Theory and Applications of Categories, Vol. 28, No. 24, 2013, pp. 780–803.

COMPLICIAL STRUCTURES IN THE NERVES OF OMEGA-CATEGORIES

RICHARD STEINER

Abstract. It is known that strict omega-categories are equivalent through the nerve functor to complicial sets and to sets with complicial identities. It follows that complicial sets are equivalent to sets with complicial identities. We discuss these equivalences. In particular we give a conceptual proof that the nerves of omega-categories are complicial sets, and a direct proof that complicial sets are sets with complicial identities.

1. Introduction

This paper is concerned with the nerves of strict ω-categories (all ω-categories in this paper will be strict ω-categories). The nerve functor, which was defined by Street in [7], takes ω-categories to simplicial sets. The nerve functor is in fact an equivalence from the category of ω-categories to a category of simplicial sets with additional structure, and this additional structure has been described in two different ways. Verity has shown in [9] that ω-categories are equivalent to complicial sets; in [5] I have shown that ω-categories are equivalent to sets with complicial identities. Here a complicial set is a simplicial set with a distinguished class of ‘thin’ elements, and a set with complicial identities is a simplicial set with additional partial binary operations; in both cases the extra structure is subject to an appropriate list of axioms. The axioms for complicial sets are simple and elegant, but their calculational implications are not very obvious; the axioms for sets with complicial identities are algebraic, in fact equational, like the axioms for cubical nerves in [1], and they allow one to make concrete calculations as in [4].

It follows from a combination of [5] and [9] that the categories of strict ω-categories, of complicial sets and of sets with complicial identities are all equivalent. The object of this paper is to shed some light on these equivalences. There are two parts. In the first part we give a conceptual proof that the nerve of an ω-category is a complicial set (Theorem 8.2). This result was first proved computationally by Street in [8] (actually in a rather stronger form), but the conceptual proof given here still seems to be enlightening. In the second part of the paper we show directly that a complicial set is a set with complicial identities (Theorem 9.2); this is straightforward, although the details are somewhat complicated.

The theory of complicial sets is based on thin fillers for a class of ‘complicial’ or ‘admissible’ horns in simplicial sets with thin elements. The process of filling an (n − 1)- dimensional complicial horn is therefore an n-ary operation with a rather complicated domain. The idea behind both parts of this paper is that an (n−1)-dimensional complicial horn in a complicial set is equivalent to a pair of arbitrary (n − 1)-dimensional elements with a common face. This idea leads to the conceptual treatment of fillers in the first part and to the binary operations in the second part. The connection with binary operations appeared long ago in unpublished work of Street [6].

Received by the editors 2013-08-03 and, in revised form, 2013-08-22.

Transmitted by Ross Street. Published on 2013-09-01.

2010 Mathematics Subject Classification: 18D05.

Key words and phrases: complicial set,complicial identities, omega-category.

\begin{matrix}c \\ ◯ \end{matrix} Richard Steiner, 2013. Permission to copy for private use granted.

The structure of the paper is as follows. In Sections 2 and 3 we recall the construction of the nerve of an ω-category as a simplicial set and we show that it is a stratified simplicial set (a simplicial set with thin elements). In Section 4 we recall the definition of complicial sets. In Section 5 we discuss horns in the nerves of ω-categories. In Section 6 we discuss a class of horns represented by pairs of faces, which are later shown to be the complicial horns, and in Section 7 we discuss the fillers for these horns. Section 7 contains the bulk of the proof that the nerves of ω-categories are complicial sets, and the proof is completed in Section 8.

The second part of the paper consists of Section 9. First we define sets with complicial identities; then we show that complicial sets are sets with complicial identities.

2. Nerves as simplicial sets

In this section we describe the nerve functor N from ω-categories to simplicial sets. It was constructed by Street in [7] in the following way: there is a sequence of ωcategories O_{0}, O_{1}, … called orientals, and there is a functor from the simplex category to ω-categories given on objects by [n] ↦ O_{n}; the nerve of an ω-category C is the simplicial set NC given by

N_{n}C = Hom(O_{n}, C).

We will describe this construction in terms of chain complexes, following [2] and [3].

We use a one-sorted description of ω-categories, so that an ω-category C is a set which serves as the set of morphisms for an infinite sequence of commuting category structures. The composition operations will be denoted #_{0}, #_{1}, …, the left identity for x under #_{n} will be denoted d_{n}^{−}x, and the right identity for x under #_{n} will be denoted d_{n}^{+}x. If n < p then the identities for #_{n} must also be identities for #_{p}; every member of C must be an identity for some operation #_{n}.

When we say that the category structures commute we mean in particular that for m ≠ n there are equalities

d_{m}^{α}d_{n}^{β}x = d_{n}^{β}d_{m}^{α}x, d_{m}^{α}(x #_{n} y) = d_{m}^{α}x #_{n} d_{m}^{α}y,

where the signs α and β are arbitrary. We deduce that the identities for any particular operation #_{m} form a sub-ω-category. This gives the following result.

2.1. Proposition. Let m be a nonnegative integer and let C be an ω-category generated by elements g such that d_{m}^{−}g = d_{m}^{+}g = g. Then d_{m}^{−}x = d_{m}^{+}x = x for all x ∈ C.

Next we recall the construction of orientals in terms of chain complexes, as in [2] and [3]. We use a category of free augmented directed complexes defined as follows.

2.2. Definition. A free augmented directed complex is an augmented chain complex of free abelian groups concentrated in nonnegative dimensions, together with a prescribed basis for each chain complex. A morphism of free augmented directed complexes is an augmentation-preserving chain map which takes sums of basis elements to sums of basis elements.

Note that the word sums is to be taken literally: a sum of basis elements is an expression

a_{1} + ⋯ + a_{k}

with k ≥ 0 such that a_{1}, …, a_{k} are basis elements; no subtraction is allowed.

The main example used in this paper is the chain complex of the n-simplex. This chain complex will be denoted Δ_{n}. The q-dimensional chain group of Δ_{n} has prescribed basis consisting of the (q + 1)-tuples of integers

[a_{0}, …, a_{q}] (0 ≤ a_{0} < a_{1} < ⋯ < a_{q} ≤ n);

the boundary of [a_{0}, …, a_{q}] for q > 0 is the alternating sum

[a_{1}, …, a_{q}] − [a_{0}, a_{2}, …, a_{q}] + ⋯ + (−1)^{q}[a_{0}, …, a_{q−1}];

the augmentation is given by ϵ[a_{0}] = 1.

We define morphisms ∂_{i}^{∨} and ϵ_{i}^{∨} of free augmented directed complexes in the obvious way, so that they correspond contravariantly to face and degeneracy operations. For n > 0 and for 0 ≤ i ≤ n the face operation

∂_{i}^{∨}: Δ_{n−1} → Δ_{n}

is given on basis elements by

∂_{i}^{∨}[a_{0}, …, a_{q}] = [a′_{0}, …, a′_{q}]

such that

a′_{j} = \begin{cases}a_{j} & (0 ≤ a_{j} ≤ i − 1), \\ a_{j+1} & (i ≤ j ≤ n − 1). \end{cases}

For 0 ≤ i ≤ n the degeneracy operation

ϵ_{i}^{∨}: Δ_{n+1} → Δ_{n}

is given on a basis element [a_{0}, …, a_{q}] as follows: if the terms a_{j} do not include both i and i + 1 then

ϵ_{i}^{∨}[a_{0}, …, a_{q}] = [a′′_{0}, …, a′′_{q}]

such that

a′_{j} = \begin{cases}a_{j} & (0 ≤ a_{j} ≤ i), \\ a_{j−1} & (i + 1 ≤ j ≤ n + 1); \end{cases}

if the terms a_{j} do include both i and i + 1 then

ϵ_{i}^{∨}[a_{0}, …, a_{q}] = 0.

