs of a B-algebra have. So what the lifting theorem does give us, when applied to K_{≤n+1}, is a functor operad L_{≤n} on G^{n}(Set)^{h(K_{≤n+1})} satisfying equation(6). Thus our desired inductive formulation of the notion weak n-category would follow if

h(K_{≤n+1}) ≅ K_{≤n}  (7)

which from the point of view of algebras says that the structure that the homs of a weak (n + 1)-category have is exactly that of a weak n-category. In this section we shall see that the presence of weak units causes (7) to be false. However we shall give a notion of “weak n-category with strict units” by producing the operads K_{≤n}^{(su)} which describe such structures, which do verify the appropriate analogue of (7). Thus the lifting theorem produces the functor operads giving a formulation of weak n-categories with strict units via iterated enrichment.

5.2. Weak units. Equation(7) isn’t true even in the case n = 1. Let B be a bicategory. Then by definition any hom B(x, y) of B is a category, whose objects are arrows x → y, an arrow f → g of B(x, y) is a 2-cell f ⇒ g in B, and composition is given by vertical composition of 2-cells. However, this is not the only structure that B(x, y) has in general. For instance one can consider composition by the identities 1_{x} and 1_{y} giving endofunctors

(−) ∘ 1_{x} : B(x, y) → B(x, y)  1_{y} ∘ (−) : B(x, y) → B(x, y),

and more complicated processes involving just composing with identities such as

1_{y} ∘ ((1_{y} ∘ 1_{y}) ∘ (((−) ∘ 1_{x}) ∘ 1_{x})) : B(x, y) → B(x, y).

Thus one has a non-trivial monoid M whose elements are such formal expressions and multiplication is given by substitution. In addition to all this the coherence isomorphisms of B give isomorphisms between all these endofunctors, and from the coherence theorem for bicategories, any diagram of such isomorphisms commutes.

Let us write ch : Set → Cat for the right adjoint to the functor ob : Cat → Set which sends a category to its set of objects. Given a set X, ch(X) is known as the indiscrete category on X, it has object set X and exactly one arrow between any 2 objects. From the previous paragraph and the coherence theorem for bicategories, the hom of a bicategory is a category equipped with a strict action by the strict monoidal category ch(M). Thus in the case n = 1, K_{≤2} is an operad whose algebras are bicategories, K_{≤1}-algebras are categories, but h(K_{≤2})-algebras are categories strictly acted on by ch(M). In particular note that the free h(K_{≤2})-algebra on a graph Z has underlying object set given by Z_{0}×M, and so h(K_{≤2}) is not even over Set.

5.3. Reduced T_{≤n}-operads. Let α : A → T_{≤n} be a T_{≤n}-collection. When A underlies a T_{≤n}-operad and p is a linear tree, then the elements of A_{p} are unital operations of some type. For example an operation of arity zU_{0} distinguishes a one cell u_{x} : x → x in any A-algebra X for all x ∈ X_{0}. As another example, an operation of arity z^{2}U_{0} will have source and target operations of arity zU_{0}, and so such an operation will distinguish one cells u_{x} and v_{x} : x → x and 2-cells u_{x} → v_{x} in any A-algebra X for all x ∈ X_{0}. In general an operation of arity z^{r}U_{s} distinguishes for each s-cell x in an A-algebra X, a single (r+s)-cell u_{x} whose s-dimensional source and target is x, and so the sources and targets of u_{x} in dimensions between s and r+s will also be part of the data determined such an operation, and in any of these intermediate dimensions, these need not be the same cell.

5.4. Definition. Let α : A → T_{≤n} be a T_{≤n}-collection. Then α is reduced when for all linear trees p ∈ Tr_{k} where k ≤ n, the set A_{p} is singleton. A reduced T_{≤n}-operad is one whose underlying T_{≤n}-collection is reduced in this sense. We denote by Rd-T_{≤n}-Coll the full subcategory of T_{≤n}-Coll consisting of the reduced T_{≤n}-collections, and by Rd-T_{≤n}-Op the full subcategory of T_{≤n}-Op consisting of the reduced T_{≤n}-operads.

Thus for a reduced T_{≤n}-operad one has a unique operation of each “unit type” (i.e., of each arity p where p is a linear tree). Examples include the terminal operad T_{≤n}, the T_{≤2}-operad for sesqui-categories, and the T_{≤3}-operad for Gray categories, and one would expect that the (yet to be defined) operads describing the higher analogues of Gray categories to be of this form. To say that a T_{≤n}-collection A is over Set is to say that A_{U_{0}} is singleton, and so reduced T_{≤n}-collections and reduced T_{≤n}-operads are in particular over Set.

The category Rd-T_{≤n}-Coll is a presheaf category, one has

Rd-T_{≤n}-Coll ≃ [R_{≤n}^{op}, Set]

where R_{≤n} is the full subcategory of el(Tr_{≤n}) consisting of those trees p which are not linear. The inclusion I : Rd-T_{≤n}-Coll ↪ T_{≤n}-Coll can also be seen as the process of right Kan extending along the inclusion i : R_{≤n} ↪ el(Tr_{≤n}), so its left adjoint L is given by restriction along i. One has

[figure: Rd-T_{≤n}-Op; T_{≤n}-Op; T_{≤n}-Coll; Rd-T_{≤n}-Coll; J; U; I; U^{(rd)}; pb]  (8)

a pullback square in CAT, where I and J are the inclusions, and U and U^{(rd)} are the forgetful functors. Each of the categories in (8) are the categories of models of finite limit sketches and so are locally finitely presentable, and the square itself is induced by a square of morphisms of limit sketches. Thus each of the functors in (8) are finitary and monadic.

While not essential for this article, it is of interest that one has explicit descriptions of the left adjoints to each of the functors participating in (8). We already gave the explicit description of the left adjoint L of I, and the construction of free operads as described in Appendix D of [Leinster, 2003] gives an explicit description of the left adjoint F of U. Given an explicit description L′ of J, F^{(rd)} = L′F I is then an explicit description of the left adjoint to U^{(rd)} as witnessed by the natural isomorphisms

Rd-T_{≤n}-Op(L′F IA, B) ≅ T_{≤n}-Coll(IA, UJB) = T_{≤n}-Coll(IA, IU^{(rd)}B)
≅ Rd-T_{≤n}-Coll(A, U^{(rd)}B).

