Theory and Applications of Categories, Vol. 28, No. 23, 2013, pp. 733–779.

THE ALGEBRA OF THE NERVES OF OMEGA-CATEGORIES

RICHARD STEINER

Abstract. We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also construct an equivalence between the categories of strict omega-categories and of sets with complical identities.

1. Introduction

This paper is concerned with the simplicial nerves of strict ω-categories, as constructed by Street [6]. The nerves are simplicial sets with additional structure, and the problem is to characterise the additional structure which can occur. One characterisation, due to Verity [7], says that the nerves are complicial sets; that is to say, they have distinguished classes of thin elements satisfying certain axioms. The object of this paper is to give a more concrete algebraic characterisation: the nerves are simplicial sets with additional operations satisfying certain identities. The result is a set with complicial identities as defined in [5]. The resultant characterisation is like the characterisation of cubical nerves given by Al-Agl, Brown and Steiner [1].

The method involves a comparison of the theories of ω-categories and of sets with complicial identities in the technical sense of universal algebra. It turns out that both theories can be expressed in terms of chain complexes and chain maps. The theory of ωcategories is represented by simple chain complexes [4]; the theory of sets with complicial identities is represented by the chain complexes of simplexes and by certain colimits of these chain complexes. The proof is based on relationships between the various chain complexes involved.

The paper is structured as follows. In Section 2 we describe ω-categories and show that their theory is represented by the class of simple ω-categories (see [2]). In Section 3 we describe sets with complicial identities. In Section 4 we describe a category of chain complexes with additional structure called augmented directed complexes and a functor ν from this category to the category of ω-categories. In Section 5 we show that simple ωcategories are the images under ν of simple chain complexes; it follows that the theory of ωcategories can be described in terms of simple chain complexes. In Section 6 we construct a functor λ from augmented directed complexes to sets with complicial identities, and in Section 7 we use this functor to express the theory of sets with complicial identities in terms of the chain complexes of simplexes. We have now described the categories of ωcategories and of sets with complicial identities in terms of augmented directed complexes, and can therefore compare the two categories. The comparison occupies Sections 8–13.

Received by the editors 2013-06-27 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 identities, omega-category.

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

The idea behind the comparison is as follows. Let X be a set with complicial identities; then there is a contravariant functor from simple chain complexes to sets given by

S ↦ Hom[λS, X].

This functor will yield an ω-category provided that it takes certain colimit diagrams to limit diagrams; we therefore need information about the sets Hom[λS, X]. We obtain this information by showing that S is a retract of the chain complex of a simplex. We begin in Section 8 by showing that S is a quotient of the chain complex of a simplex. We then show that S is a retract by constructing an idempotent endomorphism of the chain complex of the simplex with the appropriate kernel. This endomorphism represents an operation in sets with complicial identities. We construct the operation in Section 9 and give some computations concerning the induced endomorphism in Section 10; we prove that the corresponding endomorphism is idempotent with the correct kernel in Section 11; we show that the required diagrams are limit diagrams in Section 12. In Section 13 we deduce the main result (Theorem 13.3): ω-categories are equivalent to sets with complicial identities.

2. The theory of ω-categories

In this paper all ω-categories are strict ω-categories. We will use an algebraic definition with infinitely many sorts, as follows.

2.1. Definition. An ω-category C is a sequence of sets C_{0}, C_{1}, … together with the following structure.

(1) If x ∈ C_{p} then there are identity elements

i_{p}^{n}x ∈ C_{n} (p < n),

sources

d_{q}^{−}x ∈ C_{q} (q < p),

and targets

d_{q}^{+}x ∈ C_{q} (q < p).

(2) If x, y ∈ C_{p} and d_{q}^{+}x = d_{q}^{−}y for some q < p then there is a composite

x #_{q} y ∈ C_{p}.

(3) If x ∈ C_{p} then

i_{m}^{n}i_{p}^{m}x = i_{p}^{n}x (p < m < n),
d_{p}^{−}i_{p}^{n}x = d_{p}^{+}i_{p}^{n}x = x (p < n),
i_{m}^{p}d_{m}^{−}x #_{m} x = x #_{m} i_{m}^{p}d_{m}^{+}x = x (m < p),
d_{m}^{−}d_{n}^{−}x = d_{m}^{−}d_{n}^{+}x = d_{m}^{−}x (m < n < p),
d_{m}^{+}d_{n}^{−}x = d_{m}^{+}d_{n}^{+}x = d_{m}^{+}x (m < n < p).

(4) If x, y ∈ C_{p} and d_{q}^{+}x = d_{q}^{−}y for some q < p then

i_{p}^{n}(x #_{q} y) = i_{p}^{n}x #_{q} i_{p}^{n}y (p < n),
d_{m}^{−}(x #_{q} y) = d_{m}^{−}x #_{q} d_{m}^{−}y (q < m < p),
d_{m}^{+}(x #_{q} y) = d_{m}^{+}x #_{q} d_{m}^{+}y (q < m < p),
d_{q}^{−}(x #_{q} y) = d_{q}^{−}x,
d_{q}^{+}(x #_{q} y) = d_{q}^{+}y.

(5) If x, y, z ∈ C_{p} and d_{q}^{+}x = d_{q}^{−}y, d_{q}^{+}y = d_{q}^{−}z for some q < p then

(x #_{q} y) #_{q} z = x #_{q}(y #_{q} z).

(6) If x, y, z, w ∈ C_{p} and d_{q}^{+}x = d_{q}^{−}y, d_{m}^{+}y = d_{m}^{−}z, d_{q}^{+}z = d_{q}^{−}w with m < q < p then

(x #_{q} y) #_{m}(z #_{q} w) = (x #_{m} z) #_{q}(y #_{m} w).

A morphism of ω-categories f: C → D is a sequence of functions f: C_{p} → D_{p} commuting with the identity, source, target and composition operations.

2.2. Remark. In an ω-category the identity element functions i_{p}^{n} must be injective. It is therefore possible to require them to be inclusions, yielding a one-sorted description with operations d_{q}^{−}, d_{q}^{+}, #_{q}.

2.3. Remark. In axiom (6) the hypotheses d_{q}^{+}x = d_{q}^{−}y, d_{q}^{+}z = d_{q}^{−}w imply that

d_{m}^{+}x = d_{m}^{+}d_{q}^{+}x = d_{m}^{+}d_{q}^{−}y = d_{m}^{+}y, d_{m}^{−}z = d_{m}^{−}d_{q}^{+}z = d_{m}^{−}d_{q}^{−}w = d_{m}^{−}w.

One could therefore replace the single equality d_{m}^{+}y = d_{m}^{−}z by the two equalities d_{m}^{+}x = d_{m}^{−}z, d_{m}^{+}y = d_{m}^{−}w. This produces the more usual form of the hypotheses.

The domains for the axioms can be naturally indexed by sequences of nonnegative integers as follows: for (1) and (3) use the one-term sequence (p); for (2) and (4) use (p, q, p) with p > q; for (5) use (p, q, p, q, p) with p > q; for (6) use (p, q, p, m, p, q, p) with p > q > m. All of these sequences are up-down vectors in the sense of the following definition (taken from [2], 2.3).

2.4. Definition. An up-down vector is a non-empty finite sequence of nonnegative integers

(p_{0}, q_{1}, p_{1}, …, p_{k−1}, q_{k}, p_{k})

such that p_{i−1} > q_{i} and q_{i} < p_{i} for 1 ≤ i ≤ k.

The corresponding ω-categories are also taken from [2] and may be defined as follows.

2.5. Definition. Let s be an up-down vector given by

s = (p_{0}, q_{1}, p_{1}, …, p_{k−1}, q_{k}, p_{k}).

Then an s-simple ω-category is an ω-category with a presentation of the following form: the generators form an ordered list g_{0}, …, g_{k} with dim g_{i} = p_{i}; the relations are given by

d_{q_{i}}^{+}g_{i−1} = d_{q_{i}}^{−}g_{i} (1 ≤ i ≤ k).

We will usually treat simple ω-categories as iterated push-outs, using induction on the numbers of terms in up-down vectors. An up-down vector s with more than one term will therefore be written in the form

s = (s′, q, p),

so that s′ is a shorter up-down vector with last term greater than q and so that p is an integer greater than q. The corresponding push-outs are as follows.

2.6. Definition. Let s be an up-down vector with more than one term given by s = (s′, q, p); then an s-simple square of ω-categories is a push-out square

\xymatrix{C^{0} \ar[d]_{π} \ar[r]^{ρ} & C′′ \ar[d]^{τ} \\ C′ \ar[r]_{σ} & C}

such that C′, C^{0} and C′′ are s′-simple, (q)-simple and (p)-simple with final generators g′, g^{0} and g′′ and such that

πg^{0} = d_{q}^{+}g′, ρg^{0} = d_{q}^{−}g′′.

Obviously we have the following result.

2.7. Proposition. If τ: C′′ → C is the right hand vertical morphism in an s-simple square of ω-categories and if C′′ has final generator g′′ then C is an s-simple ω-category with final generator τg′′.

In the axioms for ω-categories the domains are free ω-categories on single generators and pull-backs corresponding to simple squares. We therefore get the following result.

2.8. Proposition. Let ω-cat be the category of ω-categories, let Θ be the full-subcategory of simple ω-categories, and let Θ̂ be the category of contravariant functors from Θ to sets which take simple squares to pull-back squares. Then there is an equivalence of categories

C ↦ Hom(−, C): ω-cat → Θ̂.

3. Sets with complicial identities

In this section we recall the definition of sets with complicial identities from [5]. A set with complicial identities is a simplicial set X_{0}, X_{1}, … together with additional partial binary wedge operations ∧_{i}. These operations raise dimension by 1; they correspond to the projection of an (m + 1)-simplex onto the union of the m-faces opposite vertices i and i + 2. The identities are stated here without comment, but there are illustrations in Section 7.

3.1. Definition. A set with complicial identities X is a sequence of sets

X_{0}, X_{1}, …

together with the following structure.

(1) If x ∈ X_{m} then there are faces

∂_{i}x ∈ X_{m−1} (m > 0, 0 ≤ i ≤ m)

and degeneracies

ϵ_{i}x ∈ X_{m+1} (0 ≤ i ≤ m).

(2) If x, y ∈ X_{m} and if ∂_{i}x = ∂_{i+1}y for some i with 0 ≤ i ≤ m − 1 then there is a wedge

x ∧_{i} y ∈ X_{m+1}.

(3) If x ∈ X_{m} then

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

(4) If x, y ∈ X_{m} and if ∂_{i}x = ∂_{i+1}y with 0 ≤ i < m then

∂_{j}(x ∧_{i} y) = ∂_{j}x ∧_{i−1} ∂_{j}y (0 ≤ j ≤ i − 1),
∂_{i}(x ∧_{i} y) = y,
∂_{i+2}(x ∧_{i} y) = x,
∂_{j}(x ∧_{i} y) = ∂_{j}x ∧_{i−1} ∂_{j}y (i + 3 ≤ j ≤ m + 1).