These morphisms satisfy the contravariant forms of the standard simplicial relations,

∂_{j}^{∨}∂_{i}^{∨} = ∂_{i}^{∨}∂_{j−1}^{∨} (i < j),
ϵ_{j}^{∨}∂_{i}^{∨} = ∂_{i}^{∨}ϵ_{j−1}^{∨} (i < j),
ϵ_{j}^{∨}∂_{j}^{∨} = ϵ_{j}^{∨}∂_{j+1}^{∨} = id,
ϵ_{j}^{∨}∂_{i}^{∨} = ∂_{i−1}^{∨}ϵ_{j}^{∨} (i > j + 1),
ϵ_{j}^{∨}ϵ_{i}^{∨} = ϵ_{i}^{∨}ϵ_{j+1}^{∨} (i ≤ j).

They therefore yield a functor from the simplex category to the category of augmented directed complexes.

It is straightforward to check that there is a functor ν from free augmented directed complexes to ω-categories defined as follows.

2.3. Notation. Let K be a free augmented directed complex. Then νK is the ω-category with the following structure. The members of νK are the infinite sequences

( x_{0}^{−}, x_{0}^{+} | x_{1}^{−}, x_{1}^{+} | …)

with finitely many non-zero terms such that x_{q}^{−} and x_{q}^{+} are sums of prescribed q-dimensional basis elements in K, such that

ϵx_{0}^{−} = ϵx_{0}^{+} = 1

and such that

∂x_{q}^{−} = ∂x_{q}^{+} = x_{q−1}^{+} − x_{q−1}^{−} (q > 0).

The operations d_{n}^{−} and d_{n}^{+} are given by

d_{n}^{α}( x_{0}^{−}, x_{0}^{+} | …) = ( x_{0}^{−}, x_{0}^{+} | … | x_{n−1}^{−}, x_{n−1}^{+} | x_{n}^{α}, x_{n}^{α} | 0, 0 | …).

If d_{n}^{+}x = d_{n}^{−}y = z, say, then

x #_{n} y = x − z + y,

where the addition and subtraction are performed termwise.

By composing the functor [n] ↦ Δ_{n} with the functor ν we obtain a functor from the simplex category to the category of ω-categories. The nth oriental O_{n} may be defined by the formula

O_{n} = νΔ_{n}.

The nerve of an ω-category is then defined as follows.

2.4. Definition. The nerve of an ω-category C is the simplicial set NC given by

N_{n}C = Hom(O_{n}, C) = Hom(νΔ_{n}, C).

3. Nerves as stratified simplicial sets

A complicial set is defined to be a simplicial set with a class of thin elements subject to certain conditions; in particular they must be stratified simplicial sets in the sense of Definition 3.2 below. In this section we specify the thin elements in the nerves of ω-categories and we then show that the nerves are stratified simplicial sets.

3.1. Definition. Let C be an ω-category. Then the thin elements in NC are the elements x in N_{n}C with n > 0 such that

d_{n−1}^{−}xu = d_{n−1}^{+}xu = xu

for all u ∈ νΔ_{n}

Note that the elements of νΔ_{n} are of dimension at most n, in the sense that d_{n}^{−}u = d_{n}^{+}u = u for all u ∈ νΔ_{n}. A thin element is therefore a morphism that lowers dimension.

We recall the definition of stratified simplicial sets from [9], 5.1.

3.2. Definition. A stratified simplicial set is a simplicial set together with a prescribed class of thin elements of positive dimension which includes the degenerate elements. A morphism of stratified simplicial sets is a morphism of simplicial sets taking thin elements to thin elements.

We obtain the following result.

3.3. Proposition. The nerve functor is a functor from ω-categories to stratified simplicial sets.

Proof. Let C be an ω-category. To show that NC is a stratified simplicial set we must show that the degenerate elements in NC are thin. To do this, let x be a degenerate n-dimensional element, say

x = ϵ_{i}y = y(νϵ_{i}^{∨})

with y: νΔ_{n−1} → C. We must show that d_{n−1}^{α}xu = xu for each sign α and for all u ∈ νΔ_{n}. But we have d_{n−1}^{α}v = v for all v ∈ νΔ_{n−1}, hence

d_{n−1}^{α}xu = d_{n−1}^{α}y(νϵ_{i}^{∨})u = yd_{n−1}^{α}(νϵ_{i}^{∨})u = y(νϵ_{i}^{∨})u = xu

as required.

Now let f: C → C′ be a morphism of ω-categories. To show that Nf is a morphism of stratified simplicial sets we must show that (Nf)x is thin whenever x is a thin element in N_{n}C. But for u ∈ νΔ_{n} and for each sign α we have

d_{n−1}^{α}(Nf)xu = d_{n−1}^{α}fxu = fd_{n−1}^{α}xu = fxu = (Nf)xu;

therefore (Nf)x is thin as required.

This completes the proof.

4. Complicial sets

A complicial set is a stratified simplicial set satisfying certain conditions on horn-fillers. In this section we give the definition, following [9], 6–7.

Recall that, for n > 0 and for 0 ≤ k ≤ n, an (n−1)-dimensional k-horn in an arbitrary simplicial set is a sequence of (n − 1)-dimensional elements

z = (z_{0}, …, z_{k−1}, z_{k+1}, …, z_{n})

such that

∂_{i}z_{j} = ∂_{j−1}z_{i} (i < j);

thus the elements are like the codimension one faces of an n-dimensional element x, except that the face ∂_{k}x is omitted. A filler for an (n − 1)-dimensional k-horn z = (z_{0}, …, z_{n}) is an n-dimensional element x such that ∂_{i}x = z_{i} for i ≠ k.

In a stratified simplicial set there are distinguished classes of elements and horns which are called complicial or admissible. Consider a geometrical n-simplex with vertices 0, 1, …, n. It has faces spanned by the non-empty subsets of {0, …, n}; the face spanned by the complement of a proper subset {j(1), …, j(r)} with j(1) < ⋯ < j(r) corresponds to the iterated face operator ∂_{j(1)} ⋯ ∂_{j(r)}. Given integers n and k with 0 < k < n we will say that an n-dimensional element x is k-complicial (the term k-admissible is also used) if the iterated face ∂_{j(1)} ⋯ ∂_{j(r)}x is thin whenever the complement of {j(1), …, j(r)} includes the integers k − 1, k, k + 1. We will also say that an (n − 1)-dimensional horn z = (z_{0}, …, z_{n}) is k-complicial if the elements z_{i} are configured like the codimension one faces of a k-complicial n-dimensional element x with ∂_{k}x omitted.

Formally, it is most convenient to express these definitions recursively.

4.1. Definition. Let n and k be integers with 0 < k < n.

(1) An (n−1)-dimensional k-complicial horn in a stratified simplicial set is an (n−1)- dimensional k-horn

z = (z_{0}, …, z_{k−1}, z_{k+1}, …, z_{n})

such that z_{i} is a (k −1)-complicial element for 0 ≤ i < k −1 and is a k-complicial element for k + 1 < i ≤ n.

(2) An n-dimensional k-complicial element in a stratified simplicial set is a thin filler of an (n − 1)-dimensional k-complicial horn.

Thus a 1-dimensional 1-horn is always 1-complicial, a 2-dimensional element is 1complicial if and only if it is thin, etc.

A complicial set is then defined as follows.

4.2. Definition. A complicial set is a stratified simplicial set satisfying the following conditions.

(1) Every thin 1-dimensional element is degenerate.

(2) For 0 < k < n every (n−1)-dimensional k-complicial horn has a unique thin filler.

(3) For 0 < k < n, if x is an n-dimensional k-complicial element and if ∂_{k−1}x and ∂_{k+1}x are thin, then ∂_{k}x is thin.

We aim to show that the nerves of ω-categories are complicial sets. Most of the work is in verifying the last two conditions; we can dispose of the first condition immediately.