An explicit description of L′ follows from the work of Harvey Wolff [Wolff, 1978]. Theorem(2.1) of [Wolff, 1978] deals with the following situation. One has a bipullback square in CAT as on the left in

[figure: C; A; B; D; J; U; I; V; bipb; F U; 1_{A}; P; F ILU; ε; π; F ηL; po]  (9)

where A has pushouts, I and J are fully faithful and regarded as inclusions, I has left adjoint L with unit η, and U has left adjoint F with counit ε. The conclusion of Wolff’s theorem is that the pointed endofunctor (P, π), constructed via the pushout as on the right in the previous display, is well-pointed (i.e., P π = πP) and its algebras are exactly the objects of C (i.e., those in the image of J). Thus in particular when A is locally finitely presentable and U and I are finitary, the left adjoint L′ of J is constructed as the free monad on (P, π) and is finitary. Then by virtue of well-pointedness, L′ is constructed as the colimit of the sequence

[figure: 1; P; P^{2}; ...; π; P π; P^{2}π]

– see [Kelly, 1980].

One can apply Wolff’s theorem in the case (8) since U being monadic is an isofibration, and so the pullback of (8) is also a bipullback. However we shall obtain a more basic description of L′. Denote by LinTr_{≤n} ↪ Tr_{≤n} subobject of Tr_{≤n} consisting of the linear trees. Clearly these trees are closed under substitution of trees, and so LinTr_{≤n} underlies a T_{≤n}-operad which we denote as RGr_{≤n}. To give an n-globular set X an RGr_{≤n}-algebra structure, is to give common sections z_{k} : X_{k} → X_{k+1} for its source and target maps s, t : X_{k+1} → X_{k} for 0 ≤ k < n. That is, RGr_{≤n}-algebras are exactly reflexive n-globular sets. Both as a T_{≤n}-collection and as a T_{≤n}-operad, RGr_{≤n} is subterminal, that is to say, the unique map ρ : RGr_{≤n} → T_{≤n} into the terminal collection/operad is a monomorphism. For any T_{≤n}-operad A, the operations of the T_{≤n}-operad A × RGr_{≤n} are exactly the operations of A whose arities are linear trees, and the projection p_{A} : A×RGr_{≤n} → RGr_{≤n} assigns these operations to their arities. Thus to say that A is reduced is to say that p_{A} is an isomorphism, which amounts to saying that one has a bipullback square as on the left in

[figure: Rd-T_{≤n}-Op; T_{≤n}-Op; T_{≤n}-Op/RGr_{≤n}; 1; J; Δ_{RGr_{≤n}}; ⌜1_{RGr_{≤n}}⌝; bipb; A × RGr_{≤n}; A; P A; _{RGr_{≤n}}; q_{A}; π_{A}; p_{A}; po]

where Δ_{RGr_{≤n}} is given by pulling back along ρ. Applying Wolff’s theorem to this situation, the left adjoint L′ to J is the free monad on the pointed endofunctor (P, π) constructed as on the right in the previous display, where p_{A} and q_{A} are the product projections. We are indebted to Steve Lack for pointing out that our explicit construction of L′ is an instance of Wolff’s theorem, thereby enabling us to describe this construction here more efficiently.

5.5. Defining weak n-categories with strict units. While a reduced T_{≤n}-operad has unique operations of each unit type, there is no guarantee that these really act as units with respect to the other operations. In this section we formalise this aspect, and then by mimicking the approach to defining weak n-categories in section(4.1), we shall define weak n-categories with strict units. The main idea is to work with a richer notion of collection, namely one that remembers the unit operations of a T_{≤n}-operad, and then define a stronger notion of contractibility for such collections, by isolating an appropriate algebraic weak factorisation system.

Let us first recall, for the convenience of the reader and to fix notation, substitution for T_{≤n}-operads. For a T_{≤n}-operad α : A → T_{≤n}, the substitution is encoded by the multiplication of the monad A. The object A ∘ A of T_{≤n}-Coll is the composite

A^{2} ⟶^{α^{2}} T_{≤n}^{2} ⟶^{μ} T_{≤n}

where μ here is the multiplication of the monad T_{≤n}, and then A’s multiplication underlies a morphism A ∘ A → A in T_{≤n}-Coll. To book-keep what is going on at the level of the sets of operations of A, it is convenient to use the notion of morphism of k-stage trees.

Given k-stage trees p and q, a morphism f : p → q consists of functions f^{(i)} as shown

[figure: p^{(k)}; p^{(k−1)}; ...; p^{(1)}; p^{(0)}; q^{(k)}; q^{(k−1)}; ...; q^{(1)}; q^{(0)}; ∂_{k}; ∂_{1}; ∂_{k}; ∂_{1}; f^{(k)}; f^{(k−1)}; f^{(1)}; f^{(0)}]

making the above diagram commute serially (in Set), and such that for 0 < i ≤ k, f^{(i)} is order preserving on the fibres of ∂_{i}. In this way one has a category Ω_{k}, whose set of objects is Tr_{k}, and an n-globular category Ω_{≤n} whose underlying n-globular set (i.e its object of objects when one views globular categories as category objects in G^{n}(Set)) is Tr_{≤n}. Recall from [Batanin-Street, 2000] that Ω_{≤n} has a universal property – it is the free categorical T_{≤n}-algebra containing an internal T_{≤n}-algebra.

For any given node x of q of height r one can consider a morphism x̃ : z^{k−r}U_{r} → q which picks out x and all its ancestors. Pulling the components of x̃ back along those of f produces a tree f^{−1}(x) ∈ Tr_{r}, and one has an inclusion z^{k−r}f^{−1}(x) ↪ p of k-stage trees. When x is the i-th leaf of q we denote f^{−1}(x) as p_{i}, and all the p_{i}’s together are called the fibres of f. Now the trees p_{i} provide a labelling of q in the sense that truncating p_{i} and p_{i+1} to the height of the highest common ancestor of the i-th and (i+1)-th leaves of q gives the same tree for 1 ≤ i < a, where a is the number of leaves of q. From the explicit description of T_{≤n} then, f : p → q may be identified with an element of T_{≤n}^{2}(1)_{k}, the effect of T_{≤n}(!) (where ! : T_{≤n}(1) → 1 is the unique map) on this element is q, and of μ_{1} on this element is p. In other words the result of substituting the p_{i} into q is p. Now for a T_{≤n}-operad A, the data defining the operad substitution of A amounts to sets and functions

σ_{f} : A_{q} × (A_{p_{1}} ×_{A_{p′_{1}}} ... ×_{A_{p{′ \atop a−1}}} A_{p_{a}}) ⟶ A_{p}

for all morphisms f : p → q of k-stage trees, where p′_{i} is the result of truncating p_{i} to the height of the highest common ancestor of the i-th and (i+1)-th leaves of q.