(5) If b ∈ X_{m+1} and y, z ∈ X_{m}, if ∂_{i}y = ∂_{i+1}z and ∂_{i}b = ∂_{i+1}(y ∧_{i} z) with 0 ≤ i < m, and if A = b ∧_{i} (y ∧_{i} z), then

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

(6) If x, y ∈ X_{m} and c ∈ X_{m+1}, if ∂_{i}x = ∂_{i+1}y and ∂_{i+1}(x∧_{i}y) = ∂_{i+2}c with 0 ≤ i < m, and if A = (x ∧_{i} y) ∧_{i+1} c, then

A = ∂_{i+2}A ∧_{i} (y ∧_{i} ∂_{i}c).

(7) If x, y, z ∈ X_{m} and if ∂_{i}x = ∂_{i+1}y, ∂_{i}y = ∂_{i+1}z with 0 ≤ i < m then

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

(8) If x, y, z, w ∈ X_{m}, if ∂_{i+1}x = ∂_{i+2}y, ∂_{i}y = ∂_{i+1}z, ∂_{i+1}w = ∂_{i+1}(∂_{i}x ∧_{i} ∂_{i+2}z) with 0 ≤ i ≤ m − 2, and if A = ∂_{i+2}[(x ∧_{i+1} y) ∧_{i+1} (y ∧_{i} z)], then

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

(9) If x, y, z, w ∈ X_{m} and ∂_{i}x = ∂_{i+1}y, ∂_{i}z = ∂_{i+1}w, ∂_{j−1}x = ∂_{j}z, ∂_{j−1}y = ∂_{j}w with 0 ≤ i ≤ j − 3 ≤ m − 3 then

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

A morphism of sets with complicial identities f: X → Y is a sequence of functions f: X_{m} → Y_{m} commuting with the face, degeneracy and wedge operations.

4. Augmented directed complexes

In this section we give some definitions and results based on [3].

4.1. Definition. An augmented directed complex is an augmented chain complex of abelian groups

\xymatrix{… \ar[r]^{∂} & K_{1} \ar[r]^{∂} & K_{0} \ar[r]^{ϵ} & Z,}

together with a prescribed submonoid for each chain group K_{q}. A morphism of augmented directed complexes is an augmentation-preserving chain map which takes prescribed submonoids into prescribed submonoids. A free augmented directed complex is an augmented directed complex such that each chain group is a free abelian group with a prescribed basis and such that each prescribed submonoid is generated as a monoid by the prescribed basis elements.

Let K be a free augmented directed complex. We note that the prescribed basis elements are uniquely determined as the indecomposable elements in the prescribed submonoids. We regard the union of the prescribed bases for the individual chain groups K_{q} as a prescribed graded basis for the entire chain complex K. 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.

4.2. Definition. A totally ordered directed complex is a free augmented chain complex together with a total ordering of the basis such that each basis element a satisfies the following conditions.

(1) In the ordered basis, a appears after the terms of ∂^{−}a and before the terms of ∂^{+}a.

(2) If the dimension of a is p, then

ϵ(∂^{−})^{p}a = ϵ(∂^{+})^{p}a = 1.

Given an augmented directed complex K, we define an ω-category νK as follows. The set (νK)_{p} of p-dimensional elements consists of the double sequences

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

such that x_{i}^{−} and x_{i}^{+} are i-dimensional members of the prescribed submonoids, such that

x_{i}^{−} = x_{i}^{+} = 0

for i > p, such that

ϵx_{0}^{−} = ϵx_{0}^{+} = 1,

and such that

x_{i}^{+} − x_{i}^{−} = ∂x_{i+1}^{−} = ∂x_{i+1}^{+}

for i ≥ 0. For n > p the identity element function i_{p}^{n}: (νK)_{p} → (νK)_{n} is the inclusion. For q < p, if x = ( x_{0}^{−}, x_{0}^{+} | …) as above, then

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

If x and y are p-dimensional and if d_{q}^{+}x = d_{q}^{−}y = z with q < p then

x #_{q} y = x − i_{q}^{p}z + y.

In particular let K be a totally ordered directed complex and let a be a p-dimensional basis element for K; then there is a p-dimensional element ⟨a⟩ of νK, called an atom, which is given by

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

The main results ([3], Theorems 5.11 and 6.1) can be stated as follows.

4.3. Theorem. The functor ν is a fully faithful functor from the category of totally ordered directed complexes to the category of ω-categories.

4.4. Theorem. Let K be a totally ordered directed complex. Then the ω-category νK has a presentation as follows. The generators are the atoms, such that ⟨a⟩ is a p-dimensional member of νK if a is a p-dimensional basis element. For each 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.

5. Simple chain complexes

We will now describe a class of chain complexes corresponding to simple ω-categories. The class was defined in [4]. For present purposes it is convenient to proceed inductively.

5.1. Definition. Let s be a one-term up-down vector given by s = (p). Then an ssimple chain complex with (final) generator a is a free augmented directed complex with a p-dimensional basis element a such that the basis elements can be listed as

(∂^{−})^{p}a, (∂^{−})^{p−1}a, …, ∂^{−}a, a, ∂^{+}a, …, (∂^{+})^{p−1}a, (∂^{+})^{p}a

and such that ϵ(∂^{−})^{p}a = ϵ(∂^{+})^{p}a = 1.

5.2. Definition. Let s be an up-down vector with more than one term given by s = (s′, q, p), and let p′ be the last term in s′. Then an s-simple chain complex with final generator a is an augmented directed complex K if there are s′-simple, (q)-simple and (p)-simple subcomplexes K′, K^{0} and K′′ with final generators a′, a^{0} and a such that

K = K′ + K′′,
K′ ∩ K′′ = K^{0},
(∂^{+})^{p′−q}a′ = a^{0} = (∂^{−})^{p−q}a,

and the distinguished submonoid of K is the sum of the distinguished submonoids of K′ and K′′.

5.3. Proposition. Let s be an up-down vector with last term p and let K be an s-simple chain complex with final generator a. Then K is a totally ordered directed complex whose ordered basis finishes with the elements

a, ∂^{+}a, …, (∂^{+})^{p}a.

Proof. The proof is by induction on the number of terms in s. In the case s = (p) the result is obvious. From now on, let s = (s′, q, p), let K′, K^{0} and K′′ be the subcomplexes as in the definition, and let a^{0} = (∂^{−})^{p−q}a. It follows from the inductive hypothesis that K′ is a totally ordered directed complex whose ordered basis finishes with the terms

a^{0}, ∂^{+}a^{0}, …, (∂^{+})^{q}a^{0}.

We observe that K′′ is obtained from K^{0} by adjoining the elements

(∂^{−})^{p−q−1}a, …, ∂^{−}a, a, ∂^{+}a, …, (∂^{+})^{p−q}a.

It follows that K is a totally ordered directed complex; the ordered basis is obtained from that of K′ by inserting the additional elements (∂^{−})^{p−q−1}a, …, (∂^{+})^{p−q}a immediately after a^{0}. For r > 0 we have (∂^{+})^{r}a^{0} = (∂^{+})^{p−q+r}a; the ordered basis for K therefore finishes with the elements

a, …, (∂^{+})^{p−q}a, (∂^{+})^{p−q+1}a, …, (∂^{+})^{p}a.

This completes the proof.

5.4. Proposition. Let s be an up-down vector with last term p and let K be an ssimple chain complex with final generator a. Then νK is an s-simple ω-category with final generator ⟨a⟩.

Proof. The proof is by induction on the number of terms in s.

Suppose that s = (p). According to Theorem 4.4, νK has a presentation generators

⟨(∂^{−})^{p}a⟩, …, ⟨∂^{−}a⟩, ⟨a⟩, ⟨∂^{+}a⟩, …, ⟨(∂^{+})^{p}a⟩

and with relations

d_{i−1}^{−}⟨(∂^{−})^{p−i}a⟩ = d_{i−1}^{−}⟨(∂^{+})^{p−i}a⟩ = ⟨(∂^{−})^{p−i+1}a⟩ (0 < i ≤ p),
d_{i−1}^{+}⟨(∂^{−})^{p−i}a⟩ = d_{i−1}^{+}⟨(∂^{+})^{p−i}a⟩ = ⟨(∂^{+})^{p−i+1}a⟩ (0 < i ≤ p).

Because of the axioms

d_{i−1}^{−}d_{i}^{−} = d_{i−1}^{−}d_{i}^{+} = d_{i−1}^{−}, d_{i−1}^{+}d_{i}^{−} = d_{i−1}^{+}d_{i}^{+} = d_{i−1}^{+},

this collapses to a presentation with a single p-dimensional generator ⟨a⟩ and with no relations. Therefore νK is an s-simple ω-category with final generator ⟨a⟩.

Now let s = (s′, q, p). Let K′, K^{0} and K′′ be the s′-simple, (q)-simple and (p)-simple subcomplexes with final generators a′, a^{0} and a as in Definition 5.2. Using the presentation of Theorem 4.4 and the inductive hypothesis, we see that νK is generated by νK′ and νK′′ subject to the relation d_{q}^{+}⟨a′⟩ = d_{q}^{−}⟨a⟩. This gives the result.

We also have simple squares of chain complexes, corresponding to simple squares of ω-categories.

5.5. Definition. Let s be an up-down vector with more than one term given by s = (s′, q, p) and let p′ be the last term in s′; then an s-simple square of chain complexes is a square of augmented directed complexes

\xymatrix{K^{0} \ar[d]_{π} \ar[r]^{ρ} & K′′ \ar[d]^{τ} \\ K′ \ar[r]_{σ} & K}

with the following properties: the complexes K′, K^{0} and K′′ are s′-simple, (q)-simple and (p)-simple with final generators a′, a^{0} and a′′; the morphisms π and ρ are given by

π(∂^{+})^{q−r}a^{0} = (∂^{+})^{p′−r}a′, ρ(∂^{−})^{q−r}a^{0} = (∂^{−})^{p−r}a′′ (0 ≤ r ≤ q);

the square is a push-out as a square of abelian groups; the prescribed submonoid of K is the sum of the images of the prescribed submonoids of K′ and K′′.

Obviously we have the following result.

5.6. Proposition. If τ: K′′ → K is the right hand vertical morphism in an s-simple square of chain complexes and if K′′ has final generator a′′ then K is an s-simple chain complex with final generator a such that

τ(∂^{+})^{p−r}a′′ = (∂^{+})^{p−r}a (0 ≤ r ≤ p).

Using Definition 2.6 and Proposition 2.8 we obtain the following results.

5.7. Proposition. The image under ν of an s-simple square of chain complexes is an s-simple square of ω-categories.

5.8. Proposition. Let ω-cat be the category of ω-categories, let Σ be the category of simple chain complexes and morphisms of augmented directed complexes, and let Σ̂ be the category of contravariant functors from Σ to sets which take simple squares to pull-back squares. Then there is an equivalence of categories

C ↦ Hom[ν(−), C]: ω-cat → Σ̂.

6. The chain complexes of simplexes

In this section we discuss the chain complexes of simplexes, which will simply be called simplexes. They correspond to the theory of sets with complicial identities (Section 3). The material is mostly taken from [5].