4.3. Proposition. Let x be a thin 1-dimensional element in the nerve of an ω-category C. Then x is degenerate.

Proof. Recall that x is a morphism of ω-categories from νΔ_{1} to C. It is straightforward to check that νΔ_{1} has exactly three members, which can be listed as

u = ( [0], [1] | [0, 1], [0, 1] | 0, 0 | …),
d_{0}^{−}u = ( [0], [0] | 0, 0 | …),
d_{0}^{+}u = ( [1], [1] | 0, 0 | …).

We have xd_{0}^{−}u = d_{0}^{−}xu = xu and xd_{0}^{+}u = d_{0}^{+}xu = xu because x is thin; therefore x is constant. It follows that x = x(ν∂_{0}^{∨})(νϵ_{0}^{∨}) = ϵ_{0}∂_{0}x; therefore x is degenerate.

5. Horns in nerves

Recall that the n-dimensional elements in the nerves of ω-categories are represented by the free augmented directed complex Δ_{n} via the functor ν. In this section we show that (n − 1)-dimensional k-horns are similarly represented by a subcomplex Λ_{n}^{k} of Δ_{n}.

The subcomplex Λ_{n}^{k} is as one would expect.

5.1. Notation. If n > 0 and if 0 ≤ k ≤ n then Λ_{n}^{k} is the free augmented directed subcomplex of Δ_{n} given by

Λ_{n}^{k} = ∂_{0}^{∨}Δ_{n−1} + ⋯ + ∂_{k−1}^{∨}Δ_{n−1} + ∂_{k+1}^{∨}Δ_{n−1} + ⋯ + ∂_{n}^{∨}Δ_{n−1}.

The prescribed basis for Λ_{n}^{k} is the prescribed basis for Δ_{n} with the elements

[0, 1, …, k − 1, k + 1, …, n − 1, n], [0, 1, …, n − 1, n]

omitted.

For n ≥ 2 and for 0 ≤ i < j ≤ n we have

∂_{i}^{∨}Δ_{n−1} ∩ ∂_{j}^{∨}Δ_{n−1} = ∂_{j}^{∨}∂_{i}^{∨}Δ_{n−2} = ∂_{i}^{∨}∂_{j−1}^{∨}Δ_{n−2}.

In the category of abelian groups this gives a description of Λ_{n}^{k} as a colimit. To show that Λ_{n}^{k} represents horns in nerves, we must show that νΛ_{n}^{k} is a similar colimit in the category of ω-categories. We will do this by using results from [2], which we now summarise.

Let K be a free augmented directed complex. Given a chain c in K, we write ∂^{+}c and ∂^{−}c for the positive and negative parts of the boundary ∂c; in other words, ∂^{+}c and ∂^{−}c are the sums of basis elements without common terms such that

∂c = ∂^{+}c − ∂^{−}c.

Given basis elements a, a′ of the same dimension q, we will write a < a′ if there is a basis element b of dimension q + r with r > 0 such that a is a term in (∂^{−})^{r}b and such that a′ is a term in (∂^{+})^{r}b.

5.2. Definition. A free augmented directed complex is unital if ϵ(∂^{−})^{p}a = ϵ(∂^{+})^{p}a = 1 for every basis element a, where p is the dimension of a.

A free augmented directed complex is loop-free if the transitive closure of the relation < is a strict partial order on its basis elements.

5.3. Definition. Let a be a p-dimensional basis element in a unital loop-free directed complex K. The the associated atom is the member ⟨a⟩ of νK given by

⟨a⟩ = ( (∂^{−})^{p}a, (∂^{+})^{p}a | ⋯ | ∂^{−}a, ∂^{+}a | a, a | 0, 0 | ⋯).

Note that ∂∂^{−} = ∂∂^{+} because ∂∂ = 0, hence ∂^{α}∂^{−} = ∂^{α}∂^{+} for each sign α; it follows that in Definition 5.3 ⟨a⟩ really is a member of νK.

The main result that we will use is the following ([2], Theorem 6.1).

5.4. Theorem. If K is a unital loop-free free augmented directed complex then the ωcategory νK has a presentation as follows. The generators are the atoms. For a basis element a of arbitrary dimension p there are relations

d_{p}^{−}⟨a⟩ = d_{p}^{+}⟨a⟩ = ⟨a⟩.

For a basis element a of positive dimension p there are relations

d_{p−1}^{−}⟨a⟩ = w^{−}(a), d_{p−1}^{+}⟨a⟩ = w^{+}(a),

where w^{−}(a) and w^{+}(a) are arbitrarily chosen expressions for d_{p−1}^{−}⟨a⟩ and d_{p−1}^{+}⟨a⟩ as iterated composites of atoms of dimension less than p.

We will now show that this theorem applies to simplexes, essentially by computing the atoms.

5.5. Definition. Let

a = [i_{0}, …, i_{p}]

be a p-dimensional basis element in a simplex Δ_{n}. Then a block in a is a non-empty sequence of consecutive terms

u = i_{r}, i_{r+1}, …, i_{s}.

A block is even if it has an even number of terms, it is odd if it has an odd number of terms, it is initial if it includes i_{0}, and it is final if it includes i_{p}.

5.6. Proposition. Let a be a p-dimensional element in a simplex Δ_{n} and let q be an integer with 0 ≤ q ≤ p. Then (∂^{−})^{p−q}a is the sum of the q-dimensional basis elements with expressions

[u_{0}, …, u_{k}] (k ≥ 0)

such that u_{0} is an odd initial block in a and such that u_{1}, …, u_{k} are even or final blocks in a. In the same way (∂^{+})^{p−q}a is the sum of the q-dimensional basis elements with expressions

[u_{0}, …, u_{k}] (k ≥ 0)

such that u_{0}, …, u_{k} are even or final blocks in a.

Proof. This is a straightforward downward induction on q.

5.7. Proposition. The free augmented directed complexes Δ_{n} and Λ_{n}^{k} are unital and loop-free.

Proof. We make the following deductions from Proposition 5.6.

If a = [i_{0}, …, i_{p}] is a p-dimensional basis element, then

ϵ(∂^{−})^{p}a = ϵ[i_{0}] = 1, ϵ(∂^{+})^{p}a = ϵ[i_{p}] = 1;

therefore the complexes are unital.

If a and a′ are basis elements such that a < a′, say

a = [i_{0}, …, i_{p}], a′ = [i′_{0}, …, i′_{p}]

and if i_{0} = i′_{0}, …, i_{q−1} = i′_{q−1}, i_{q} ≠ i′_{q}, then (−1)^{q}i_{q} < (−1)^{q}i′_{q}. The transitive closure of < is therefore contained in a total ordering of the basis elements and it follows that the complexes are loop-free.

For n ≥ 2 it now follows from Theorem 5.4 that νΛ_{n}^{k} is the obvious colimit of copies of νΔ_{n−1} and νΔ_{n−2}. If n = 1 then νΛ_{n}^{k} is obviously just a copy of νΔ_{n−1}. We deduce the following result.

5.8. Theorem. Let n and k be integers with n > 0 and 0 ≤ k ≤ n, and let C be an ω-category. Then the function

Hom(νΛ_{n}^{k}, C) → Hom(νΔ_{n−1}, C) × ⋯ × Hom(νΔ_{n−1}, C)

induced by ( ν∂_{i}^{∨} : i ≠ k) is a bijection whose image consists of the (n − 1)-dimensional k-horns in the nerve of C. The function

Hom(νΔ_{n}, C) → Hom(νΛ_{n}^{k}, C)

induced by the inclusion νΛ_{n}^{k} → νΔ_{n} sends n-dimensional elements to the k-horns which they fill.

6. Horns represented by pairs of faces

Let n and k be integers with 0 < k < n. We have constructed a subcomplex Λ_{n}^{k} in Δ_{n} which represents (n − 1)-dimensional k-horns (Theorem 5.8). We will now construct a subcomplex V_{n}^{k} of Λ_{n}^{k} and a retraction

Π_{n}^{k}: Δ_{n} → V_{n}^{k}.