5.6. Definition. A morphism f : p → q of k-stage trees is an inclusion when its components – the f^{(i)} – are injective functions. We denote by Ω_{≤n}^{(incl)} the sub-n-globular category consisting of all the trees and the inclusions between them.

It is also worth noting that the categories Ω_{≤n} and Ω_{≤n}^{(incl)} sit naturally within Joyal’s category Θ_{n}. In the language of [Berger, 2002], Ω_{≤n} (resp. Ω_{≤n}^{(incl)}) is dual to the subcategory of Θ_{n} consisting of all the objects and the covers (resp. degeneracies) between them.

Clearly a morphism of k-stage trees is an inclusion iff its fibres are linear. Thus σ_{f} for f an inclusion encodes the substitution of operations of unit type into operations of (an arbitrary) arity q. In this case when A is reduced, the sets A_{p_{i}} and A_{p′_{i}} are all singletons, and so σ_{f} may be regarded simply as a function σ_{f} : A_{q} → A_{p}. It is these substitutions that we wish to include in our richer notion of collection. Denote by Ψ_{≤n} the full subcategory of el(Ω_{≤n}^{(incl)}) consisting of the non-linear trees.

5.7. Definition. A pointed reduced T_{≤n}-collection is a presheaf A : Ψ_{≤n}^{op} → Set. We denote by PtRd-T_{≤n}-Coll = [Ψ_{≤n}^{op}, Set] the category of pointed reduced T_{≤n}-collections.

Thus a pointed reduced collection A contains a set of operations of arity p for any tree p that is not linear, just as with a reduced collection, and in addition one has extra functions between those sets that codify the substitution of unique operations of unit type into arbitrary operations. Recall that Rd-T_{≤n}-Coll ≃ [R_{≤n}^{op}, Set] where R_{≤n} is the full subcategory of el(Tr_{≤n}) consisting of the trees which are not linear, and so there is by definition an identity on objects functor i : R_{≤n} → Ψ_{≤n}, with the morphisms of Ψ_{≤n} not in the image of i coming from inclusions of trees. Denoting by j : R_{≤n} → el(Tr_{≤n}) the fully faithful inclusion, we have adjunctions

[figure: PtRd-T_{≤n}-Coll; Rd-T_{≤n}-Coll; T_{≤n}-Coll; ^{res^{i}}; ^{ran^{j}}; ^{lan^{i}}; ^{res^{j}}; ⊥; ⊥]

given by left Kan extension and restriction along i, and right Kan extension and restriction along j. Note in particular the effect of res_{j} and lan_{i} on representables. One has res_{j}(p) = 0 if p is a linear tree, otherwise one can take res_{j}(p) = p, and one has lan_{i}(p) = p for all representables. Moreover from the description of boundaries of representables as pushouts in T_{≤n}-Coll, it is clear that res_{j}(∂p) = 0 if tr(p) is linear.

Let us now describe the appropriate cofibrantly generated algebraic weak factorisation system on PtRd-T_{≤n}-Coll. We shall define the functor

J : V_{≤n} ⟶ (PtRd-T_{≤n}-Coll)^{[1]}

where V_{≤n} is the subcategory of Ψ_{≤n} consisting of all the non-linear trees, and just the inclusions between them. Note that the category el(Ω_{≤n}^{(incl)}), as the domain of a split fibration into 𝔾_{≤n}, comes with a strict factorisation system in which the left class are the vertical arrows, and the right class are the chosen cartesian arrows (i.e., chosen by the given split cleavage). Thus Ψ_{≤n} inherits such a factorisation system, R_{≤n} is the subcategory of morphisms in the right class – consisting of the cosource and cotarget morphisms, and V_{≤n} is the subcategory of morphisms in the left class – the vertical morphisms, which in this case are the inclusions of trees. We shall define the object and arrow maps of J in an inductive manner.

Recall that sources and targets for Tr_{≤n} agree and are given by truncation. For 0 ≤ r ≤ k ≤ n let us denote by tr^{k−r} : Tr_{k} → Tr_{r} the function which sends p ∈ Tr_{k} to its height-r truncation, that is to say, to the tree p^{(r)} → ... → p^{(0)}. When k − r = 1 this was denoted above simply as tr. The non-linear height ht(p) of p ∈ Tr_{k} is greatest h ∈ ℕ such that tr^{k−h}(p) is a linear tree. Since tr^{k}(p) = U_{0} is linear, ht(p) is well-defined for all p ∈ Tr_{k}, and by definition a tree p is linear iff ht(p) = 0.

For a non-linear tree p we write J(p) = ϕ_{p} : ∂(p) → p in PtRd-T_{≤n}-Coll, and these maps are defined by induction on ht(p). For the base case ht(p) = 1, we take ∂(p) to be initial giving us a unique map ϕ_{p} : ∂(p) → p. Similarly as in section(4.7) ∂(p) and ϕ_{p}, for p of non-linear height > 1, are defined by

[figure: ∂tr(p); tr(p); ∂(p); tr(p); ϕ_{tr(p)}; τ′_{p}; σ′_{p}; ϕ_{tr(p)}; po; p; σ_{p}; τ_{p}; ϕ_{p}]

though now in the category PtRd-T_{≤n}-Coll. Note that the image of the object map of J just described agrees with applying lan_{i}res_{j} to I′_{≤n}, except for the presence of copies of 1_{0} (coming from ϕ_{p} and ϕ′_{p} in T_{≤n}-Coll with p linear).

Given an inclusion f : p → q of non-linear k-stage trees we shall define ∂(f) as on the left in

∂(f) : ∂(p) → ∂(q)  J(f) = (∂(f), f) : ϕ_{p} → ϕ_{q}

such that ϕ_{q}∂(f) = fϕ_{p}, which thus gives a morphism J(f) in (PtRd-T_{≤n}-Coll)^{[1]} as on the right in the previous display. We define ∂(f) by induction on the non-linear height of p. For the base case where ht(p) = 1, ∂(p) = 0 and so ∂(f) is uniquely defined and one has ϕ_{q}∂(f) = fϕ_{p} by initiality. When ht(p) > 1, ∂(f) is constructed in