6.1. Definition. For m = 0, 1, 2, … the m-simplex Δ(m) is the free augmented directed complex constructed as follows. The basis elements correspond to the sequences of integers

a_{0}, …, a_{q}

with 0 ≤ q ≤ m and 0 ≤ a_{0} < a_{1} < … < a_{q} ≤ m. The basis element corresponding to a_{0}, …, a_{q} is written [a_{0}, …, a_{q}] and has dimension q. If q > 0 then the boundary of [a_{0}, …, a_{q}] 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 will now show that the simplexes are totally ordered directed complexes by expressing them as joins.

6.2. Definition. Let K and L be augmented directed complexes. Then the join K ∗ L is the direct sum of abelian groups

K ∗ L = K ⊕ (K × L) ⊕ L

with the following structure. The grading is given by

(K ∗ L)_{q} = K_{q} ⊕ [ ⨁_{i+j=q−1}(K_{i} ⊗ L_{j})] ⊕ L_{q}.

The inclusions of K and L in K ∗ L commute with the boundary and augmentation homomorphisms. The boundary on K_{i} ⊗ L_{j} is given by

∂(x ⊗ y) = \begin{cases}(ϵx)y − (ϵy)x & (i = j = 0), \\ (ϵx)y − x ⊗ ∂y & (i = 0, j > 0), \\ ∂x ⊗ y − (−1)^{i}(ϵy)x & (i > 0, j = 0), \\ ∂x ⊗ y − (−1)^{i}x ⊗ ∂y & (i, j > 0). \end{cases}

The prescribed submonoid of K ∗ L is generated by the elements of the prescribed submonoids of K and L and by the tensor products of these elements.

6.3. Example. The m-simplex Δ(m) is the join of m + 1 copies of Δ(0).

6.4. Proposition. If K and L are totally ordered directed complexes, then K ∗ L is a totally ordered directed complex.

Proof. One can check that K ∗ L has a suitably ordered basis consisting of the basis elements of K and L and of the tensor products of these basis elements. The ordering of the basis for K ∗ L is obtained as follows. Take the basis elements of K in order followed by the basis elements of L in order. If a is an odd-dimensional basis element in K, then the basis elements of the form a ⊗ b are inserted before a in the order given by the second factor; if a is an even-dimensional basis element in K, then the basis elements of the form a ⊗ b are inserted after a in the reverse of the order given by the second factor.

6.5. Proposition. A simplex is a totally ordered directed complex.

Proof. Obviously Δ(0) is a totally ordered directed complex. The result now follows from Example 6.3 and Proposition 6.4.

We will now use simplexes to construct a functor λ from augmented directed complexes to sets with complicial identities. The m-dimensional elements in λK will be the morphisms of augmented directed complexes from Δ(m) to K. An operation θ in sets with complicial identities will be contravariantly represented by a morphism θ^{∨} between simplexes. In particular there are the obvious morphisms corresponding to the face and degeneracy operations.

6.6. Notation. The face and degeneracy morphisms

∂_{i}^{∨}: Δ(m − 1) → Δ(m) (m > 0, 0 ≤ i ≤ m),
ϵ_{i}^{∨}: Δ(m + 1) → Δ(m) (0 ≤ i ≤ m)

are defined on basis elements as follows.

If a = [a_{0}, …, a_{q}] is a basis element for Δ(m − 1) then ∂_{i}^{∨}a = [a′_{0}, …, a′_{q}] with

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

If b = [b_{0}, …, b_{q}] is a basis element for Δ(m + 1) including both the terms i and i + 1 then ϵ_{i}^{∨}b = 0.

If b = [b_{0}, …, b_{q}] is a basis element for Δ(m + 1) not including both the terms i and i + 1 then ϵ_{i}^{∨}b = [b′′_{0}, …, b′′_{q}] with

b′′_{q} = \begin{cases}b_{q} & (0 ≤ b_{q} ≤ i), \\ b_{q} − 1 & (i + 1 ≤ b_{q} ≤ m + 1). \end{cases}

Recall from Definition 4.1 that morphisms of augmented directed complexes are augmentation-preserving chain maps taking prescribed submonoids into prescribed submonoids. Recall also that the prescribed submonoid of a free augmented directed complex is the submonoid generated by the prescribed basis elements. We obviously have the following result.

6.7. Proposition. The face and degeneracy morphisms are morphisms of augmented directed complexes.

Less obviously we also have the following result.

6.8. Proposition. Let K be an augmented directed complex and let

x, y: Δ(m) → K

be morphisms of augmented directed complexes such that x∂_{i}^{∨} = y∂_{i+1}^{∨} for some i with 0 ≤ i < m. Then there is a morphism of augmented directed complexes

z: Δ(m + 1) → K

given by

z = xϵ_{i+1}^{∨} − x∂_{i}^{∨}(ϵ_{i}^{∨})^{2} + yϵ_{i}^{∨} = xϵ_{i+1}^{∨} − y∂_{i+1}^{∨}(ϵ_{i}^{∨})^{2} + yϵ_{i}^{∨}.

Proof. It is clear that z is an augmentation preserving chain map; it therefore suffices to prove that za is in the prescribed submonoid of K for each basis element a in Δ(m + 1). We do this by considering three cases: if a has no term i + 2 then ∂_{i+1}^{∨}(ϵ_{i}^{∨})^{2}a = ϵ_{i}^{∨}a, hence za = xϵ_{i+1}^{∨}a; if a has no term i then ∂_{i}^{∨}(ϵ_{i}^{∨})^{2}a = ϵ_{i+1}^{∨}a, hence za = yϵ_{i}^{∨}a; if a has terms i and i + 2 then (ϵ_{i}^{∨})^{2}a = 0, hence

za = xϵ_{i+1}^{∨}a + yϵ_{i}^{∨}a.

The definition of λ is now as follows.

6.9. Notation. Let K be an augmented directed complex. Then λK is the graded set given by

(λK)_{m} = Hom[Δ(m), K].

If x ∈ (λK)_{m} with m > 0 and if 0 ≤ i ≤ m then

∂_{i}x = x∂_{i}^{∨}.

If x ∈ (λK)_{m} and 0 ≤ i ≤ m then

ϵ_{i}x = xϵ_{i}^{∨}.

If x, y ∈ (λK)_{m} and ∂_{i}x = ∂_{i+1}y for some i with 0 ≤ i < m then

x ∧_{i} y = ϵ_{i+1}x − ϵ_{i}^{2}∂_{i}x + ϵ_{i}y = ϵ_{i+1}x − ϵ_{i}^{2}∂_{i+1}y + ϵ_{i}y.

6.10. Proposition. If K is an augmented directed complex then λK is a set with complicial identities.

Proof. We see that the operations are well-defined. The axioms follow straightforwardly from computations with chain maps.

We conclude this section with the main result of [5] (Theorem 8.7).

6.11. Theorem. Let O be the full subcategory of the category of augmented directed complexes with objects Δ(0), Δ(1), …. For n ≥ 0 let ι_{n} be the identity endomorphism of Δ(n). Then λ is a fully faithful embedding of O in the category of sets with complicial identities such that λΔ(n) is freely generated by the n-dimensional element λι_{n}.

7. Complicial identities in terms of chain complexes

In the last section we constructed a functor λ from augmented directed complexes to sets with complicial identities (see Proposition 6.10). A set with complicial identities X therefore defines a contravariant set-valued functor

K ↦ Hom(λK, X)

on the category adc of augmented directed complexes. We will now reverse this process: we will show that sets with complicial identities can be obtained from contravariant setvalued functors on a suitable subcategory of adc, provided that they take certain diagrams to limit diagrams.

The objects and diagrams correspond to the domains in the axioms for sets with complicial identities (see Definition 3.1), and we will now consider the various axioms.

The augmented directed complexes associated to axioms (1) and (3) are the simplexes Δ(m).

In the remaining cases we use diagrams of augmented directed complexes which are colimit diagrams as diagrams of abelian groups. The prescribed submonoid of the target object is always the sum of the images of the prescribed submonoids of the other objects in the diagram.

For axioms (2) and (4) we use diagrams

\xymatrix{Δ(m − 1) \ar[d]_{∂_{i}^{∨}} \ar[r]^{∂_{i+1}^{∨}} & Δ(m) \ar[d]^{η_{y}} \\ Δ(m) \ar[r]_{η_{x}} & Δ_{(2)}(m, i)}

with 0 ≤ i < m. Since these diagrams are to be colimit diagrams as diagrams of abelian groups, we have

Δ_{(2)}(m, i) ≅ \frac{Δ(m) ⊕ Δ(m)}{\{ (∂_{i}^{∨}z, −∂_{i+1}^{∨}z) : z ∈ Δ(m − 1) \}}.

If X is a set with complicial identities then Δ_{(2)}(m, i) corresponds to the limit

{ (x, y) ∈ X_{m} × X_{m} : ∂_{i}x = ∂_{i+1}y }.

We will also need the morphisms

v_{i}^{∨}: Δ(m + 1) → Δ_{(2)}(m, i)

corresponding to the wedge operations; these are given by

v_{i}^{∨} = η_{x}ϵ_{i+1}^{∨} − η_{x}∂_{i}^{∨}(ϵ_{i}^{∨})^{2} + η_{y}ϵ_{i}^{∨} = η_{x}ϵ_{i+1}^{∨} − η_{y}∂_{i+1}^{∨}(ϵ_{i}^{∨})^{2} + η_{y}ϵ_{i}^{∨}.

For axiom (5) we use similar diagrams

\xymatrix{Δ(m) \ar[dd]_{∂_{i}^{∨}} \ar[r]^{v_{i}^{∨}∂_{i+1}^{∨}} & Δ_{(2)}(m, i) \ar[dd] \\  &  & Δ(m) \ar[ul] \ar[dl]_{η_{y}} & Δ(m − 1) \ar[l]_{∂_{i}^{∨}} \ar[r]^{∂_{i+1}^{∨}} & Δ(m) \ar[ulll] \ar[dlll]^{η_{z}} \\ Δ(m + 1) \ar[r]_{η_{b}} & Δ_{(5)}(m, i)}

with 0 ≤ i < m.

For axiom (6) we use diagrams

\xymatrix{ &  &  & Δ_{(2)}(m, i) \ar[dd] & Δ(m) \ar[l]^{v_{i}^{∨}∂_{i+1}^{∨}} \ar[dd]_{∂_{i+2}^{∨}} \\ Δ(m) \ar[urrr] \ar[drrr]_{η_{x}} & Δ(m − 1) \ar[l]_{∂_{i}^{∨}} \ar[r]^{∂_{i+1}^{∨}} & Δ(m) \ar[ur] \ar[dr]^{η_{y}} \\  &  &  & Δ_{(6)}(m, i) & Δ(m + 1) \ar[l]^{η_{c}}}

with 0 ≤ i < m.

For axiom (7) we use diagrams

\xymatrix{Δ(m − 1) \ar[d]_{∂_{i}^{∨}} \ar[r]^{∂_{i+1}^{∨}} & Δ(m) \ar[d]_{η_{y}} & Δ(m − 1) \ar[l]_{∂_{i}^{∨}} \ar[d]^{∂_{i+1}^{∨}} \\ Δ(m) \ar[r]_{η_{x}} & Δ_{(7)}(m, i) & Δ(m) \ar[l]^{η_{z}}}

with 0 ≤ i < m.