Eventually we will show that V_{n}^{k} represents n-dimensional k-complicial elements, and that Π_{n}^{k} represents the inclusion of these elements in the set of all n-dimensional elements. We will also show that V_{n}^{k} represents (n − 1)-dimensional k-complicial horns and that the restriction

π_{n}^{k} = Π_{n}^{k}|Λ_{n}^{k}: Λ_{n}^{k} → V_{n}^{k}

represents the inclusion of these horns in the set of all (n − 1)-dimensional k-horns. We will now define the subcomplex V_{n}^{k} and the retraction Π_{n}^{k}.

6.1. Notation. If n > 0 and 0 < k < n then V_{n}^{k} is the free augmented directed subcomplex of Δ_{n} given by

V_{n}^{k} = ∂_{k−1}^{∨}Δ_{n−1} + ∂_{k+1}^{∨}Δ_{n−1}.

The prescribed basis elements for V_{n}^{k} are the prescribed basis elements [i_{0}, …, i_{q}] for Δ_{n} not including both k − 1 and k + 1.

The results for Δ_{n} and Λ_{n}^{k} given in Proposition 5.7 obviously apply to Λ_{n}^{k} as well.

6.2. Proposition. The free augmented directed complex V_{n}^{k} is unital and loop-free.

6.3. Notation. Let n and k be integers with 0 < k < n. Then

Π_{n}^{k}: Δ_{n} → V_{n}^{k}

is the morphism given on basis elements a = [i_{0}, …, i_{q}] as follows. If a does not include k − 1 or does not include k + 1 then Π_{n}^{k}a = a. If a includes k − 1, k and k + 1 then Π_{n}^{k}a = 0. If a includes k − 1 and k + 1 but not k, say

a = [u, k − 1, k + 1, v],

then

Π_{n}^{k}a = [u, k, k + 1, v] + [u, k − 1, k, v].

We also write π_{n}^{k} for the restriction

π_{n}^{k} = Π_{n}^{k}|Λ_{n}^{k}: Λ_{n}^{k} → V_{n}^{k}.

It is straightforward to show Π_{n}^{k} and π_{n}^{k} are augmentation-preserving chain maps. Since they take basis elements to sums of basis elements, they are morphisms of free augmented directed complexes. It is also straightforward to check that they have the following properties.

6.4. Proposition. The morphisms of free augmented directed complexes Π_{n}^{k} and π_{n}^{k} are retractions such that

π_{n}^{k}∂_{i}^{∨} = ∂_{i}^{∨}Π_{n−1}^{k−1} (0 ≤ i < k − 1),
π_{n}^{k}∂_{k−1}^{∨} = ∂_{k−1}^{∨},
π_{n}^{k}∂_{k+1}^{∨} = ∂_{k+1}^{∨},
π_{n}^{k}∂_{i}^{∨} = ∂_{i}^{∨}Π_{n−1}^{k} (k + 1 < i ≤ n).

We immediately obtain an analogue for Definition 4.1(1).

6.5. Proposition. Let C be an ω-category, let z be a morphism in Hom(νΛ_{n}^{k}, C), and let

z_{i} = z(ν∂_{i}^{∨}) (0 ≤ i ≤ n, i ≠ k).

Then z is of the form y(νπ_{n}^{k}) if and only if z_{i} is of the form y_{i}(νΠ_{n−1}^{k−1}) for 0 ≤ i < k − 1 and of the form y_{i}(νΠ_{n−1}^{k}) for k + 1 < i ≤ n.

This shows that composites with morphisms of the forms νπ_{n}^{k} and νΠ_{n−1}^{k} are somewhat analogous to (n − 1)-dimensional complicial horns and (n − 1)-dimensional complicial elements (see Definition 4.1(1)). To complete the analogy we must relate composites with νΠ_{n}^{k} to thin fillers of composites with νπ_{n}^{k}. We will do this in the next section.

7. Fillers for horns represented by pairs of faces

In this section let n and k be fixed integers with 0 < k < n. We have seen in Theorem 5.8 that the process of filling (n − 1)-dimensional k-horns in the nerves of ω-categories is represented by the inclusion of Λ_{n}^{k} in Δ_{n}. In Section 6 we have constructed a retraction π_{n}^{k} from Λ_{n}^{k} to V_{n}^{k} which is intended to represent the inclusion of the set of (n−1)-dimensional k-complicial horns in the set of arbitrary (n − 1)-dimensional k-horns. We will now construct an extension W of V_{n}^{k} and a morphism Π: Δ_{n} → W extending π_{n}^{k} such that the square

\xymatrix{νΛ_{n}^{k} \ar[d]_{νπ_{n}^{k}} \ar[r]^{⊂} & νΔ_{n} \ar[d]^{νΠ} \\ νV_{n}^{k} \ar[r]^{⊂} & νW}

is a push-out square. The fillers for horns represented by V_{n}^{k} will then be represented by W. We will also show that any such horn has a unique thin filler.

We will use the following notation.

7.1. Notation. In Δ_{n} let s, t_{0}, …, t_{n} be the basis elements of dimensions n and n − 1 given by

s = [0, 1, …, n],
t_{i} = [0, …, i − 1, i + 1, …, n] (0 ≤ i ≤ n).

Let κ be the sign given by

κ = (−)^{k}.

Note that Δ_{n} is obtained from Λ_{n}^{k} by adjoining the basis elements t_{k} and s. Note also that

∂s = ∑_{i=0}^{n}(−1)^{i}t_{i}

with

π_{n}^{k}t_{k−1} = t_{k−1},
π_{n}^{k}t_{k+1} = t_{k+1},
π_{n}^{k}t_{i} = 0 (i ≠ k − 1, k, k + 1).

We can therefore construct a free augmented directed complex W and a morphism of free augmented directed complexes Π: Δ_{n} → W in the following way.

7.2. Notation. Let W be the free augmented directed complex obtained from V_{n}^{k} by adjoining an (n − 1)-dimensional basis element t′_{k} and an n-dimensional basis element s′ with

∂t′_{k} = π_{n}^{k}∂t_{k},
∂s′ = (−1)^{k}(t′_{k} − t_{k−1} − t_{k+1}).

Let Π: Δ_{n} → W be the morphism given by

Π|Λ_{n}^{k} = π_{n}^{k}, Πt_{k} = t′_{k}, Πs = s′.

We will obtain properties of νW by using Theorem 5.4. In order to do this, we must show that W is unital and loop-free. We therefore make the following calculations.

7.3. Proposition. Let a be a basis element in W which is a term in Π(∂^{α})^{r+1}s′ or Π(∂^{α})^{r}t′_{k} for some sign α. If a is a basis element in V_{n}^{k} omitting k − 1 then a is a term in (∂^{α})^{r}t_{k−1}; if a is a basis element in V_{n}^{k} omitting k + 1 then a is a term in (∂^{α})^{r}t_{k+1}.

Proof. Let b be a basis element in Δ_{n} which is a term in (∂^{α})^{r+1}s or (∂^{α})^{r}t_{k}; then b consists of blocks of consecutive entries in s or t_{k} satisfying the conditions of Proposition 5.6. It is straightforward to check that any term of Πb in V_{n}^{k} omitting k − 1 consists of blocks of consecutive entries in t_{k−1} satisfying the same conditions and is therefore a term in (∂^{α})^{r}t_{k−1}. A similar argument applies to terms omitting k + 1.

7.4. Proposition. Let α be any sign; then

(∂^{α})^{r}s′ = Π(∂^{α})^{r}s (0 ≤ r ≤ n),
(∂^{α})^{r}t′_{k} = Π(∂^{α})^{r}t_{k} (0 ≤ r ≤ n − 1).

Proof. We first prove the result for (∂^{α})^{r}s by induction on r. The result holds for r = 0 and for r = 1 because

s′ = Πs,
∂^{κ}s′ = t′_{k} = Π∂^{κ}s,
∂^{−κ}s′ = t_{k−1} + t_{k+1} = Π∂^{−κ}s.