[figure: ∂tr(p); tr(p); ∂(p); tr(p); ϕ_{tr(p)}; τ′_{p}; σ′_{p}; ϕ_{tr(p)}; po; ∂tr(q); tr(q); ∂(q); tr(q); ϕ_{tr(q)}; τ′_{q}; σ′_{q}; ϕ_{tr(q)}; ∂(tr(f)); tr(f); tr(f); ∂(f)]

in which the existence of ∂(tr(f)) and ϕ_{tr(q)}∂(tr(f)) = tr(f)ϕ_{tr(p)} follow by induction, and so ∂(f) is defined uniquely so that ∂(f)σ′_{p} = σ′_{q}tr(f) and ∂(f)τ′_{p} = τ′_{q}tr(f). In order that this induction goes through we must verify ϕ_{q}∂(f) = fϕ_{p}. Since (σ′_{p}, τ′_{p}) are jointly epic, it suffices to show ϕ_{q}∂(f)σ′_{p} = fϕ_{p}σ′_{p} and ϕ_{q}∂(f)τ′_{p} = fϕ_{p}τ′_{p}. One has

ϕ_{q}∂(f)σ′_{p} = ϕ_{q}σ′_{q}tr(f) = σ_{q}tr(f) = fσ_{p} = fϕ_{p}σ′_{p},

ϕ_{q}∂(f)τ′_{p} = fϕ_{p}τ′_{p} follows similarly. The functoriality of J is clear by construction. Finally we define the set S of strict objects of V_{≤n} to be the non-linear n-stage trees.

5.8. Proposition. The algebraic weak factorisation system (L, R) on PtRd-T_{≤n}-Coll cofibrantly generated by (S, J) defined above is finitary.

Proof. By lemma(4.5) it suffices to show that the domains and codomains of arrows of PtRd-T_{≤n}-Coll in the image of J_{S} are all finitely presentable. The domains are nonlinear trees viewed as objects of PtRd-T_{≤n}-Coll, are representables, and so are finitely presentable objects. The codomains are clearly finite colimits of representables, and so are also finitely presentable.

5.9. Definition. A choice of unital contractions for a pointed reduced T_{≤n}-collection A is an R_{1}-algebra structure on A for the algebraic weak factorisation system (L, R) of proposition(5.8). A choice of unital contractions for a reduced T_{≤n}-operad is one for its underlying pointed reduced collection.

Explicitly a choice of unital contractions for A amounts to

• choice of contractions in the usual sense for the underlying collection, and

• a compatibility condition on these choices with respect to the process of substituting the unique unit operations to general operations.

For p an object of Ψ_{≤n}, a morphism p → A in PtRd-T_{≤n}-Coll is by Yoneda an operation of A of arity p, and a morphism ∂(p) → A is, by the definition of ∂(p), a pair of operations (a, b) of (the underlying T_{≤n}-collection of) A both of arity tr(p), whose sources and targets agree (called “parallel operations” in [Leinster, 2003]). So within PtRd-T_{≤n}-Coll a choice of contractions γ amounts to choices of fillers as in

[figure: ∂(p); A; 1; p; (a,b); ϕ_{p}; γ(p,a,b)]  (10)

for all 0 < k ≤ n, non-linear k-stage trees p, and parallel operations (a, b) as shown.

We remind the reader that as pointed out in [Leinster, 2003], the choice γ encodes two aspects of higher category theory in one go – compositions and coherences. For example taking p to be the 1-stage tree on the left

[figure: •; •; •]  z ([figure: •; •; •; •])

then an operation of arity p is a binary composition of 1-cells, and in (10) (a, b) are determined uniquely (i.e., a and b are both the operadic unit in dimension 0), and so the contraction in this instance gives a distinguished binary composition of 1-cells as part of the resulting algebraic structure. On the other hand taking p to be the 2-stage tree on the right in the previous display, then (a, b) of (10) will be a pair of operations that compose chains of 1-cells of length 3, and then the filler will give distinguished coherence 2-cells between these. So in general γ distinguishes a higher categorical operation of each arity.

The compatibility for a choice of unital contractions for A says that if one substitutes some unique unit operations into a given distinguished operation – either one that gives some kind of composition, or one that gives some kind of coherence – then the result is another distinguished operation. For example part of the data of a choice of contractions for a reduced operad A will be a choice of operation that composes chains of one-cells of length n for all n ≥ 2. Compatibility says in particular that, if one starts with a chain

w ⟶^{f} x ⟶^{g} y ⟶^{h} z

of one-cells in a given A-algebra, and then regards it as a longer chain by adding in some identity arrows

w ⟶^{f} x ⟶^{1_{x}} x ⟶^{1_{x}} x ⟶^{g} y ⟶^{h} z ⟶^{1_{z}} z

which are part of the structure by virtue of the reducedness of A (i.e., what we denote here as identity one-cells are provided by the unique operation of A of arity z(U_{0})), then applying the chosen method of composition to this longer chain should agree with applying the chosen method of composition to the original chain.

As the foregoing discussion indicates, the existence of a choice of unital contractions for a given reduced T_{≤n}-operad A amounts to saying that one can distinguish operations of A of each arity, in such a way that this choice is compatible with the process of substituting in unit operations. While this may seem like a strong condition at first glance, we shall see now that such choices can be exhibited in low-dimensional cases of interest, and moreover one has universal such T_{≤n}-operads that we will use to define weak n-categories with strict units. In section(6) we shall see that this choice of unital contractions is exactly what we need to get an alternative iterative formulation.

5.10. Example. Let A be the T_{≤2}-operad whose algebras are bicategories in the sense of Bénabou [Bénabou, 1967] for which the unit coherence 2-cells are identities, and we write “∘” for the given binary horizontal composition of 1-cells. The strictness of the units ensures that A is reduced. The coherence theorem for bicategories ensures that one has unique choices of contractions γ(p, a, b) when p is a 2-stage tree. So to give a chosen contraction for A we must choose a distinguished composite of chains of 1-cells of length n, for n ≥ 2. The set of all such compositions may be identified with the set of binary bracketings of n symbols, so for each n we must choose one of these. There are two obvious choices, to bracket completely to the left or completely to the right, as illustrated in

x_{0} ⟶^{f_{1}} x_{1} ⟶^{f_{2}} x_{2} ⟶^{f_{3}} x_{3} ⟶^{f_{4}} x_{4}  ((f_{4} ∘ f_{3}) ∘ f_{2}) ∘ f_{1}  f_{4} ∘ (f_{3} ∘ (f_{2} ∘ f_{1}))

in the case n = 4. Clearly each of these schemes is compatible with units, for example

(((((1_{x_{4}} ∘ f_{4}) ∘ f_{3}) ∘ 1_{x_{2}}) ∘ 1_{x_{2}}) ∘ f_{2}) ∘ f_{1} = ((f_{4} ∘ f_{3}) ∘ f_{2}) ∘ f_{1},

and so in two ways, one can exhibit a choice of unital contractions for A.