For axiom (8) we use diagrams

\xymatrix{Δ(m − 1) \ar[ddd]_{∂_{i}^{∨}} \ar[r] & Δ_{(2)}(m − 1, 1) & Δ(m − 1) \ar[l] \ar[ddd]^{∂_{i+2}^{∨}} \\  & Δ(m − 1) \ar[u]^{v_{i}^{∨}∂_{i+1}^{∨}} \ar[d]_{∂_{i+1}^{∨}} \\  & Δ(m) \ar[d]_{η_{w}} \\ Δ(m) \ar[r]^{η_{x}} & Δ_{(7)}(m, i) & Δ(m) \ar[l]_{η_{z}} \\ Δ(m − 1) \ar[u]^{∂_{i+1}^{∨}} \ar[r]_{∂_{i+2}^{∨}} & Δ(m) \ar[u]^{η_{y}} & Δ(m − 1) \ar[l]^{∂_{i}^{∨}} \ar[u]_{∂_{i+1}^{∨}}}

with 0 ≤ i ≤ m − 2.

For axiom (9) we use diagrams

\xymatrix{Δ(m − 1) \ar[r]^{∂_{i+1}^{∨}} \ar[d]_{∂_{i}^{∨}} & Δ(m) \ar[d]_{η_{y}} & Δ(m − 1) \ar[l]_{∂_{j−1}^{∨}} \ar[d]^{∂_{j}^{∨}} \\ Δ(m) \ar[r]^{η_{x}} & Δ_{(9)}(m, i, j) & Δ(m) \ar[l]_{η_{w}} \\ Δ(m − 1) \ar[r]_{∂_{j}^{∨}} \ar[u]^{∂_{j−1}^{∨}} & Δ(m) \ar[u]^{η_{z}} & Δ(m − 1) \ar[l]^{∂_{i}^{∨}} \ar[u]_{∂_{i+1}^{∨}}}

with 0 ≤ i ≤ j − 3 ≤ m − 3.

We can evidently obtain sets with complicial identities from contravariant set-valued functors in the following way.

7.1. Proposition. Let Π be the full subcategory of the category of augmented directed complexes given by the objects in the diagrams associated to the axioms for sets with complicial identities. Let X be a contravariant set-valued functor on Π taking each of the diagrams to a limit diagram. Then there is a set with complicial identities functorial in X such that the m-dimensional elements are the members of X[Δ(m)] and such that the operations are induced by the morphisms ∂_{i}^{∨}, ϵ_{i}^{∨} and v_{i}^{∨}.

In particular let C be an ω-category; then there is a contravariant set-valued functor on the category Π of this definition given by Hom[ν(−), C]. We want this functor to yield a set with complicial identities. In order to do this, we must show that the images under ν of the diagrams associated to the axioms are colimit diagrams of ω-categories. We will do this by showing that the objects of Π are totally ordered directed complexes; the colimit properties will then be consequences of the presentations in terms of atoms (Theorem 4.4).

7.2. Proposition. If K is an object in a diagrams associated to an axiom for sets with complicial identities, then K is a totally ordered directed complex.

Proof. We already know from Proposition 6.5 that the simplexes Δ(m) are totally ordered directed complexes, because they are joins of copies of Δ(0) and because Δ(0) is a totally ordered directed complex. We will prove the result for the other complexes involved in a similar way, by expressing them as joins. It is convenient to write Δ(−1) for the zero chain complex, which serves as an identity for the join construction; it is then straightforward to verify that

Δ_{(k)}(m, i) ≅ Δ(i − 1) ∗ Δ_{(k)}(1, 0) ∗ Δ(m − i − 2) (k = 2, 5, 6, 7),
Δ_{(8)}(m, i) ≅ Δ(i − 1) ∗ Δ_{(8)}(2, 0) ∗ Δ(m − i − 3),
Δ_{(9)}(m, i, j)
≅ Δ(i − 1) ∗ Δ_{(2)}(1, 0) ∗ Δ(j − i − 4) ∗ Δ_{(2)}(1, 0) ∗ Δ(m − j − 1).

It now suffices to show that Δ_{(k)}(1, 0) is a totally ordered directed complex for k = 2, 5, 6, 7 and that Δ_{(8)}(2, 0) is a totally ordered directed complex. We will do this in each case by drawing a figure and listing the basis elements in the correct order.

For Δ_{(2)}(1, 0) the figure is

[figure: •; •; •; η_{x}; η_{y}]

and the ordered basis is

η_{x}[0], η_{x}[0, 1], η_{x}[1] = η_{y}[0], η_{y}[0, 1], η_{y}[1].

For Δ_{(5)}(1, 0) the figure is

[figure: •; •; •; •; η_{b}; η_{y}; η_{z}]

and the ordered basis is

η_{b}[0], η_{b}[0, 2], η_{b}[0, 1, 2], η_{b}[0, 1], η_{b}[1] = η_{y}[0],
η_{y}[0, 1], η_{y}[1] = η_{z}[0], η_{z}[0, 1], η_{z}[1],

with

∂^{+}η_{b}[0, 1, 2] = η_{b}[0, 1] + η_{b}[1, 2] = η_{b}[0, 1] + η_{y}[0, 1] + η_{z}[0, 1].

For Δ_{(6)}(1, 0) the figure is

[figure: •; •; •; •; η_{x}; η_{y}; η_{c}]

and the ordered basis is

η_{c}[0], η_{c}[0, 2], η_{c}[0, 1, 2], η_{x}[0, 1], η_{x}[1] = η_{y}[0], η_{y}[0, 1],
η_{y}[1] = η_{c}[1], η_{c}[1, 2], η_{c}[2],

with

∂^{+}η_{c}[0, 1, 2] = η_{c}[0, 1] + η_{c}[1, 2] = η_{x}[0, 1] + η_{y}[0, 1] + η_{c}[1, 2].

For Δ_{(7)}(1, 0) the figure is

[figure: •; •; •; •; η_{x}; η_{y}; η_{z}]

and the ordered basis is

η_{x}[0], η_{x}[0, 1], η_{x}[1] = η_{y}[0], η_{y}[0, 1], η_{y}[1] = η_{z}[0], η_{z}[0, 1], η_{z}[1].

For Δ_{(8)}(2, 0) the figure is

[figure: •; •; •; •; •; •; η_{x}; η_{y}; η_{z}; η_{w}]

and the ordered basis is

η_{y}[0], η_{y}[0, 2], η_{y}[0, 1, 2], η_{y}[0, 1] = η_{x}[0, 2], η_{x}[0, 1, 2], η_{x}[0, 1],
η_{x}[1], η_{x}[1, 2], η_{x}[2] = η_{z}[0], η_{z}[0, 2], η_{z}[0, 1, 2], η_{z}[0, 1],
η_{w}[0, 1, 2], η_{w}[0, 1], η_{w}[1], η_{w}[1, 2], η_{w}[2] = η_{z}[1], η_{z}[1, 2], η_{z}[2],

with

∂^{−}η_{w}[0, 1, 2] = η_{w}[0, 2] = η_{x}[1, 2] + η_{z}[0, 1].

This completes the proof.

Now let Π be the category of Proposition 7.1 and let C be an ω-category. We have shown in Proposition 7.2 that the objects of Π are totally ordered directed complexes. It follows from the atomic presentations (Theorem 4.4) that the images under ν of the diagrams of Proposition 7.1 are colimit diagrams of ω-categories. It therefore follows from Proposition 7.1 that one obtains a set with complicial identities from the functor

K ↦ Hom[νK, C].

We will use the following notation.

7.3. Notation. Let α be the functor from ω-categories to sets with complicial identities defined on an ω-category C as follows. The set of m-dimensional elements is given by

(αC)_{m} = Hom[νΔ(m), C].

If x ∈ (αC)_{m} then

∂_{i}x = x(ν∂_{i}^{∨}) (m > 0, 0 ≤ i ≤ m),
ϵ_{i}x = x(νϵ_{i}^{∨}) (0 ≤ i ≤ m).

If x, y ∈ (αC)_{m} and ∂_{i}x = ∂_{i+1}y with 0 ≤ i < m then

x ∧_{i} y = zv_{i}^{∨},

where z is the member of Hom[νΔ_{(2)}(m, i), C] with

z(νη_{x}) = x, z(νη_{y}) = y.

8. Simple chain complexes as quotients of simplexes

At the end of Section 7 we have constructed a functor α from ω-categories to sets with complicial identities. We also need a functor in the opposite direction. Equivalently (Proposition 5.8), given sets with complicial identities, we need contravariant set-valued functors with suitable properties on the category of simple chain complexes. We will again use the functor λ of Proposition 6.10 from augmented directed complexes to sets with complicial identities; the functor on simple chain complexes corresponding to a set with complicial identities X will be given by

S ↦ Hom[λS, X].

We must show that these functors take simple squares of chain complexes to pull-backs (see Proposition 5.8). We will obtain information about the sets Hom[λS, X] by showing that simple chain complexes are retracts of simplexes. In this section, as a first step, we show that an s-simple chain complex can be expressed as a quotient

S_{s} = Δ(|s|)/U_{s},

where Δ(|s|) is a simplex of a suitable dimension. We will also show an s-simple square of chain complexes can be obtained from a commutative square of simplexes Q_{s}.

We will now define our notations.

8.1. Notation. Let s be an up-down vector given by

s = (p_{0}, q_{1}, p_{1}, …, p_{k−1}, q_{k}, p_{k}).

Then

|s| = p_{0} − q_{1} + p_{1} − … + p_{k−1} − q_{k} + p_{k}.

We make the following observation, which will be used frequently in inductive arguments, mostly without comment.

8.2. Proposition. Let s be an up-down vector with more than one term given by s = (s′, q, p), and let p′ be the last term in s′. Then

|s| − p = |s′| − q > |s′| − p′.

Proof. This holds because |s| = |s′| − q + p and because p′ > q.

8.3. Notation. Let s = (s′, q, p) be an up-down vector with more than one term. Then Q_{s} is the commutative square

\xymatrix{Δ(q) \ar[d]_{(∂_{0}^{∨})^{|s|−p}} \ar[r]^{(∂_{1}^{∨})^{p−q}} & Δ(p) \ar[d]^{(∂_{0}^{∨})^{|s|−p}} \\ Δ(|s′|) \ar[r]_{(∂_{|s|−p+1}^{∨})^{p−q}} & Δ(|s|).}

8.4. Notation. For p ≥ 0 let U_{(p)} be the subcomplex of Δ(p) generated by the basis elements [i_{0}, …, i_{m}] with m > 0 and i_{1} ≤ p − m.

For s = (s′, q, p) let V_{s} be the subcomplex of Δ(|s|) generated by the basis elements [i_{0}, …, i_{r−1}, |s| − p, i_{r+1}, …, i_{m}] with 0 < r < m and

0 ≤ i_{r−1} < |s| − p < i_{r+1} ≤ |s| − q,

and let

U_{s} = (∂_{|s|−p+1}^{∨})^{p−q}U_{s′} + (∂_{0}^{∨})^{|s|−p}U_{(p)} + V_{s}.

8.5. Notation. For an arbitrary up-down vector s, let

S_{s} = Δ(|s|)/U_{s}.