In cases with 2 ≤ r ≤ n assume that (∂^{−})^{r−1}s′ = Π(∂^{−})^{r−1}s. Then

∂(∂^{−})^{r−1}s′ = ∂Π(∂^{−})^{r−1}s = Π∂(∂^{−})^{r−1}s = Π(∂^{+})^{r}s − Π(∂^{−})^{r}s.

By construction, (∂^{+})^{r}s′ and (∂^{−})^{r}s′ are sums of basis elements without common terms such that

∂(∂^{−})^{r−1}s′ = (∂^{+})^{r}s′ − (∂^{−})^{r}s′.

It follows from Proposition 7.3 that Π(∂^{+})^{r}s and Π(∂^{−})^{r}s are also sums of basis elements without common terms, and it then follows that

(∂^{α})^{r}s′ = Π(∂^{α})^{r}s

for each sign α.

A similar argument applies to (∂^{α})^{r}t′_{k}, starting with t′_{k} = Πt_{k}.

7.5. Proposition. Let a and b be basis elements in W such that a is a term in (∂^{α})^{r}s′ or (∂^{α})^{r}t′_{k} for some α and such that a is also a term in (∂^{−α})^{q}b. Then r = 0 or q = 0.

Proof. By construction, if r > 0 then (∂^{−})^{r}s′ and (∂^{+})^{r}s′ have no common terms. For r > 1 we have

(∂^{−})^{r}s′ = (∂^{−})^{r−1}t′_{k}, (∂^{+})^{r}s′ = (∂^{+})^{r−1}t′_{k},

because ∂^{κ}s′ = t′_{k}. This gives the result for b = s′ and for b = t′_{k}.

Now let b be a basis element in V_{n}^{k} omitting k − 1. Since a is a term in (∂^{−α})^{q}b, it follows that a is also a basis element in V_{n}^{k} omitting k − 1. By Proposition 7.3, a is a term in (∂^{α})^{r−1}t_{k−1} or (∂^{α})^{r}t_{k−1}. It now follows from Proposition 5.6 that a is a term in (∂^{α})^{q}b, because terms of a which are consecutive in t_{k−1} are also consecutive in b. This makes a a term in (∂^{α})^{q}b and in (∂^{−α})^{q}b; therefore q = 0.

A similar argument applies when b is a basis element in V_{n}^{k} omitting k + 1.

7.6. Theorem. The free augmented directed complex W is unital and loop-free.

Proof. By Proposition 6.2 the subcomplex V_{n}^{k} is unital and loop-free; it therefore suffices to consider the effect of the additional basis elements s′ and t′_{k}.

Since Π is augmentation-preserving and Δ_{n} is unital, it follows from Proposition 7.4 that

ϵ(∂^{α})^{n}s′ = ϵΠ(∂^{α})^{n}s = ϵ(∂^{α})^{n}s = 1

and it similarly follows that ϵ(∂^{α})^{n−1}t′_{k} = 1; therefore W is unital.

Next we consider the relations a < a′ between basis elements of W. The transitive closure of the relations coming from V_{n}^{k} is a partial ordering because V_{n}^{k} is loop-free. If a < a′ is a relation coming from the extra basis elements s′ and t′_{k}, then it follows from Proposition 7.5 that there are no relations of the form c < a or a′ < c′. The transitive closure therefore remains a partial ordering even after the extra relations are included, and it follows that W is loop-free.

This completes the proof.

It follows from Theorems 7.6 and 5.4 that νW has a presentation in terms of its atoms. For these atoms we have the following results.

7.7. Proposition. The atoms ⟨s′⟩ and ⟨t′_{k}⟩ in νW are such that

⟨s′⟩ = (νΠ)⟨s⟩, ⟨t′_{k}⟩ = (νΠ)⟨t_{k}⟩

and

d_{n−1}^{κ}⟨s′⟩ = ⟨t′_{k}⟩, d_{n−1}^{−κ}⟨s′⟩ ∈ νV_{n}^{k}.

Proof. The equalities ⟨s′⟩ = (νΠ)⟨s⟩ and ⟨t′_{k}⟩ = (νΠ)⟨t_{k}⟩ are restatements of Proposition 7.4. The equality d_{n−1}^{κ}⟨s′⟩ = ⟨t′_{k}⟩ holds because ∂^{κ}s′ = t′_{k}. We have d_{n−1}^{−κ}⟨s′⟩ ∈ νV_{n}^{k} because

∂^{−κ}s′ = t_{k−1} + t_{k+1} ∈ V_{n}^{k}.

It follows that νW is a push-out in the required way.

7.8. Theorem. The square of ω-categories

\xymatrix{νΛ_{n}^{k} \ar[d]_{νπ_{n}^{k}} \ar[r]^{⊂} & νΔ_{n} \ar[d]^{νΠ} \\ νV_{n}^{k} \ar[r]^{⊂} & νW}

is a push-out square. The ω-category νW is generated as an extension of νV_{n}^{k} by the atom ⟨s′⟩.

Proof. It follows from Theorem 5.4, Theorem 7.6 and Proposition 7.7 that νW is generated as an extension of νV_{n}^{k} by ⟨s′⟩ and ⟨t′_{k}⟩ subject to relations which are the images under νΠ of relations in νΔ_{n}. This shows that the square is a push-out square. Since ⟨t′_{k}⟩ = d_{n−1}^{κ}⟨s′⟩, the single atom ⟨s′⟩ suffices to generate νW as an extension of νV_{n}^{k}.

We can now prove the main theorem of this section.

7.9. Theorem. Let C be an ω-category and let y be a morphism in Hom(νV_{n}^{k}, C). Then the composite

y(νΠ_{n}^{k}) ∈ Hom(νΔ_{n}, C)

is the unique thin filler for the (n − 1)-dimensional k-horn represented by

y(νπ_{n}^{k}) ∈ Hom(νΛ_{n}^{k}, C).

Proof. The morphism y(νΠ_{n}^{k}) is certainly a filler for y(νπ_{n}^{k}) because Π_{n}^{k}|Λ_{n}^{k} = π_{n}^{k}. It is a thin filler because d_{n−1}^{α}v = v for all v ∈ νV_{n}^{k}.

It now suffices to show that there is at most one thin filler. To do this, let x be an arbitrary thin filler. By Theorem 7.8 there is a morphism

w ∈ Hom(νW, C)

such that

w|νV_{n}^{k} = y, w(νΠ) = x.

By Proposition 7.7, w⟨s′⟩ = x⟨s⟩. Since x is thin, w⟨s′⟩ = d_{n−1}^{−κ}w⟨s′⟩. By another application of Proposition 7.7,

d_{n−1}^{−κ}w⟨s′⟩ = wd_{n−1}^{−κ}⟨s′⟩ = yd_{n−1}^{−κ}⟨s′⟩;

therefore

w⟨s′⟩ = yd_{n−1}^{−κ}⟨s′⟩.

Since νW is generated by ⟨s′⟩ as an extension of νV_{n}^{k} and since x is w(νΠ) it follows that x is uniquely determined by y. This completes the proof.

8. Nerves as complicial sets

We have shown in Proposition 3.3 that the nerves of ω-categories are naturally stratified simplicial sets. We will now show that they are complicial sets, with complicial elements represented by the complexes V_{n}^{k}.

8.1. Theorem. Let C be an ω-category. For integers n and k with 0 < k < n, an (n−1)-dimensional k-horn in the nerve of C is k-complicial if and only if it is represented by a composite y(νπ_{n}^{k}) with y ∈ Hom(νV_{n}^{k}, C), and an n-dimensional element in the nerve of C is k-complicial if and only if it is represented by a composite y(νΠ_{n}^{k}) with y ∈ Hom(νV_{n}^{k}, C).

Proof. The proof is by induction on n.