5.11. Example. Let A be the T_{≤3}-operad for Gray categories. The underlying 1-operad is terminal, A’s contractibility ensures that one has unique choices of contractions γ(p, a, b) when p is a 2-stage tree, and so the only choices that aren’t forced concern what one takes to be the distinguished composites of non-degenerate 2-dimensional pasting diagrams in a Gray category. The set of all ways to compose a given such pasting diagram, say

[figure: •; •; •; •; •; α_{11}; α_{21}; α_{13}; α_{14}; α_{24}; α_{34}; α_{44}]

to a 2-cell, that one has in a Gray category, can be identified with linear orders ≤ on the set of α_{ij} such that i_{1} ≤ i_{2} implies α_{i_{1}j} ≤ α_{i_{2}j}, or in other words, with (2, 1, 4)-shuffles. There are two evident unit-compatible choices. In terms of linear orders one can define α_{i_{1}j_{1}} < α_{i_{2}j_{2}} iff j_{1} < j_{2} or (j_{1} = j_{2} and i_{1} < i_{2}); or alternatively α_{i_{1}j_{1}} < α_{i_{2}j_{2}} iff j_{1} > j_{2} or (j_{1} = j_{2} and i_{1} < i_{2}), which for the above example correspond to the linear orders

(α_{11}, α_{21}, α_{13}, α_{14}, α_{24}, α_{34}, α_{44})  (α_{14}, α_{24}, α_{34}, α_{44}, α_{13}, α_{11}, α_{21})

which correspond to “moving” down each column completely, and from left-to-right or right-to-left respectively.

Not every “systematic looking” choice of compositions need give a unit-compatible choice of contractions. For instance in this last example another choice of contractions could be to move from left to right and row by row, just as one reads text in European languages, so that in the case of the above pasting diagram one would choose the order

(α_{11}, α_{13}, α_{14}, α_{21}, α_{24}, α_{34}, α_{44}).

The choice of contractions so determined is not unit-compatible, because the distinguished composite for the pasting diagram on the left

[figure: •; •; •; f; g; •; •; •; 1; f; g; 1]

corresponds to the order (f, g), whereas substituting in identities to the distinguished composite as indicated on the right in the previous display, gives the operation that corresponds to the other order (g, f).

We now proceed to give the definition of weak n-category with strict units, following the approach of section(4).

5.12. Proposition. There is a finitary monad on PtRd-T_{≤n}-Coll whose algebras are reduced operads equipped with a choice of unital contractions.

Proof. It suffices to give two finitary monads on PtRd-T_{≤n}-Coll, one whose algebras are reduced collections equipped with chosen unital contractions, and the other whose algebras are reduced operads, because then the required monad will just be the coproduct of these. In the first case one has the monad R_{1} for the finitary algebraic weak factorisation system (L, R) of proposition(5.8). In the second case one can use a general finite limit sketch argument to establish the existence of the required finitary monad. Alternatively for a construction, one has forgetful functors

[figure: Rd-T_{≤n}-Op; PtRd-T_{≤n}-Coll; Rd-T_{≤n}-Coll,; U; ^{res^{i}}]

res_{i} is cocontinuous and monadic by a standard application of the Beck Theorem (using that i is a bijective on objects functor), we noted the finitariness of U^{(rd)} = res_{i}U and constructed explicitly its left adjoint in section(5.3), and so Dubuc’s adjoint triangle theorem [Dubuc, 1970] gives an explicit construction of the left adjoint of U. Moreover U is finitary since both res_{i} and U^{(rd)} create filtered colimits, and monadic by another standard application of the Beck theorem.

Thus the category of reduced operads equipped with a choice of unital contractions is locally finitely presentable, and we denote its initial object as K_{≤n}^{(su)}.

5.13. Definition. A weak n-category with strict units is a K_{≤n}^{(su)}-algebra.

Since as we noted above chosen unital contractions are in particular chosen contractions, there is an operad morphism K_{≤n} → K_{≤n}^{(su)} by the universal property of K_{≤n}, and thus every weak n-category with strict units has an underlying weak n-category.

6. Strictly unital weak n-categories via iterated enrichment

The bulk of this section will be occupied with the proof of

6.1. Lemma. For all n ∈ ℕ one has an isomorphism of T_{≤n}-operads

h(K_{≤n+1}^{(su)}) ≅ K_{≤n}^{(su)}.

As explained in section(5.1), from lemma(6.1) and the lifting theorem one obtains the following result.

6.2. Theorem. For all n ∈ ℕ there is a distributive functor operad L_{≤n} on the category G^{n}(Set)^{K_{≤n}^{(su)}} of algebras of K_{≤n}^{(su)} and an isomorphism of categories

L_{≤n}-Cat ≅ G^{n+1}(Set)^{K_{≤n+1}^{(su)}}  (11)

over G^{n+1}(Set).

Thus the lifting theorem has produced the tensor products – the L_{≤n}’s – enabling us to recover the notion of weak n-category with strict units of definition(5.13) inductively by successive enrichment – as expressed by (11) at the level of objects.

In general we have functors

h : T_{≤n+1}-Op_{0} ⟶ T_{≤n}-Op  r : T_{≤n}-Op ⟶ T_{≤n+1}-Op_{0}.

Recall from section(5.1) that for a T_{≤n+1}-operad B over Set, the operad h(B) describes the structure that the homs of a B-algebra have, and is constructed formally by taking the unary part of the T_{≤n}^{×}-multitensor associated to B by [Weber, 2013] theorem(6.1.1). On the other hand given a T_{≤n}-operad A, r(A) is the T_{≤n+1}-operad Γ(A^{×}), whose algebras are categories enriched in G^{n}(Set)^{A} using the cartesian product by proposition(3.2.1) of [Weber, 2013] and proposition(2.8) of [Batanin-Weber, 2011]. Since (Γ(A^{×}))_{1} = A one has that rh = 1.

We shall prove lemma(6.1) by first proving

6.3. Lemma. The functors h and r defined above restrict to an adjunction

[figure: Rd-T_{≤n+1}-Op; Rd-T_{≤n}-Op; h; r; ⊥]

between categories of reduced operads.

and then denoting by CtRd-T_{≤n}-Op the category of reduced T_{≤n}-operads with chosen unital contractions, we shall prove

6.4. Lemma. The adjunction of lemma(6.3) lifts to an adjunction

[figure: CtRd-T_{≤n+1}-Op; CtRd-T_{≤n}-Op; h; r; ⊥]

between categories of reduced operads with chosen unital contractions.

Since lemma(6.1) is then just the statement that this lifted h preserves the initial object, it follows immediately from lemma(6.4).