8.6. Remark. As an abelian group, U_{s} is generated by basis elements of positive dimension and by their boundaries. It follows that the quotient S_{s} is naturally an augmented chain complex and the quotient homomorphism

Δ(|s|) → Δ(|s|)/U_{s} = S_{s}

is augmentation-preserving. We make S_{s} into an augmented directed complex by taking the images of the basis elements for Δ(|s|) as generators for the prescribed submonoid of S_{s}. This makes the quotient homomorphism into a morphism of augmented directed complexes.

We will now consider the one-term case.

8.7. Proposition. Let p be a nonnegative integer and let

a_{m}^{i} = [i, p − m + 1, p − m + 2, …, p] (0 ≤ i ≤ p − m ≤ p).

Then S_{(p)} is a (p)-simple chain complex with generator a such that

a_{m}^{i} + U_{(p)} = (∂^{−})^{p−m}a (0 ≤ i < p − m ≤ p),
a_{m}^{p−m} + U_{(p)} = (∂^{+})^{p−m}a (0 ≤ m ≤ p).

Proof. We use Definition 5.1. Note that U_{(p)} is the subcomplex of Δ(p) generated by the basis elements not of the form a_{m}^{i}. If b is a generator of U_{(p)} of the form

b = [j, i, p − m + 1, p − m + 2, p − m + 3, …, p] (0 ≤ j < i < p − m ≤ p),

then

∂b = a_{m}^{i} − a_{m}^{j} + u

with u ∈ U_{(p)}; if b is any other generator for U_{(p)} then ∂b ∈ U_{(p)}. As an abelian group, U_{(p)} is therefore generated by the basis elements not of the form a_{m}^{i} and by the differences

a_{m}^{i} − a_{m}^{0} (0 < i < p − m ≤ p).

It follows that S_{(p)} is a free augmented directed complex with basis

a_{0}^{0} + U_{(p)}, …, a_{p−1}^{0} + U_{(p)}, a_{p}^{0} + U_{(p)}, a_{p−1}^{1} + U_{(p)}, …, a_{0}^{p} + U_{(p)}

and that

a_{m}^{i} + U_{(p)} = a_{m}^{0} + U_{(p)} (0 ≤ i < p − m ≤ p).

It is straightforward to check that

∂(a_{m+1}^{0} + U_{(p)}) = a_{m}^{p−m} − a_{m}^{0} + U_{(p)} (0 ≤ m < p),
∂(a_{m+1}^{p−m−1} + U_{(p)}) = a_{m}^{p−m} − a_{m}^{p−m−1} + U_{(p)} (0 ≤ m < p),

from which it follows that

a_{m}^{i} + U_{(p)} = (∂^{−})^{p−m}(a_{p}^{0} + U_{(p)}) (0 ≤ i < p − m ≤ p),
a_{m}^{p−m} + U_{(p)} = (∂^{+})^{p−m}(a_{p}^{0} + U_{(p)}) (0 ≤ m ≤ p).

We also have

ϵ(a_{0}^{0} + U_{(p)}) = ϵ(a_{0}^{p} + U_{(p)}) = 1;

therefore S_{(p)} is (p)-simple with generator a_{p}^{p} + U_{(p)}, and the images a_{m}^{i} + U_{(p)} are as described.

We will now consider up-down vectors with more than one term. We first show that we can pass to quotients in the squares Q_{s}.

8.8. Proposition. Let s = (s′, q, p). Then the morphisms in the square Q_{s} restrict to morphisms between the subcomplexes U_{s′}, U_{(q)}, U_{(p)}, U_{s}.

Proof. We have

(∂_{|s|−p+1}^{∨})^{p−q}U_{s′} ⊂ U_{s}, (∂_{0}^{∨})^{|s|−p}U_{(p)} ⊂ U_{s}

by definition. We also have (∂_{1}^{∨})^{p−q}U_{(q)} ⊂ U_{(p)} by considering generators. It therefore remains to show that (∂_{0}^{∨})^{|s|−p}U_{(q)} ⊂ U_{s′}. To do this, let p′ be the last term of s′, so that

(∂_{0}^{∨})^{|s|−p}U_{(q)} = (∂_{0}^{∨})^{|s′|−q}U_{(q)} = (∂_{0}^{∨})^{|s′|−p′}(∂_{0}^{∨})^{p′−q}U_{(q)}.

We have (∂_{0}^{∨})^{p′−q}U_{(q)} ⊂ U_{(p′)} by considering generators. We also have

(∂_{0}^{∨})^{|s′|−p′}U_{(p′)} ⊂ U_{s′}

trivially (if s′ = (p′)) or by definition (if s′ has more than one term). Therefore (∂_{0}^{∨})^{|s|−p}U_{(q)} ⊂ U_{s′} as required.

It therefore makes sense to use the following notation.

8.9. Notation. Let s = (s′, q, p) be an up-down vector with more than one term. Then R_{s} is the commutative square

\xymatrix{S_{(q)} \ar[d] \ar[r] & S_{(p)} \ar[d] \\ S_{s′} \ar[r] & S_{s}}

induced by Q_{s}.

We want to show that these squares are simple in the sense of Definition 5.5. In particular we want to show that they are push-outs as squares of abelian groups, and we begin with the following computations.

8.10. Proposition. Let s = (s′, q, p) and let T′, T^{0}, T′′ be the subcomplexes of Δ(|s|) given by

T′ = (∂_{|s|−p+1}^{∨})^{p−q}Δ(|s′|),
T^{0} = (∂_{|s|−p+1}^{∨})^{p−q})(∂_{0}^{∨})^{|s|−p}Δ(q) = (∂_{0}^{∨})^{|s|−p}(∂_{1}^{∨})^{p−q})Δ(q),
T′′ = (∂_{0}^{∨})^{|s|−p}Δ(p).

Then

T′ ∩ T′′ = T^{0}, Δ(|s|) = (T′ + T′′) ⊕ V_{s},

and every standard basis element for Δ(|s|) is congruent modulo V_{s} to a sum of basis elements in T′ + T′′.

Proof. We consider various sets of basis elements of Δ(|s|). Let

J′ = {0, 1, …, |s| − p − 1}, J′′ = {|s| − p + 1, |s| − p + 2, …, |s| − q},

let A′ be the set of basis elements with no terms in J′′, and let A′′ be the set of basis elements with no terms in J′. We see that T′, T′′ and T^{0} have bases A′, A′′ and A′ ∩ A′′ respectively; therefore T′ ∩ T′′ = T^{0}.

Further, let B be the set of generators for V_{s}; that is, B is the set of basis elements containing terms in both J′ and J′′ and also containing a term |s|−p. We see that A′ ∪A′′ and B are disjoint; we also see that the boundary of a member of B has exactly one term not in A′ ∪ A′′ ∪ B, and that each basis element not in A′ ∪ A′′ ∪ B arises in this way from exactly one member of B; therefore Δ(|s|) = (T′ + T′′) ⊕ V_{s}.

It now suffices to show that every basis element c not in A′ ∪ A′′ ∪ B is congruent to a sum of members of A′ ∪A′′ modulo V_{s}. To do this, note that c has terms in both J′ and J′′ but has no term |s| − p. Let b be the member of B obtained from c by inserting |s| − p, so that c is a term in ∂b, and let u′, u′′ be the terms of ∂b adjacent to c. We see that

∂b = ±(u′ − c + u′′) + v

such that v is a linear combination of members of B; therefore c is congruent to u′ + u′′ modulo V_{s}. We also see that u′ and u′′ are in A′ ∪ A′′ ∪ B; therefore c is congruent modulo V_{s} to a sum of basis elements in T′ + T′′ as required.

8.11. Proposition. Let s = (s′, q, p) be an up-down vector with more than one term. Then the square R_{s} is a push-out as a square of abelian groups. The prescribed submonoid in the target object S_{s} is generated by the images of the prescribed submonoids in S_{s′} and S_{(p)}.

Proof. Recall that

U_{s} = (∂_{|s|−p+1}^{∨})^{p−q}U_{s′} + (∂_{0}^{∨})^{|s|−p}U_{(p)} + V_{s}.

From Proposition 8.10, a morphism θ of abelian groups with domain Δ(|s|) such that θ|U_{s} = 0 is equivalent to a pair of morphisms θ′ and θ′′ with domains Δ(|s′|) and Δ(p) such that

θ′|U_{s′} = 0, θ′′|U_{(p)} = 0, θ′(∂_{0}^{∨})^{|s|−p} = θ′′(∂_{1}^{∨})^{p−q}.

From this it follows that a morphism χ of abelian groups with domain S_{s} is equivalent to a pair of morphisms χ′ and χ′′ with domains S_{s′} and S_{(p)} which agree on S_{(q)}. Therefore R_{s} is a push-out as a square of abelian groups.

It also follows from Proposition 8.10 that the prescribed submonoid in S_{s} is generated by the images of the prescribed submonoids in S_{s′} and S_{(p)}, because the basis elements in Δ(|s|) are congruent modulo V_{s} to sums of basis elements in

(∂_{|s|−p+1}^{∨})^{p−q}Δ(|s′|) + (∂_{0}^{∨})^{|s|−p}Δ(p).

This completes the proof.

We can now give the main result in this section.

8.12. Theorem. Let s be an up-down vector with last term p and let

a_{m} = [ |s| − m, |s| − m + 1, |s| − m + 2, …, |s|] (0 ≤ m ≤ p).

Then S_{s} is an s-simple chain complex with final generator a such that

a_{m} + U_{s} = (∂^{+})^{p−m}a (0 ≤ m ≤ p).

If s has more than one term, then R_{s} is an s-simple square of chain complexes.

Proof. The proof is by induction on the number of terms in s. Proposition 8.7 gives the result for the case s = (p). From now on let s = (s′, q, p), let p′ be the last term of s′, and recall that the square R_{s} has the form

\xymatrix{S_{(q)} \ar[d]_{π} \ar[r]^{ρ} & S_{(p)} \ar[d]^{τ} \\ S_{s′} \ar[r]_{σ} & S_{s}.}

From the inductive hypothesis and the one-term case we see that S_{s′}, S_{(q)} and S_{(p)} are s′- simple, (q)-simple and (p)-simple; let the final generators be a′, a^{0} and a′′. For 0 ≤ m ≤ q it follows from the equalities

(∂_{0}^{∨})^{|s|−p}[q − m, q − m + 1, …, q] = [ |s′| − m, |s′| − m + 1, …, |s′|],
(∂_{1}^{∨})^{p−q}[0, q − m + 1, …, q] = [0, p − m + 1, …, p]

that

π(∂^{+})^{q−m}a^{0} = (∂^{+})^{p′−m}a′,
π(∂^{−})^{q−m}a^{0} = (∂^{−})^{p−m}a′′.

It now follows from Proposition 8.11 that R_{s} is an s-simple square (see Definition 5.5). By Proposition 5.6, this makes S_{s} an s-simple chain complex with final generator a such that

τ(∂^{+})^{p−m}a′′ = (∂^{+})^{p−m}a (0 ≤ m ≤ p).

For 0 ≤ m ≤ p let

a′′_{m} = [p − m, p − m + 1, …, p] ∈ Δ(p),

so that a′′_{m} + U_{(p)} = (∂^{+})^{p−m}a′′ by Proposition 8.7; then

a_{m} + U_{s} = (∂_{0}^{∨})^{|s|−p}a′′_{m} + U_{s} = τ(a′′_{m} + U_{(p)}) = τ(∂^{+})^{p−m}a′′ = (∂^{+})^{p−m}a.