Suppose that (n − 1)-dimensional elements are (k − 1)-complicial if and only if they factor through νΠ_{n−1}^{k−1} and that they are k-complicial if and only if they factor through νΠ_{n−1}^{k}. Then it follows from Proposition 6.4 that an (n − 1)-dimensional k-horn

(z_{0}, …, z_{k−1}, z_{k+1}, …, z_{n})

factors through νπ_{n}^{k} if and only if z_{i} is (k − 1)-complicial for 0 ≤ i < k − 1 and z_{i} is k-complicial for k + 1 < i ≤ n. This means that an (n − 1)-dimensional k-horn factors through νπ_{n}^{k} if and only if it is k-complicial. It now follows from Theorem 7.9 that an n-dimensional element is a thin filler of an (n − 1)-dimensional k-complicial horn if and only if it factors through νΠ_{n}^{k}; therefore an n-dimensional element is k-complicial if and only if it factors through νΠ_{n}^{k}. This completes the proof.

8.2. Theorem. If C is an ω-category then the nerve of C is a complicial set.

Proof. By Proposition 3.3 the nerve is a stratified simplicial set. By Proposition 4.3 every thin 1-dimensional element in the nerve is degenerate. By Theorems 7.9 and 8.1 complicial horns in the nerve have unique thin fillers. It remains to verify the condition of Definition 4.2(3).

Let x be an n-dimensional k-complicial element in the nerve such that ∂_{k−1}x and ∂_{k+1}x are thin; we must show that ∂_{k}x is thin. Equivalently, for each sign α, we must show that d_{n−2}^{α}x(ν∂_{k}^{∨}) = x(ν∂_{k}^{∨}). We will do this by showing that d_{n−2}^{α}x = x. Since x factors through the retraction νΠ_{n}^{k}: νΔ_{n} → νV_{n}^{k} it suffices to show that d_{n−2}^{α}xv = xv for all v ∈ νV_{n}^{k}. Since νV_{n}^{k} is generated by its atoms (see Theorem 5.4 and Proposition 6.2) it suffices by Proposition 2.1 to show that d_{n−2}^{α}x⟨a⟩ = x⟨a⟩ for every basis element a in V_{n}^{k}. For basis elements a of dimension less than n − 1 this is immediate. For a basis element a of dimension n − 1 we have

x⟨a⟩ = x⟨t_{k±1}⟩ ∈ im x(ν∂_{k±1}^{∨}) = im ∂_{k±1}x,

hence d_{n−2}^{α}x⟨a⟩ = x⟨a⟩ because ∂_{k±1}x is thin. This completes the proof.

9. Complicial sets and complicial identities

We have shown in Theorem 8.2 that the nerves of ω-categories are complicial sets. In this section we will show that complicial sets are sets with complicial identities in the sense of the following definition ([4], Definition 6.1).

9.1. Definition. A set with complicial identities is a simplicial set X, together with wedges

x ∧_{i} y ∈ X_{m+1},

defined when x, y ∈ X_{m} and ∂_{i}x = ∂_{i+1}y, such that the following axioms hold.

(1) If x ∧_{i} y is defined with x, y ∈ X_{m}, then

∂_{j}(x ∧_{i} y) = ∂_{j}x ∧_{i−1} ∂_{j}y (0 ≤ j < i),
∂_{i}(x ∧_{i} y) = y,
∂_{i+2}(x ∧_{i} y) = x,
∂_{j}(x ∧_{i} y) = ∂_{j−1}x ∧_{i} ∂_{j−1}y (i + 2 < j ≤ m + 1).

(2) If x ∈ X_{m} and 0 ≤ i < m then

ϵ_{i}x = ϵ_{i}∂_{i+1}x ∧_{i} x, ϵ_{i+1}x = x ∧_{i} ϵ_{i}∂_{i}x.

(3) If A is of the form b ∧_{i} (y ∧_{i} z) then

A = (∂_{i+2}b ∧_{i} y) ∧_{i+1} ∂_{i+1}A.

(4) If A is of the form (x ∧_{i} y) ∧_{i+1} c then

A = ∂_{i+2}A ∧_{i} (y ∧_{i} ∂_{i}c).

(5) The equality

[x ∧_{i} ∂_{i+1}(y ∧_{i} z)] ∧_{i} (y ∧_{i} z) = (x ∧_{i} y) ∧_{i+1} [∂_{i+1}(x ∧_{i} y) ∧_{i} z]

holds whenever either side is defined.

(6) If A is of the form ∂_{i+2}[(x ∧_{i+1} y) ∧_{i+1} (y ∧_{i} z)], then the equality

A ∧_{i} (w ∧_{i+1} ∂_{i}A) = (∂_{i+3}A ∧_{i} w) ∧_{i+2} A

holds whenever either side is defined.

(7) If i ≤ j − 3 then the equality

(x ∧_{i} y) ∧_{j} (z ∧_{i} w) = (x ∧_{j−1} z) ∧_{i} (y ∧_{j−1} w)

holds whenever either side is defined.

The object of this section is to prove the following result.

9.2. Theorem. Let X be a complicial set. Then there is a unique family of wedge operations making X into a set with complicial identities such that the wedges x ∧_{i} y are (i + 1)-complicial elements.

Since complicial elements in complicial sets are uniquely determined by their faces, the construction of Theorem 9.2 is functorial.

The proof of Theorem 9.2 uses the method appearing in Street’s notes [6]. It consists of a sequence of lemmas. In all of these lemmas, X is a complicial set.

9.3. Lemma. There is a unique way to construct wedges

x ∧_{i} y ∈ X_{m+1}

such that x ∧_{i} y is defined for x, y ∈ X_{m} with ∂_{i}x = ∂_{i+1}y, such that x ∧_{i} y is (i + 1)- complicial, and such that the conditions of Definition 9.1(1) are satisfied.

Proof. The proof is by induction on m. Let x, y ∈ X_{m} be such that ∂_{i}x = ∂_{i+1}y and assume that we have constructed wedges with the required properties for pairs of elements with dimensions less than m. We will show that there is an m-dimensional (i + 1)-complicial horn

(z_{0}, …, z_{i}, z_{i+2}, …, z_{m+1})

given by

z_{j} = ∂_{j}x ∧_{i−1} ∂_{j}y (0 ≤ j < i),
z_{i} = y,
z_{i+2} = x,
z_{j} = ∂_{j−1}x ∧_{i} ∂_{j−1}y (i + 2 < j ≤ m + 1).

The conditions on x ∧_{i} y amount to requiring it to be a thin filler of this horn. Since complicial horns in X have unique thin fillers, this means that there will be a unique wedge x ∧_{i} y with the required properties.

It remains to show that (z_{0}, …, z_{m+1}) really is an m-dimensional (i + 1)-complicial horn. It is straightforward to verify that ∂_{k}z_{j} = ∂_{j−1}z_{k} for k < j by using Definition 9.1(1) inductively. It also follows from the inductive hypothesis that z_{j} is i-complicial for j < i and that z_{j} is (i + 1)-complicial for j > i + 2. Therefore (z_{0}, …, z_{m+1}) is indeed an m-dimensional (i + 1)-complicial horn. This completes the proof.

We must now verify the conditions of Definition 9.1(2)–(7). Essentially, we need to show that elements of certain forms are wedges. We will write im ∧_{i} for the subset of X consisting of the wedges x ∧_{i} y and we will use the following result.

9.4. Lemma. Let A be a thin (m + 1)-dimensional element of X such that ∂_{j}A ∈ im ∧_{i−1} for 0 ≤ j < i and such that ∂_{j}A ∈ im ∧_{i} for i + 2 < j ≤ m + 1. Then

A = ∂_{i+2}A ∧_{i} ∂_{i}A.