We now describe the object maps of h and r in more elementary terms. Recall from [Weber, 2013] section(3), that given a category V with an initial object, a finite sequence (X_{1}, ..., X_{m}) of objects of V can be regarded as a V -graph whose set of objects is {0, ..., m}, (X_{1}, ..., X_{m})(i, i+1) = X_{i} and the other homs are initial. This simple construction enables an inductive definition of “k-dimensional globular pasting diagram”:

• base case k = 0: the representable globular set 0, with one vertex and no edges, is the unique 0-dimensional pasting diagram.

• inductive step: a (k+1)-dimensional pasting diagram p is a sequence (p_{1}, ..., p_{m}) of k-dimensional pasting diagrams.

which makes the connection with trees very transparent. So a (k+1)-stage tree p can be regarded as a sequence of maps in Δ_{+} as on the left

p^{(k+1)} → ... → p^{(1)} → p^{(0)}=1  ∑_{i=1}^{m} p_{i}^{(k)} → ... → ∑_{i=1}^{m} p_{i}^{(0)} → 1

or as a sequence of k-stage trees (p_{1}, ..., p_{m}) as indicated on the right in the previous display where p^{(1)} = m. In particular, the tree (p) is obtained from the tree p by adding a new root and an edge from the new root to the old root. With these preliminaries in hand one has, by unpacking the definitions, the formulae

h(B)_{p} = B_{(p)}  r(A)_{p} = ∏_{i=1}^{k} A_{p_{i}}

for p = (p_{1}, ..., p_{m}) ∈ Tr_{k+1}, expressing the effect of h and r at the level of sets of operations.

For a given morphism of (k+1)-stage trees f : q → p with fibres (r_{1}, ..., r_{s}) we shall need, for the proof of lemma(6.3) below, to unpack a description of corresponding substitution function

σ_{f}^{rh(B)} : rh(B)_{p} × (rh(B)_{r_{1}} ×_{rh(B)_{r′_{1}}} ... ×_{rh(B)_{r{′ \atop s−1}}} rh(B)_{r_{s}}) ⟶ rh(B)_{q}

for rh(B) in terms of substitution functions for B. To do so, we must describe the morphism f : q → p of (k+1)-trees in the same inductive terms as we did trees in the previous paragraph. So we let q^{(1)} = l and denote by (l_{1}, ..., l_{m}) the cardinality of the fibres of f^{(1)} : l → m which is order preserving by definition. Now let us reindex the sequence of fibres (r_{1}, ..., r_{s}) of f to keep track of over which p_{i} a given fibre of f lives, so we write

(r_{1}, ..., r_{s}) = (r_{11}, ..., r_{ms_{m}})

that is to say, r_{ij} for 1 ≤ i ≤ m and 1 ≤ j ≤ s_{i}, is the j-th fibre of f that lives over p_{i}. Moreover we shall describe each of the r_{ij} as sequences of trees one dimension lower as in

r_{ij} = (r_{ij1}, ..., r_{ijl_{i}}).

Thus the inductive description of the components of f is as

f^{(δ+1)} = (f_{11}^{(δ)}, ..., f_{1l_{1}}^{(δ)}) + ... + (f_{m1}^{(δ)}, ..., f_{ml_{m}}^{(δ)})

where f_{iα} : q_{iα} → p_{i} for 1 ≤ i ≤ m and 1 ≤ α ≤ l_{m}, is the morphism of k-stage trees whose fibres are (r_{i1α}, ..., r_{is_{i}α}). By this notation the number of leaves of p is s, and the number of leaves of p_{i} is s_{i}. Let us denote by t_{ijα} the number of leaves of r_{ijα}. Thus r_{ij} has ∑_{α} t_{ijα} leaves, and q has ∑_{i,j,α} t_{ijα} leaves. With these details in hand, one may readily verify that σ_{f}^{rh(B)} may be rewritten as a product, varying over i and α, of the functions

σ_{f_{iα}}^{B} : B_{(p_{i})} × (B_{(r_{i1α})} ×_{B_{(r{′ \atop ilα})}} ... ×_{B_{(r{′ \atop is_{i−1}α})}} B_{(r_{is_{i}α})}) ⟶ B_{(q_{iα})}.

Proof. (of lemma(6.3)): by the above explicit formulae, h and r send reduced operads to reduced operads. Given a reduced T_{≤n+1}-operad B, we shall describe a morphism as on the left in

ν_{B} : B ⟶ rh(B)  ν_{B,p,i} : B_{p} ⟶ B_{(p_{i})}

of T_{≤n+1}-operads, naturally in B, such that hν_{B} = id and ν_{B}r = id, so that ν is the unit of the desired adjunction in which the counit is the identity. To give the morphism of underlying T_{≤n+1}-collections, it suffices by the explicit descriptions of r and h spelled out above, to give functions ν_{B,p,i} as indicated on the right in the previous display, where 0 < k ≤ n, p = (p_{1}, ..., p_{m}) ∈ Tr_{k+1} and 1 ≤ i ≤ m, satisfying appropriate naturality conditions. Denoting by π_{i} : (p_{i}) → p the canonical inclusion of trees, we define ν_{B,p,i} from the base change function in the underlying pointed reduced collection, namely as ν_{B,p,i} = B(π_{i}). This clearly defines a morphism of T_{≤n}-collections.

Intuitively, ν_{B} is compatible with the operad structures since it was defined by substitution of unit operations. Formally, for any morphism f : q → p of trees whose fibres are denoted as (r_{1}, ..., r_{s}), we must show that

[figure: B_{p} × (B_{r_{1}} ×_{B_{r′_{1}}} ... ×_{B_{r{′ \atop s−1}}} B_{r_{s}}); B_{q}; rh(B)_{q}; rh(B)_{p} × (rh(B)_{r_{1}} ×_{rh(B)_{r′_{1}}} ... ×_{rh(B)_{r{′ \atop s−1}}} rh(B)_{r_{s}}); σ_{f}^{B}; ν_{B,q}; σ_{f}^{rh(B)}; ν_{B,p}×(ν_{B,r_{1}}×_{ν_{B,r′_{1}}} ...×_{ν_{B,r{′ \atop s−1}}} ν_{B,rs})]  (12)

commutes. It suffices to verify this commutativity after composition with each projection proj_{iα} : rh(B)_{q} → B_{(q_{iα})}. We do this now using the notation established above for f : q → p. For

(b, b_{11}, ..., b_{ms_{m}}) ∈ B_{p} × (B_{r_{1}} ×_{B_{r′_{1}}} ... ×_{B_{r{′ \atop s−1}}} B_{r_{s}})

the effect of proj_{iα} composed with the top path of (12) on this element is

(b[b_{11}, ..., b_{ms_{m}}]) [\underbrace{v, ..., v,}_{∑_{X} t_{i′jα′}} \underbrace{u, ..., u,}_{∑_{j} t_{ijα}} \underbrace{v, ..., v}_{∑_{Y} t_{i′jα′}} ]  (13)

where

X = {(i′, α′) : i′ < i or (i′ = i and α′ < α)}
Y = {(i′, α′) : i′ > i or (i′ = i and α′ > α)}.