This completes the proof.

9. Combined operations in sets with complicial identities

In Theorem 8.12 we have constructed an s-simple chain complex S_{s} as a quotient of a simplex,

S_{s} = Δ(|s|)/U_{s}.

We really want to express S_{s} as a retract of Δ(|s|); that is, we want to construct an idempotent endomorphism of Δ(|s|) with kernel U_{s}. In this section we construct the corresponding operation on |s|-dimensional elements in sets with complicial identities; this operation will be denoted Ψ_{s}. We use the axioms of Definition 3.1 throughout.

We will construct Ψ_{s} by iterating wedge operations. There are two basic families of iterated wedges, and we will now describe the first of these families.

9.1. Notation. For 0 < i ≤ i + j ≤ m, let ϕ̃_{i,j} and ϕ_{i,j} be the operations on mdimensional elements in sets with complicial identities given by

ϕ̃_{i,0}x = ϵ_{i−1}x,
ϕ̃_{i,j}x = ϕ̃_{i,j−1}∂_{i+1}x ∧_{i} x (j > 0),
ϕ_{i,j}x = ∂_{i+1}ϕ̃_{i,j}x.

The wedge in the formula for ϕ̃_{i,j}x exists by an induction on j: if ϕ̃_{i,j−1}x exists and is given by the stated formula, then

∂_{i}ϕ̃_{i,j−1}∂_{i+1}x = ∂_{i+1}x,

and the wedge ϕ̃_{i,j−1}∂_{i+1}x ∧_{i} x therefore exists.

We will now compute some faces and some fixed point sets.

9.2. Proposition. The operations ϕ_{i,j} and ϕ̃_{i,j} are such that

∂_{i}ϕ_{i,j} = ∂_{i}, ∂_{i+1}^{j}ϕ_{i,j} = ∂_{i+1}^{j+1}ϕ̃_{i,j} = ϵ_{i−1}∂_{i}^{j+1}.

Proof. The first formula holds because

∂_{i}ϕ_{i,j}x = ∂_{i}∂_{i+1}ϕ̃_{i,j}x = ∂_{i}∂_{i}ϕ̃_{i,j}x = ∂_{i}x.

The second formula holds by induction on j: we certainly have

ϕ_{i,0}x = ∂_{i+1}ϕ̃_{i,0}x = ∂_{i+1}ϵ_{i−1}x = ϵ_{i−1}∂_{i}x,

and for j > 0 we have

∂_{i+1}^{j}ϕ_{i,j}x = ∂_{i+1}^{j+1}ϕ̃_{i,j}x
= ∂_{i+1}^{j}∂_{i+2}(ϕ̃_{i,j−1}∂_{i+1}x ∧_{i} x)
= ∂_{i+1}^{j}ϕ̃_{i,j−1}∂_{i+1}x
= ϵ_{i−1}∂_{i}^{j}∂_{i+1}x
= ϵ_{i−1}∂_{i}^{j+1}x.

9.3. Proposition. Let x be a member of a set with complicial identities. Then

ϕ̃_{i,j}x = ϵ_{i}x ⟺ ϕ_{i,j}x = x ⟺ ∂_{i+1}^{j}x ∈ im ϵ_{i−1}.

Proof. Suppose that ϕ̃_{i,j}x = ϵ_{i}x. Then

ϕ_{i,j}x = ∂_{i+1}ϵ_{i}x = x.

Suppose that ϕ_{i,j}x = x. Then

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

It now suffices to show that

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

We argue by induction on j.

Suppose that x ∈ im ϵ_{i−1}. Since ϵ_{i−1}ϵ_{i−1} = ϵ_{i}ϵ_{i−1}, it follows that ϕ̃_{i,0}x = ϵ_{i}x.

Now suppose that ∂_{i+1}^{j}x ∈ im ϵ_{i−1} for some j > 0. It follows from the inductive hypothesis that ϕ̃_{i,j−1}∂_{i+1}x = ϵ_{i}∂_{i+1}x, and it then follows that

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

This completes the proof.

We now consider the second basic family of iterated wedges.

9.4. Notation. For integers k, l ≥ 0 let ∧_{k,l} be the partial binary operation in sets with complicial identities such that x ∧_{k,l} y is defined when

∂_{0}^{k}x = ∂_{1}^{l}y

and such that

x ∧_{k,l} y = x (l = 0),
x ∧_{k,l} y = y (k = 0),
x ∧_{k,l} y = (x ∧_{k,l−1} ∂_{1}y) ∧_{k−1} (∂_{k−1}x ∧_{k−1,l} y) (k, l > 0),
∂_{k−1}(x ∧_{k,l} y) = ∂_{k−1}x ∧_{k−1,l} y (k > 0),
∂_{k+1}(x ∧_{k,l} y) = x ∧_{k,l−1} ∂_{1}y (l > 0).

To justify this definition, let x and y be such that ∂_{0}^{k}x = ∂_{1}^{l}y; we must show that the stated conditions make sense and are consistent. We do this by induction on k and l.

Suppose that k = l = 0. Then x = y, so the conditions x ∧_{k,l} y = x and x ∧_{k,l} y = y are consistent.

Suppose that k > 0 and l = 0. Then ∂_{0}^{k−1}∂_{k−1}x = ∂_{0}^{k}x = ∂_{1}^{l}y; hence, by the inductive hypothesis, ∂_{k−1}x ∧_{k−1,l} y exists and is equal to ∂_{k−1}x. The conditions x ∧_{k,l} y = x and ∂_{k−1}(x ∧_{k,l} y) = ∂_{k−1}x ∧_{k−1,l} y therefore make sense and are consistent.

Suppose that k = 0 and l > 0. Then ∂_{0}^{k}x = ∂_{1}^{l}y = ∂_{1}^{l−1}∂_{1}y; hence x ∧_{k,l−1} ∂_{1}y exists and is equal to ∂_{1}y. The conditions x ∧_{k,l} y = y and ∂_{k+1}(x ∧_{k,l} y) = ∂_{1}y therefore make sense and are consistent.

Finally suppose that k > 0 and l > 0. As in the previous cases, the expressions ∂_{k−1}x ∧_{k−1,l} y and x ∧_{k,l−1} ∂_{1}y make sense. We also have

∂_{k−1}(x ∧_{k,l−1} ∂_{1}y) = ∂_{k−1}x ∧_{k−1,l−1} ∂_{1}y = ∂_{k}(∂_{k−1}x ∧_{k−1,l} y),

so the conditions

x ∧_{k,l} y = (x ∧_{k,l−1} ∂_{1}y) ∧_{k−1} (∂_{k−1}x ∧_{k−1,l} y),
∂_{k−1}(x ∧_{k,l} y) = ∂_{k−1}x ∧_{k−1,l} y,
∂_{k+1}(x ∧_{k,l} y) = x ∧_{k,l−1} ∂_{1}y

all make sense. It is also clear that they are consistent.

Each of these binary operations determines its own factors.

9.5. Proposition. If x ∧_{k,l} y is defined, then

∂_{k+1}^{l}(x ∧_{k,l} y) = x, ∂_{0}^{k}(x ∧_{k,l} y) = y.

Proof. The first equality is proved by induction on l; for l = 0 it is obvious, and for l > 0 we have

∂_{k+1}^{l}(x ∧_{k,l} y) = ∂_{k+1}^{l−1}∂_{k+1}(x ∧_{k,l} y) = ∂_{k+1}^{l−1}(x ∧_{k,l−1} ∂_{1}y) = x.

The second equality is similarly proved by induction on k; for k = 0 it is obvious, and for k > 0 we have

∂_{0}^{k}(x ∧_{k,l} y) = ∂_{0}^{k−1}∂_{k−1}(x ∧_{k,l} y) = ∂_{0}^{k−1}(∂_{k−1}x ∧_{k,l} y) = y.

Because of the simplicial identities ∂_{0}^{k}∂_{k+1}^{l} = ∂_{1}^{l}∂_{0}^{k} there are everywhere defined unary operations as follows.

9.6. Notation. If k and l are nonnegative integers and if x is an element of dimension at least k + l in a set with complicial identities, then

w_{k,l}x = ∂_{k+1}^{l}x ∧_{k,l} ∂_{0}^{k}x.

9.7. Proposition. The operations w_{k,l} are idempotent operations such that

∂_{k+1}^{l}w_{k,l} = ∂_{k+1}^{l}, ∂_{0}^{k}w_{k,l} = ∂_{0}^{k}.

If X is a set with complicial identities and if q ≥ 0 then the square

\xymatrix{w_{k,l}X_{k+l+q} \ar[d]_{∂_{k+1}^{l}} \ar[r]^{∂_{0}^{k}} & X_{l+q} \ar[d]^{∂_{1}^{l}} \\ X_{k+q} \ar[r]_{∂_{0}^{k}} & X_{q}}

is a pull-back square.

Proof. For all x it follows from Proposition 9.5 that

∂_{k+1}^{l}w_{k,l}x = ∂_{k+1}^{l}(∂_{k+1}^{l}x ∧_{k,l} ∂_{0}^{k}x) = ∂_{k+1}^{l}x,
∂_{0}^{k}w_{k,l}x = ∂_{0}^{k}(∂_{k+1}^{l}x ∧_{k,l} ∂_{0}^{k}x) = ∂_{0}^{k}x;

therefore ∂_{k+1}^{l}w_{k,l} = ∂_{k+1}^{l} and ∂_{0}^{k}w_{k,l} = ∂_{0}^{k}. It then follows that

w_{k,l}w_{k,l}x = ∂_{k+1}^{l}w_{k,l}x ∧_{k,l} ∂_{0}^{k}w_{k,l}x = ∂_{k+1}^{l}x ∧_{k,l} ∂_{0}^{k}x = w_{k,l}x;

therefore w_{k,l} is idempotent. If x ∈ w_{k,l}X_{k+l+q} then certainly ∂_{0}^{k}∂_{k+1}^{l}x = ∂_{1}^{l}∂_{0}^{k}x. Conversely, if y ∈ X_{k+q} and z ∈ X_{l+q} are such that ∂_{0}^{k}y = ∂_{1}^{l}z then

x = w_{k,l}(y ∧_{k+l} z)

is a member of w_{k,l}X_{k+l+q} such that ∂_{k+1}^{l}x = y and ∂_{0}^{k}x = z and it is clearly the unique such member; therefore the square is a pull-back square.

We will now combine the operations ϕ_{i,j} and w_{k,l}. Let s be an up-down vector. We will construct operators ψ_{i,s} on |s|-dimensional elements for 0 < i < |s|. We begin with large values of i and work downwards.

9.8. Notation. Let s be an up-down vector with last term p and let x be an |s|- dimensional element in a set with complicial identities. Then

ψ_{i,s}x = ϕ_{i,1}x (|s| − p < i < |s|).