Proof. There is a wedge ∂_{i+2}A ∧_{i} ∂_{i}A, because ∂_{i}∂_{i+2}A = ∂_{i+1}∂_{i}A, and this wedge is an (i + 1)-complicial element. The element A is also (i + 1)-complicial, because elements of im ∧_{i−1} are i-complicial and elements of im ∧_{i} are (i + 1)-complicial. Since (i + 1)- complicial elements in the complicial set X are uniquely determined by the (i + 1)-horns which they fill, it suffices to prove that

∂_{j}A = ∂_{j}(∂_{i+2}A ∧_{i} ∂_{i}A) (j ≠ i + 1).

We use the conditions of Definition 9.1(1): for 0 ≤ j < i we have ∂_{j}A = x_{j} ∧_{i−1} y_{j} such that

x_{j} = ∂_{i+1}(x_{j} ∧_{i−1} y_{j}) = ∂_{i+1}∂_{j}A = ∂_{j}∂_{i+2}A,
y_{j} = ∂_{i−1}(x_{j} ∧_{i−1} y_{j}) = ∂_{i−1}∂_{j}A = ∂_{j}∂_{i}A,

hence

∂_{j}A = ∂_{j}∂_{i+2}A ∧_{i−1} ∂_{j}∂_{i}A = ∂_{j}(∂_{i+2}A ∧_{i} ∂_{i}A);

we certainly have

∂_{i}(∂_{i+2}A ∧_{i} ∂_{i}A) = ∂_{i}A,
∂_{i+2}(∂_{i+2}A ∧_{i} ∂_{i}A) = ∂_{i+2}A;

for i + 2 < j ≤ m + 1 we have ∂_{j}A = x_{j} ∧_{i} y_{j} such that

x_{j} = ∂_{i+2}(x_{j} ∧_{i} y_{j}) = ∂_{i+2}∂_{j}A = ∂_{j−1}∂_{i+2}A,
y_{j} = ∂_{i}(x_{j} ∧_{i} y_{j}) = ∂_{i}∂_{j}A = ∂_{j−1}∂_{i}A,

hence

∂_{j}A = ∂_{j−1}∂_{i+2}A ∧_{i} ∂_{j−1}∂_{i}A = ∂_{j}(∂_{i+2}A ∧_{i} ∂_{i}A).

This completes the proof.

When applying this result, we need to show that certain elements are thin. We will use the following result.

9.5. Lemma. Let x ∧_{i} y be a wedge in X; then x ∧_{i} y is thin. If also x and y are thin then ∂_{i+1}(x ∧_{i} y) is thin.

Proof. By construction, x ∧_{i} y is an (i + 1)-complicial element; therefore x ∧_{i} y is thin. If x and y are thin then ∂_{i}(x ∧_{i} y) and ∂_{i+2}(x ∧_{i} y) are the thin elements y and x, and it follows from Definition 4.2(3) that ∂_{i+1}(x ∧_{i} y) is thin.

We will now verify the conditions of Definition 9.1(2)–(7).

9.6. Lemma. The wedges in X satisfy the conditions of Definition 9.1(2)–(4).

Proof. In each case we proceed by induction on dimension and apply Lemma 9.4. We use Lemma 9.5 to recognise thin elements and Definition 9.1(1) to compute faces.

(2) Suppose that x ∈ X_{m} and that 0 ≤ i < m. The elements ϵ_{i}x and ϵ_{i+1}x are thin because they are degenerate (Definition 3.2). For 0 ≤ j < i it follows from the inductive hypothesis that

∂_{j}ϵ_{i}x = ϵ_{i−1}∂_{j}x ∈ im ∧_{i−1}, ∂_{j}ϵ_{i+1}x = ϵ_{i}∂_{j}x ∈ im ∧_{i−1};

for i + 2 < j ≤ m + 1 it similarly follows that

∂_{j}ϵ_{i}x = ϵ_{i}∂_{j−1}x ∈ im ∧_{i}, ∂_{j}ϵ_{i+1}x = ϵ_{i+1}∂_{j−1}x ∈ im ∧_{i}.

By Lemma 9.4,

ϵ_{i}x = ∂_{i+2}ϵ_{i}x ∧_{i} ∂_{i}ϵ_{i}x = ϵ_{i}∂_{i+1}x ∧_{i} x,
ϵ_{i+1}x = ∂_{i+2}ϵ_{i+1}x ∧_{i} ∂_{i}ϵ_{i+1}x = x ∧_{i} ϵ_{i}∂_{i}x.

(3) Let A be an element of the form b ∧_{i} (y ∧_{i} z). Then A is thin because A is a wedge. For j < i it follows from the inductive hypothesis that

∂_{j}A = ∂_{j}b ∧_{i−1} (∂_{j}y ∧_{i−1} ∂_{j}z) ∈ im ∧_{i},

we certainly have

∂_{i}A = y ∧_{i} z ∈ im ∧_{i},

and for j > i + 3 it follows from the inductive hypothesis that

∂_{j}A = ∂_{j−1}b ∧_{i} (∂_{j−2}y ∧_{i} ∂_{j−2}z) ∈ im ∧_{i+1};

therefore

A = ∂_{i+3}A ∧_{i+1} ∂_{i+1}A = (∂_{i+2}b ∧_{i} y) ∧_{i+1} ∂_{i+1}A.

(4) Let A be an element of the form (x ∧_{i} y) ∧_{i+1} c. Then A is thin because A is a wedge. For j < i it follows from the inductive hypothesis that

∂_{j}A = (∂_{j}x ∧_{i−1} ∂_{j}y) ∧_{i} ∂_{j}c ∈ im ∧_{i−1},

we certainly have

∂_{i+3}A = x ∧_{i} y ∈ im ∧_{i},

and for j > i + 3 it follows from the inductive hypothesis that

∂_{j}A = (∂_{j−2}x ∧_{i} ∂_{j−2}y) ∧_{i+1} ∂_{j−1}c ∈ im ∧_{i};

therefore

A = ∂_{i+2}A ∧_{i} ∂_{i}A = ∂_{i+2}A ∧_{i} (y ∧_{i} ∂_{i}c).

9.7. Lemma. The wedges in X satisfy the conditions of Definition 9.1(5).

Proof. Note that the conditions for the two sides of Definition 9.1(5) to be defined are the same, namely ∂_{i}x = ∂_{i+1}y and ∂_{i}y = ∂_{i+1}z. We may therefore assume that the left side is defined. Let

A = b ∧_{i} (y ∧_{i} z)

with

b = x ∧_{i} ∂_{i+1}(y ∧_{i} z);

we must show that

A = (x ∧_{i} y) ∧_{i+1} [∂_{i+1}(x ∧_{i} y) ∧_{i} z].

But it follows from Definition 9.1(3) (already proved in Lemma 9.6) that

A = (∂_{i+2}b ∧_{i} y) ∧_{i+1} ∂_{i+1}A = (x ∧_{i} y) ∧_{i+1} ∂_{i+1}A,

and it therefore suffices to prove that

∂_{i+1}A = ∂_{i+1}(x ∧_{i} y) ∧_{i} z.

This we do by induction in the usual way. We have

∂_{i+1}A = ∂_{i+1}[b ∧_{i} (y ∧_{i} z)]

such that b and y ∧_{i} z are thin; therefore ∂_{i+1}A is thin. For j < i it follows from the inductive hypothesis that ∂_{j}∂_{i+1}A is in im ∧_{i−1} because

∂_{j}∂_{i+1}A = ∂_{i}[∂_{j}b ∧_{i−1} (∂_{j}y ∧_{i−1} ∂_{j}z)]

with

∂_{j}b = ∂_{j}x ∧_{i−1} ∂_{i}(∂_{j}y ∧_{i−1} ∂_{j}z),

and for j > i + 2 it follows from the inductive hypothesis that ∂_{j}∂_{i+1}A is in im ∧_{i} because

∂_{j}∂_{i+1}A = ∂_{i+1}[∂_{j}b ∧_{i} (∂_{j−1}y ∧_{i} ∂_{j−1}z)]

with

∂_{j}b = ∂_{j−1}x ∧_{i} ∂_{i+1}(∂_{j−1}y ∧_{i} ∂_{j−1}z);

therefore

∂_{i+1}A = ∂_{i+2}∂_{i+1}A ∧_{i} ∂_{i}∂_{i+1}A = ∂_{i+1}(x ∧_{i} y) ∧_{i} z

as required.