On the other hand the effect on (b, b_{11}, ..., b_{ms_{m}}) of proj_{iα} composed with the bottom path of (12) is

(b[\underbrace{v, ..., v,}_{∑_{i′<i,j} s_{i′j}} \underbrace{u, ..., u,}_{∑_{j} s_{ij}} \underbrace{v, ..., v}_{∑_{i′>i,j} s_{i′j}}]) [b_{i1}[\underbrace{v, ..., v,}_{∑_{α′<α} t_{i1α′}} \underbrace{u, ..., u,}_{t_{i1α}} \underbrace{v, ..., v}_{∑_{α′>α} t_{i1α′}}], ...

..., b_{is_{i}}[\underbrace{v, ..., v,}_{∑_{α′<α} t_{is_{i}α′}} \underbrace{u, ..., u,}_{t_{is_{i}α}} \underbrace{v, ..., v}_{∑_{α′>α} t_{is_{i}α′}}]]  (14)

Using the associativity of substitution, and that for all operations x of B one has x[v, ..., v] = v by reducedness, (13) equals

b [\underbrace{v, ..., v,}_{∑_{i′<i,j} s_{i′j}} b_{i1}[\underbrace{v, ..., v,}_{∑_{α′<α} t_{i1α′}} \underbrace{u, ..., u,}_{t_{i1α}} \underbrace{v, ..., v}_{∑_{α′>α} t_{i1α′}}], ...

..., b_{is_{i}}[\underbrace{v, ..., v,}_{∑_{α′<α} t_{is_{i}α′}} \underbrace{u, ..., u,}_{t_{is_{i}α}} \underbrace{v, ..., v}_{∑_{α′>α} t_{is_{i}α′}}], \underbrace{v, ..., v}_{∑_{i′>i,j} s_{i′j}} ]  (15)

and by applying operad associativity and one of the operad unit laws to (14), one obtains (14)=(15).

Proof. (of lemma(6.4)): we must explain how the functors r and h of lemma(6.3) lift to the level of reduced operads with chosen unital contractions, and then show that the operad map ν_{B} is compatible with the chosen unital contractions.

Recall that the data of a choice γ of contractions for B ∈ Rd-T_{≤n}-Op amounts to morphisms

γ(p, a, b) : p ⟶ B

in PtRd-T_{≤n}-Coll, for all 0 ≤ k ≤ n, non-linear p ∈ Tr_{k+1} and a, b ∈ B_{tr(p)} such that s(a) = s(b) and t(a) = t(b), and that this choice is unital when it satisfies

γ(q, atr(f), btr(f)) = γ(p, a, b)f

for all p, a, b as above and tree inclusions f : q → p. Given γ and recalling the explicit description of h(B) given above, γ′(p, a, b) = γ((p), a, b) defines a choice γ′ of contractions for h(B). The unitality of γ′ with respect to an inclusion f : q → p follows from that of γ with respect to the evident inclusion (f) : (q) → (p). Clearly if F : B → B′ preserves chosen contractions, then so does h(F). Thus h lifts to the level of operads with chosen unital contractions.

Suppose that a choice ψ of unital contractions for A ∈ Rd-T_{≤n}-Op is given. Recalling the explicit description of r(A) given above,

ψ′(p, (a_{1}, ..., a_{m}), (b_{1}, ..., b_{m})) = (ψ(p_{1}, a_{1}, b_{1}), ..., ψ(p_{m}, a_{m}, b_{m}))

where p = (p_{1}, ..., p_{m}), defines a choice of contractions for r(A). Note that the inductive description of a tree morphism f : q → p given above simplifies in the case where f is an inclusion, because f^{(1)} is injective and so l_{i} = 0 or 1, and so one may write f as a composite

(q_{1}, ..., q_{l}) ⟶^{(f_{1},...,f_{l})} (p_{f^{(1)}(1)}, ..., p_{f^{(1)}(l)}) ⟶^{ss_{f^{(1)}}} (p_{1}, ..., p_{m})

where ss_{f^{(1)}} is the subsequence inclusion corresponding to the injection f^{(1)} interpreted as an inclusion of trees in the evident way, and the f_{i} are themselves inclusions of trees. By definition ψ′ is compatible with such subsequence inclusions. Since ψ is unital, ψ′ is compatible with tree inclusions of the form (f_{1}, ..., f_{l}) as above, and so ψ′ is itself unital. Clearly if G : A → A′ preserves chosen contractions, then so does r(G). Thus r lifts to the level of operads with chosen unital contractions.

Let us now verify that ν_{B} is compatible with chosen contractions. So given 0 ≤ k ≤ n, p = (p_{1}, ..., p_{m}) ∈ Tr_{k+1} and a, b ∈ B_{tr(p)} such that s(a) = s(b) and t(a) = t(b), we must verify

ν_{B}(γ(p, a, b)) = γ′′(p, ν_{B}(a), ν_{B}(b))

and it suffices to verify this equation for all 1 ≤ i ≤ m after applying the projection proj_{i} : rh(B)_{p} → B_{(p_{i})}. Recalling the inclusion π : (p_{i}) → p of trees, by the definition of ν_{B} we have

proj_{i}ν_{B}(γ(p, a, b)) = γ(p, a, b)π_{i}

whereas

proj_{i}γ′′(p, ν_{B}(a), ν_{B}(b)) = γ′(p_{i}, atr(π_{i}), btr(π_{i})) = γ((p_{i}), atr(π_{i}), btr(π_{i}))
= γ(p, a, b)π_{i}

in which the first two equalities follow from the definitions, and the last follows from the unitality of γ with respect to π_{i}.