9.9. Notation. Let s = (s′, q, p) and let x be an |s|-dimensional element in a set with complicial identities. Then

ψ_{|s|−p,s}x = \begin{cases}w_{|s|−p,p−q}x & (q = 0), \\ w_{|s|−p,p−q}ϕ_{|s|−p,p−q+1}x & (q > 0). \end{cases}

9.10. Proposition. If s is an up-down vector with last term p and if ψ_{i,s} is an operator with |s| − p ≤ i < |s|, then

∂_{0}^{j}ψ_{i,s} = ∂_{0}^{j} (i < j ≤ |s|).

Proof. In cases with i > |s| − p the result holds by Proposition 9.2 because

∂_{0}^{j}ψ_{i,s} = ∂_{0}^{j−1}∂_{i}ϕ_{i,1} = ∂_{0}^{j−1}∂_{i} = ∂_{0}^{j}.

In cases with i = |s| − p we have

ψ_{i,s} = w_{|s|−p,p−q}

or we have

ψ_{i,s} = w_{|s|−p,p−q}ϕ_{|s|−p,p−q+1}

for some q. The result now holds because

∂_{0}^{j}w_{|s|−p,p−q} = ∂_{0}^{j−|s|+p}∂_{0}^{|s|−p}w_{|s|−p,p−q} = ∂_{0}^{j−|s|+p}∂_{0}^{|s|−p} = ∂_{0}^{j}

by Proposition 9.7 and because ∂_{0}^{j}ϕ_{|s|−p,p−q+1} = ∂_{0}^{j} by Proposition 9.2 as before.

In the remaining cases we use induction on the number of terms in s.

9.11. Notation. Let s = (s′, q, p), let x be an |s|-dimensional element in a set with complicial identities, and let i be an integer with 0 < i < |s| − p. Then ψ_{i,s}x is the element such that

ψ_{i,s}x = ψ_{i,s′}∂_{|s|−p+1}^{p−q}x ∧_{|s|−p,p−q} ∂_{0}^{|s|−p}x

and

∂_{0}^{j}ψ_{i,s}x = ∂_{0}^{j}x (|s| − p ≤ j ≤ |s|).

To justify this definition, we must show that the two conditions make sense and are consistent. We argue by induction on the number of terms in s. Let p′ be the last term in s′, so that

|s| − p = |s′| − q > |s′| − p′.

If i ≥ |s′| − p′ then ∂_{0}^{|s|−p}ψ_{i,s′} = ∂_{0}^{|s|−p} by Proposition 9.10; if i < |s| − p′ then ∂_{0}^{|s|−p}ψ_{i,s′} = ∂_{0}^{|s|−p} by the inductive hypothesis. In all cases it follows that

∂_{0}^{|s|−p}ψ_{i,s′}∂_{|s|−p+1}^{p−q}x = ∂_{0}^{|s|−p}∂_{|s|−p+1}^{p−q}x = ∂_{1}^{p−q}∂_{0}^{|s|−p}x.

The condition

ψ_{i,s}x = ψ_{i,s′}∂_{|s|−p+1}^{p−q}x ∧_{|s|−p,p−q} ∂_{0}^{|s|−p}x

therefore makes sense because the iterated wedge exists. This condition actually implies the other condition by Proposition 9.7, because for |s| − p ≤ j ≤ |s| we have

∂_{0}^{j}(ψ_{i,s′}∂_{|s|−p+1}^{p−q}x ∧_{|s|−p,p−q} ∂_{0}^{|s|−p}x)
= ∂_{0}^{j−|s|+p}∂_{0}^{|s|−p}(ψ_{i,s′}∂_{|s|−p+1}^{p−q}x ∧_{|s|−p,p−q} ∂_{0}^{|s|−p}x)
= ∂_{0}^{j−|s|+p}∂_{0}^{|s|−p}x
= ∂_{0}^{j}x.

Finally we construct the operation Ψ_{s} as an iterated composite.

9.12. Notation. Let s be an up-down vector. Then Ψ_{s} is the operation on |s|- dimensional elements in sets with complicial identities given by

Ψ_{s} = (ψ_{1,s})(ψ_{2,s}ψ_{1,s})(ψ_{3,s}ψ_{2,s}ψ_{1,s}) ⋯ (ψ_{|s|−1,s} ⋯ ψ_{2,s}ψ_{1,s})

(to be interpreted as the identity when |s| ≤ 1).

10. The induced morphisms between simplexes

In Section 9 we have constructed operations in sets with complicial identities. We will now give some results concerning the induced morphisms between simplexes.

10.1. Proposition. If θ_{1} and θ_{2} are operations in sets with complicial identities such that ∂_{i}θ_{1} = ∂_{i+1}θ_{2} and if θ is the operation given by

θx = θ_{1}x ∧_{i} θ_{2}x,

then

θ^{∨} = θ_{1}^{∨}ϵ_{i+1}^{∨} − θ_{1}^{∨}∂_{i}^{∨}(ϵ_{i}^{∨})^{2} + θ_{2}^{∨}ϵ_{i}^{∨} = θ_{1}^{∨}ϵ_{i+1}^{∨} − θ_{2}^{∨}∂_{i+1}^{∨}(ϵ_{i}^{∨})^{2} + θ_{2}^{∨}ϵ_{i}^{∨}.

Proof. This follows from the construction of wedges in sets with complicial identities of the form λK; see Notation 6.9.

10.2. Proposition. If j > 0 then

ϕ_{i,j}^{∨} = (∂_{i+1}^{∨})^{j}[ϵ_{i−1}^{∨} − ϵ_{i}^{∨}](ϵ_{i+1}^{∨})^{j−1} + id .

Proof. Recall from Notation 9.1 that ϕ_{i,j} = ∂_{i+1}ϕ̃_{i,j} with

ϕ̃_{i,0}x = ϵ_{i−1}x,
ϕ̃_{i,j}x = ϕ̃_{i,j−1}∂_{i+1}x ∧_{i} x (j > 0).

It follows from Proposition 10.1 by induction on j that

ϕ̃_{i,j}^{∨} = (∂_{i+1}^{∨})^{j}[ϵ_{i−1}^{∨} − ϵ_{i}^{∨})](ϵ_{i+1}^{∨})^{j} + ϵ_{i}^{∨} (j ≥ 0),

and for j > 0 it then follows that

ϕ_{i,j}^{∨} = ϕ̃_{i,j}^{∨}∂_{i+1}^{∨} = (∂_{i+1}^{∨})^{j}[ϵ_{i−1}^{∨} − ϵ_{i}^{∨})](ϵ_{i+1}^{∨})^{j−1} + id .

10.3. Proposition. If θ_{1} and θ_{2} are operations in sets with complicial identities such that ∂_{0}^{k}θ_{1} = ∂_{1}^{l}θ_{2} and if θ is the operation given by

θx = θ_{1}x ∧_{k,l} θ_{2}x,

then

θ^{∨} = θ_{1}^{∨}(ϵ_{k}^{∨})^{l} − θ_{1}^{∨}(∂_{0}^{∨})^{k}(ϵ_{0}^{∨})^{k+l} + θ_{2}^{∨}(ϵ_{0}^{∨})^{k}
= θ_{1}^{∨}(ϵ_{k}^{∨})^{l} − θ_{2}^{∨}(∂_{1}^{∨})^{k}(ϵ_{0}^{∨})^{k+l} + θ_{2}^{∨}(ϵ_{0}^{∨})^{k}.

Proof. This follows by induction from Proposition 10.1 using the formulae

θ_{1}x ∧_{k,0} θ_{2}x = θ_{1}x,
θ_{1}x ∧_{0,l} θ_{2}x = θ_{2}x,

θ_{1}x ∧_{k,l} θ_{2}x = (θ_{1}x ∧_{k,l−1} ∂_{1}θ_{2}x) ∧_{k−1} (∂_{k−1}θ_{1}x ∧_{k−1,l} θ_{2}x) (k, l > 0)

(see Notation 9.4).

10.4. Proposition. The morphisms w_{k,l}^{∨} are given by

w_{k,l}^{∨} = (∂_{k+1}^{∨})^{l}(ϵ_{k}^{∨})^{l} − (∂_{k+1}^{∨})^{l}(∂_{0}^{∨})^{k}(ϵ_{0}^{∨})^{k+l} + (∂_{0}^{∨})^{l}(ϵ_{0}^{∨})^{l}.

Proof. This follows from Proposition 10.3, because

w_{k,l}x = ∂_{k+1}^{l}x ∧_{k,l} ∂_{0}^{k}x

(see Notation 9.6).

10.5. Proposition. If s = (s′, q, p) and if 0 < i < |s| − p, then

ψ_{i,s}^{∨} − w_{|s|−p,p−q}^{∨} = (∂_{|s|−p+1}^{∨})^{p−q}[ψ_{i,s′}^{∨} − id](ϵ_{|s|−p}^{∨})^{p−q},
ψ_{i,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q}ψ_{i,s′}^{∨},
ψ_{i,s}^{∨}(∂_{0}^{∨})^{|s|−p} = (∂_{0}^{∨})^{|s|−p}.

Proof. By Definition (see Notation 9.11),

ψ_{i,s}x = ψ_{i,s′}∂_{|s|−p+1}^{p−q}x ∧_{|s|−p,p−q} ∂_{0}^{|s|−p}x.

The formula for ψ_{i,s}^{∨}−w_{|s|−p,p−q}^{∨} follows from Propositions 10.3 and 10.4. The other two formulae hold because ∂_{|s|−p+1}^{p−q}ψ_{i,s} = ψ_{i,s′}∂_{|s|−p+1}^{p−q} and ∂_{0}^{|s|−p}ψ_{i,s} = ∂_{0}^{|s|−p} (see Proposition 9.5).

10.6. Proposition. If s = (s′, q, p) with q = 0, then

ψ_{|s|−p,s}^{∨} − w_{|s|−p,p−q}^{∨} = 0,
ψ_{|s|−p,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q},
ψ_{|s|−p,s}^{∨}(∂_{0}^{∨})^{|s|−p} = (∂_{0}^{∨})^{|s|−p}.

Proof. By Notation 9.9, ψ_{|s|−p,s} = w_{|s|−p,p−q}. The last two formulae follow from Proposition 10.4.

10.7. Proposition. If s = (s′, q, p) with q > 0, then

ψ_{|s|−p,s}^{∨} − w_{|s|−p,p−q}^{∨} = (∂_{|s|−p+1}^{∨})^{p−q}[ψ_{|s|−p,s′}^{∨} − id](ϵ_{|s|−p+1}^{∨})^{p−q}w_{|s|−p,p−q}^{∨},
ψ_{|s|−p,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q}ψ_{|s|−p,s′}^{∨},
[ψ_{|s|−p,s}^{∨} − id](∂_{0}^{∨})^{|s|−p}
= (∂_{|s|−p+1}^{∨})^{p−q+1}[ϵ_{|s|−p−1}^{∨} − ϵ_{|s|−p}^{∨}](ϵ_{|s|−p+1}^{∨})^{p−q}(∂_{0}^{∨})^{|s|−p}.