9.8. Lemma. The wedges in X satisfy the conditions of Definition 9.1(6).

Proof. Again one side is defined if and only if the other is; we may therefore assume that the left side is defined. Let

A = ∂_{i+2}[(x ∧_{i+1} y) ∧_{i+1} (y ∧_{i} z)]

and let

B = A ∧_{i} (w ∧_{i+1} ∂_{i}A);

we must show that

B = (∂_{i+3}A ∧_{i} w) ∧_{i+2} A

and we apply the usual inductive argument. We write A_{j} = ∂_{j}A, so that

A_{j} = \begin{cases}∂_{i+1}[(∂_{j}x ∧_{i} ∂_{j}y) ∧_{i} (∂_{j}y ∧_{i−1} ∂_{j}z)] & (j < i), \\ ∂_{i+2}[(∂_{j−1}x ∧_{i+1} ∂_{j−1}y) ∧_{i+1} (∂_{j−1}y ∧_{i} ∂_{j−1}z)] & (j > i + 3); \end{cases}

in each of these cases A_{j} is of the same form as A. For j < i it follows from the inductive hypothesis that ∂_{j}B is in im ∧_{i+1} because

∂_{j}B = A_{j} ∧_{i−1} (∂_{j}w ∧_{i} ∂_{i−1}A_{j}),

we certainly have

∂_{i}B = w ∧_{i+1} ∂_{i}A ∈ im ∧_{i+1},

and for j > i + 4 it follows from the inductive hypothesis that ∂_{j}B is in im ∧_{i+2} because

∂_{j}B = A_{j−1} ∧_{i} (∂_{j−2}w ∧_{i+1} ∂_{i}A_{j−1}).

If we can show that ∂_{i+1}B ∈ im ∧_{i+1} then it will follow that

B = ∂_{i+4}B ∧_{i+2} ∂_{i+2}B = (∂_{i+3}A ∧_{i} w) ∧_{i+2} A

as required.

To show that ∂_{i+1}B ∈ im ∧_{i+1} we again apply the usual inductive argument. By Lemma 9.5 A is thin, hence ∂_{i+1}B is thin. For j < i it follows from the inductive hypothesis that ∂_{j}∂_{i+1}B is in im ∧_{i} because

∂_{j}∂_{i+1}B = ∂_{i}[A_{j} ∧_{i−1} (∂_{j}w ∧_{i} ∂_{i−1}A_{j})],

we certainly have

∂_{i}∂_{i+1}B = ∂_{i}w ∧_{i} ∂_{i}∂_{i}A ∈ im ∧_{i},

and for j > i + 3 it follows from the inductive hypothesis that ∂_{j}∂_{i+1}B is in im ∧_{i+1} because

∂_{j}∂_{i+1}B = ∂_{i+1}[A_{j} ∧_{i} (∂_{j−1}w ∧_{i+1} ∂_{i}A_{j})];

therefore ∂_{i+1}B ∈ im ∧_{i+1} as required.

This completes the proof.

9.9. Lemma. The wedges in X satisfy the conditions of Definition 9.1(7).

Proof. Again we find that one side is defined if and only if the other is, so we may assume that the left side is defined. Let

A = (x ∧_{i} y) ∧_{j} (z ∧_{i} w)

with i ≤ j − 3; we will apply the usual inductive argument to show that

A = (x ∧_{j−1} z) ∧_{i} (y ∧_{j−1} w).

For k < i it follows from the inductive hypothesis that

∂_{k}A = (∂_{k}x ∧_{i−1} ∂_{k}y) ∧_{j−1} (∂_{k}z ∧_{i−1} ∂_{k}w) ∈ im ∧_{i−1},

for i + 2 < k < j it follows from the inductive hypothesis that

∂_{k}A = (∂_{k−1}x ∧_{i} ∂_{k−1}y) ∧_{j−1} (∂_{k−1}z ∧_{i} ∂_{k−1}w) ∈ im ∧_{i},

we certainly have

∂_{j}A = z ∧_{i} w ∈ im ∧_{i}

and

∂_{j+2}A = x ∧_{i} y ∈ im ∧_{i},

and for k > j + 2 it follows from the inductive hypothesis that

∂_{k}A = (∂_{k−2}x ∧_{i} ∂_{k−2}y) ∧_{j} (∂_{k−2}z ∧_{i} ∂_{k−2}w) ∈ im ∧_{i}.

If we can show that ∂_{j+1}A ∈ im ∧_{i} then it will follow that

A = ∂_{i+2}A ∧_{i} ∂_{i}A = (x ∧_{j−1} z) ∧_{i} (y ∧_{j−1} w).

To show that ∂_{j+1}A ∈ im ∧_{i} we again apply the usual inductive argument. It follows from Lemma 9.5 that ∂_{j+1}A is thin. For k < i it follows from the inductive hypothesis that

∂_{k}∂_{j+1}A = ∂_{j}[(∂_{k}x ∧_{i−1} ∂_{k}y) ∧_{j−1} (∂_{k}z ∧_{i−1} ∂_{k}w)] ∈ im ∧_{i−1},

for i + 2 < k < j it follows from the inductive hypothesis that

∂_{k}∂_{j+1}A = ∂_{j}[(∂_{k−1}x ∧_{i} ∂_{k−1}y) ∧_{j−1} (∂_{k−1}z ∧_{i} ∂_{k−1}w)] ∈ im ∧_{i},

we certainly have

∂_{j}∂_{j+1}A = ∂_{j−1}z ∧_{i} ∂_{j−1}w ∈ im ∧_{i}

and

∂_{j+1}∂_{j+1}A = ∂_{j}x ∧_{i} ∂_{j}y ∈ im ∧_{i},

and for k > j + 1 it follows from the inductive hypothesis that

∂_{k}∂_{j+1}A = ∂_{j+1}[(∂_{k−1}x ∧_{i} ∂_{k−1}y) ∧_{j} (∂_{k−1}z ∧_{i} ∂_{k−1}w)] ∈ im ∧_{i};

therefore ∂_{j+1}A ∈ im ∧_{i} as required.

This completes the proof.

It follows from Lemmas 9.6–9.9 that the wedges of Lemma 9.3 satisfy all the axioms for sets with complicial identities. This completes the proof of Theorem 9.2.

References

[1] F. A. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence of a globular and a cubical approach, Adv. Math. 170 (2002), no. 1, 71–118.

[2] R. Steiner, Omega-categories and chain complexes, Homology Homotopy Appl. 6 (2004), no. 1, 175–200.

[3] R. Steiner, Orientals, in Categories in algebra, geometry and mathematical physics, 427–439, Contemp. Math., 431 Amer. Math. Soc., Providence, RI.

[4] R. Steiner, The algebraic structure of the universal complicial sets, J. Pure Appl. Algebra 216 (2012), no. 8-9, 1976–1993.

[5] R. Steiner, The algebra of the nerves of omega-categories, Theory and Applications of Categories 28(2013), 733-779.

[6] R. Street, Higher dimensional nerves, http://maths.mq.edu.au/∼street/HDN1982.pdf (1982).

[7] R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), no. 3, 283–335.

[8] R. Street, Fillers for nerves, in Categorical algebra and its applications (Louvain-La-Neuve, 1987), 337–341, Lecture Notes in Math., 1348 Springer, Berlin.

[9] D. Verity, Complicial sets characterising the simplicial nerves of strict ω-categories, Mem. Amer. Math. Soc. 193 (2008), no. 905, xvi+184 pp.

School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow, Great Britain G12 8QW

Email: Richard.Steiner@glasgow.ac.uk

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