6.5. Remark. For a reduced T_{≤n+1}-operad B we have the component ν_{B} : B → rh(B) of the unit of the above adjunctions. If we denote by E the T_{≤n}^{×}-multitensor such that ΓE = B, then E_{1}^{×} is the T_{≤n}^{×}-multitensor such that ΓE_{1}^{×} = rh(B), and then ν_{B} is the effect of Γ on a T_{≤n}^{×}-multitensor map κ_{E} : E → E_{1}^{×}, whose unary part is the identity. The natural transformation κ_{E} can be regarded as the coherence data for a lax monoidal functor

(1, κ_{E}) : (G^{n}(Set), E_{1}^{×}) ⟶ (G^{n}(Set), E),

applying C from proposition(3.14) to this gives

(1, κ′_{E}) : (G^{n}(Set)^{E_{1}}, ∏) ⟶ (G^{n}(Set)^{E_{1}}, E′)

and the components of κ′_{E} are of the form

(κ′_{E})_{Z_{1},...,Z_{n}} : E′_{i=1}^{n} Z_{i} ⟶ ∏_{i=1}^{n} Z_{i}

defined for all finite sequences (Z_{1}, ..., Z_{n}) of E_{1}-algebras. In particular one has comparison maps between the functor operads L_{≤n} of theorem(6.2) and the cartesian product of weak n-categories with strict units.

Acknowledgements

We would like to acknowledge Clemens Berger, Richard Garner, André Joyal, Steve Lack, Joachim Kock, Jean-Louis Loday, Paul-André Melliès, Ross Street and Dima Tamarkin for interesting discussions on the substance of this paper. In particular the third author would like to thank André Joyal, who already in 2006 during some inspiring discussions indicated that a strictly unital version of operadic higher category theory ought to be better behaved, and gave some indication at that time already of how the theory of reduced operads should work. We would also like to acknowledge the Centre de Recerca Matemàtica in Barcelona for the generous hospitality and stimulating environment provided during the thematic year 2007-2008 on Homotopy Structures in Geometry and Algebra.

The first author would like to acknowledge the financial support on different stages of this project of the Scott Russell Johnson Memorial Foundation, the Australian Research Council grant No.DP0558372, and L’Université Paris 13. For the later stages of the project, the first and third authors acknowledge the financial support of the Australian Research Council grant No.DP1095346. The second author would like to acknowledge the support of the ANR grant no. ANR-07-BLAN-0142. The third author would like to acknowledge the laboratory PPS (Preuves Programmes Systèmes) in Paris, the Max Planck Institute in Bonn, the IHES and the Macquarie University Mathematics Department for the excellent working conditions he enjoyed during this project.

References

M. Aguiar and S. Mahajan. Monoidal functors, species and Hopf algebras. Number 29 in CRM Monograph Series. American Mathematical Society, 2010.

M. Barr and C. Wells. Toposes triples and theories. Number 12 in TAC Reprints. Theory and applications of categories, 2005.

M. Batanin. Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Mathematics, 136:39–103, 1998.

M. Batanin, D-C. Cisinski, and M. Weber. Algebras of higher operads as enriched categories II. ArXiv:0909.4715.

M. Batanin and M. Markl. Centres and homotopy centres in enriched monoidal categories. Advances in Mathematics, 230:1811–1858, 2012.

M. Batanin and R. Street. The universal property of the multitude of trees. JPAA, 154:3–13, 2000.

M. Batanin and M. Weber. Algebras of higher operads as enriched categories. Applied Categorical Structures, 19(1):93–135, 2011.

J. Bénabou. Introduction to bicategories, part I. Lecture Notes in Math., 47:1–77, 1967.

C. Berger. A cellular nerve for higher categories. Advances in Mathematics, 169:118–175, 2002.

C. Berger, P-A. Melliès, and M. Weber. Monads with arities and their associated theories. J. Pure Appl. Algebra, 216:2029–2048, 2012.

A. K. Bousfield. Constructions of factorization systems in categories. JPAA, 9:207–220, 1977.

A. Burroni. T-categories (catégories dans un triple). Cahiers Topologie Géom. Différentielle Catégoriques, 12:215–321, 1971.

Eugenia Cheng. Iterated distributive laws. Math. Proc. Cambridge Philos. Soc., 150(3):459–487, 2011.

S. Crans. A tensor product for Gray categories. Theory and applications of categories, 5:12–69, 1999.

S. Crans. On braidings, syllepses, and symmetries. Cahiers Topologie Géom. Différentielle Catégoriques, 41:2–74, 2000.

B. Day. Closed categories of functors. Lecture Notes in Math., 137:1–38, 1970.

B. Day and R. Street. Abstract substitution in enriched categories. JPAA, 179:49–63, 2003.

E. Dubuc. Kan extensions in enriched category theory. Number 145 in SLNM. Springer Verlag, 1970.

R. Garner. A homotopy-theoretic universal property of leinster’s operad for weak ωcategories. Mathematical Proceedings Cambridge Philosophical Society, 147(3):615– 628, 2009.

R. Garner. Understanding the small object argument. Applied categorical structures, 17(3):247–285, 2009.

R. Garner. Homomorphisms of higher categories. Advances in Mathematics, 224(6):2269– 2311, 2010.

V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Math. Journal, 76(1):203–272, 1994.

M. Grandis and W. Tholen. Natural weak factorisation systems. Archivum Mathematicum, 42:397–408, 2006.

J.W. Gray. Formal Category Theory: Adjointness for 2-categories. Number 391 in SLNM. Springer Verlag, 1974.

C. Hermida. Representable multicategories. Advances in Mathematics, 151:164–225, 2000.

G.M. Kelly. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc., 22, 1980.

T. Leinster. Higher operads, higher categories. Lecture note series. London Mathematical Society, 2003.

J. McClure and J. Smith. Cosimplicial objects and little n-cubes I. Amer. J. Math, 126:1109–1153, 2004.

P-A. Melliès. Segal condition meets computational effects. In Proceedings of the IEEE Symposium on Logic in Computer Science. LICS, 2010.

R. Street. The formal theory of monads. J. Pure Appl. Algebra, 2:149–168, 1972.

R. Street. Fibrations and Yoneda’s lemma in a 2-category. Lecture Notes in Math., 420:104–133, 1974.

M. Weber. Generic morphisms, parametric representations, and weakly cartesian monads. Theory and applications of categories, 13:191–234, 2004.

M. Weber. Familial 2-functors and parametric right adjoints. Theory and applications of categories, 18:665–732, 2007.

M. Weber. Multitensors and monads on categories of enriched graphs. Theory and applications of categories, 25:857–932, 2004.

H. Wolff. Free monads and the orthogonal subcategory problem. J. Pure Appl. Algebra, 13:233–242, 1978.

Department of Mathematics, Macquarie University (Batanin) Université Paul Sabatier, Institut de Mathématiques de Toulouse (Cisinski) Department of Mathematics, Macquarie University (Weber)

Email: mark.weber.math@gmail.com

This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/25/28-25.{dvi,ps,pdf}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.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