Proof. In this case ψ_{|s|−p,s} = w_{|s|−p,p−q+1}ϕ_{|s|−p,p−q+1} (see Notation 9.9), hence

ψ_{|s|−p,s}^{∨} − w_{|s|−p,p−q}^{∨}
= [ϕ_{|s|−p,p−q+1}^{∨} − id]w_{|s|−p,p−q}^{∨}
= (∂_{|s|−p+1}^{∨})^{p−q+1}[ϵ_{|s|−p−1}^{∨} − ϵ_{|s|−p}^{∨}](ϵ_{|s|−p+1}^{∨})^{p−q}w_{|s|−p,p−q}^{∨}
= (∂_{|s|−p+1}^{∨})^{p−q}[ϕ_{|s|−p,1}^{∨} − id](ϵ_{|s|−p+1}^{∨})^{p−q}w_{|s|−p,p−q}^{∨}

by Proposition 10.2. We also have ϕ_{|s|−p,1} = ψ_{|s|−p,s′} by Notation 9.8 because |s| − p > |s′| − p′, where p′ is the last term in s′. The results follow.

10.8. Proposition. If s = (s′, q, p) then

ψ_{i,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q} (|s| − p < i < |s| − q),
ψ_{|s|−q,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q} (q > 0),
ψ_{i,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q}ψ_{i−p+q,s′}^{∨} (|s| − q + 1 < i < |s|),
w_{|s|−p,p−q}^{∨}ψ_{|s|−q+1,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q} = (∂_{|s|−p+1}^{∨})^{p−q}ψ_{|s|−p+1,s′}^{∨}
+ (∂_{0}^{∨})^{|s|−p}[ϕ_{p−q+1,1}^{∨}(∂_{1}^{∨})^{p−q} − (∂_{1}^{∨})^{p−q}ϕ_{1,1}^{∨}](ϵ_{0}^{∨})^{|s|−p} (q > 1),
ψ_{|s|−q,s}^{∨}(∂_{0}^{∨})^{|s|−p} = (∂_{0}^{∨})^{|s|−p}ψ_{i−|s|+p,(p)}^{∨} (|s| − p < i < |s|).

Proof. According to Notation 9.8 we have

ψ_{i,s} = ϕ_{i,1}, ψ_{i−|s|+p,(p)} = ϕ_{i−|s|+p,1}

for |s| − p < i < |s|. We also have ψ_{i−p+q,s′} = ϕ_{i−p+q,1} for |s| − q < i < |s| because

i − p + q > |s| − p > |s′| − p′,

where p′ is the last term in s′. The results now follow from Propositions 10.2 and 10.4.

Finally in this section, we consider the action of ψ_{i,s}^{∨} on V_{s} in the case s = (s′, q, p); we recall from Notation 8.4 that V_{s} is the subcomplex of Δ(|s|) generated by the basis elements

[i_{0}, …, i_{r−1}, |s| − p, i_{r+1}, …, i_{m}]

with 0 ≤ i_{r−1} < |s| − p < i_{r+1} ≤ |s| − q.

10.9. Proposition. If s = (s′, q, p) then

ψ_{i,s}^{∨}V_{s} = 0 (0 < i ≤ |s| − p),
ψ_{i,s}^{∨}V_{s} ⊂ V_{s} (|s| − p < i < |s).

Proof. For 0 < i < |s| − p we have

ψ_{i,s}^{∨} = (∂_{|s|−p+1}^{∨})^{p−q}ψ_{i,s′}^{∨}(ϵ_{|s|−p}^{∨})^{p−q} − (∂_{|s|−p+1}^{∨})^{p−q}(∂_{0}^{∨})^{|s|−p}(ϵ_{0}^{∨})^{|s|−q} + (∂_{0}^{∨})^{|s|−p}(ϵ_{0}^{∨})^{|s|−p}

(see Notation 9.11 and Proposition 9.3), hence ψ_{i,s}^{∨}V_{s} = 0. In the same way w_{|s|−p,p−q}^{∨}V_{s} = 0 by Proposition 10.4; hence, by Notation 9.9, ψ_{|s|−p,s}^{∨}V_{s} = 0. For |s| − p < i < |s| we have ψ_{i,s} = ϕ_{i,1} by Notation 9.8, hence ψ_{i,s}^{∨}V_{s} ⊂ V_{s} by Proposition 10.2.

11. Simple chain complexes as retracts of simplexes

Given an up-down vector s, we have shown in Section 8 that the simplex Δ(|s|) has an s-simple quotient

S_{s} = Δ(|s|)/U_{s}.

In Section 9 we have constructed an operation Ψ_{s} on |s|-dimensional elements in sets with complicial identities. By Theorem 6.11 there is a corresponding endomorphism Ψ_{s}^{∨} of Δ(|s|). We will now show that S_{s} is a retract of Δ(|s|) by showing that Ψ_{s}^{∨} is idempotent with kernel U_{s}.

The method is as follows. By construction (see Notation 9.12), Ψ_{s}^{∨} is a composite,

Ψ_{s}^{∨} = (ψ_{1,s}^{∨}ψ_{2,s}^{∨} ⋯ ψ_{|s|−1,s}^{∨}) ⋯ (ψ_{1,s}^{∨}ψ_{2,s}^{∨}ψ_{3,s}^{∨})(ψ_{1,s}^{∨}ψ_{2,s}^{∨})(ψ_{1,s}^{∨}).

We will construct subcomplexes U_{s}^{j} of Δ(|s|) for 0 ≤ j < |s| such that

U_{s} = U_{s}^{0} + ⋯ + U_{s}^{|s|−1},
ψ_{i,s}^{∨}U_{s}^{j} ⊂ U_{s}^{j} (0 < i < j < |s|),
ψ_{j,s}^{∨}U_{s}^{j} ⊂ U_{s}^{j−1} (0 < j < |s|),
U_{s}^{0} = 0,

from which it will follow that U_{s} ⊂ ker Ψ_{s}^{∨}. We will also show that

(ψ_{i,s}^{∨} − id)Δ(|s|) ⊂ U_{s} (0 < i < |s|),

from which it will follow that (Ψ_{s}^{∨} − id)Δ(|s|) ⊂ U_{s}. From these inclusions it will indeed follow that Ψ_{s}^{∨} is idempotent with kernel U_{s} as required.

The subcomplexes U_{s}^{j} are defined by induction on the number of terms in s. In the many-term case s = (s′, q, p) recall from Notation 8.4 that

U_{s} = (∂_{|s|−p+1}^{∨})^{p−q}U_{s′} + (∂_{0}^{∨})^{|s|−p}U_{(p)} + V_{s},

where V_{s} is the subcomplex of Δ(|s|) generated by the basis elements

[i_{0}, …, i_{r−1}, |s| − p, i_{r+1}, …, i_{m}]

with 0 ≤ i_{r−1} < |s| − p < i_{r+1} ≤ |s| − q.

11.1. Notation. For 0 ≤ j < p let U_{(p)}^{j} be the subcomplex of Δ(p) generated by the basis elements [i_{0}, …, i_{m}] with at least two terms less than or equal to j and with no term equal to j + 1.

For s = (s′, q, p) with q = 0, let

U_{s}^{j} = \begin{cases}(∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j} & (0 ≤ j < |s| − p), \\ (∂_{0}^{∨})^{|s|−p}U_{(p)}^{j−|s|+p} + V_{s} & (|s| − p ≤ j < |s|). \end{cases}

For s = (s′, q, p) with q > 0, let

U_{s}^{j} = \begin{cases}(∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j} & (0 ≤ j < |s| − p), \\ (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{|s|−p} \\ + (∂_{0}^{∨})^{|s|−p}U_{(p)}^{j−|s|+p} + V_{s} & (|s| − p ≤ j < |s| − q), \\ (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j−p+q} \\ + (∂_{0}^{∨})^{|s|−p}U_{(p)}^{j−|s|+p} + V_{s} & (|s| − q ≤ j < |s|). \end{cases}

We begin with the following result.

11.2. Proposition. Let s be an up-down vector. Then

U_{s} = U_{s}^{0} + ⋯ + U_{s}^{|s|−1},
U_{s}^{0} = 0.

Proof. We use induction on the number of terms in s. The inductive step is obvious; we will therefore consider the one-term case s = (p).

Recall from Notation 8.4 that U_{(p)} is the subcomplex of Δ(p) generated by the basis elements [i_{0}, …, i_{m}] with m > 0 and i_{1} ≤ p − m. It is easy to see that the generating set for U_{(p)} is the union of the generating sets for U_{(p)}^{0}, …, U_{(p)}^{p−1}; therefore U_{(p)} = U_{(p)}^{0} + … + U_{(p)}^{p−1}. It is also easy to see that the generating set for U_{(p)}^{0} is empty; therefore U_{(p)}^{0} = 0.

This completes the proof.

We will now give three lemmas aimed at describing the subcomplexes V_{s} more explicitly.

11.3. Lemma. If k is a fixed integer with 0 ≤ k ≤ m then Δ(m) is generated as a chain complex by the basis elements [j_{0}, …, j_{m}] including a term equal to k.

Proof. Let a be a basis element not including k, and let b be the basis element obtained by inserting a term equal to k in a. Then ∂b has a term equal to a, and every other term of ∂b includes k. The result follows.

11.4. Lemma. If s = (s′, q, p) then

[w_{|s|−p,p−q}^{∨} − id]Δ(|s|) ⊂ V_{s}.

Proof. Let a be a basis element for Δ(|s|) which includes the term |s| − p. By Proposition 10.4, if a is a generator for V_{s} then w_{|s|−p,p−q}^{∨}a = 0; if a is not a generator for V_{s} then w_{|s|−p,p−q}^{∨}a = a. The result now follows because of Lemma 11.3.

11.5. Lemma. Let s be an up-down vector with last term p and let A be the set of basis elements [j_{0}, …, j_{m}] for Δ(|s|) with at least two terms in the set

{ 0, 1, …, |s| − p }.

For 0 ≤ j < |s| let B^{j} be the set of basis elements with at least two terms in the set

{0, 1, …, j}

and with no term j + 1, and let C^{j} be the set of basis elements with at least two terms in the set

{ |s| − p, |s| − p + 1, …, j}

and with no term j + 1. Then U_{s}^{j} is generated as a chain complex by a subset of A ∪ B^{j}, and U_{s}^{j} contains every member of C^{j}.

Proof. The proof is by induction on the number of terms in s.

If s = (p) then the results hold because U_{s}^{j} is generated by the members of C^{j} and because B^{j} = C^{j}.

From now on let s = (s′, q, p) and let p′ be the last term of s′. We will first show that U_{s}^{j} is generated by some of the members of A ∪ B^{j}. We do this by considering the various constituents of U_{s}^{j}.

Suppose that 0 ≤ j < |s| − p. Since |s′| − p′ < |s| − p, it follows from the inductive hypothesis that U_{s′}^{j} is generated by basis elements with at least two terms less than or equal to |s| − p, and it then follows that (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j} is generated by members of A.

Suppose that q > 0 and |s| − p ≤ j < |s| − q. Then (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{|s|−p} is generated by members of A as in the previous case.

Suppose that |s| − q ≤ j < |s|. Then U_{s′}^{j−p+q} is generated by basis elements with at least two terms less than or equal to |s| − p, or with at least two terms less than or equal to j − p + q and with no term j − p + q + 1. It follows that (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j−p+q} is generated by members of A ∪ B^{j}.

For |s| − p ≤ j < |s| it is clear that (∂_{0}^{∨})^{|s|−p}U_{(p)}^{j−|s|+p} is generated by members of B^{j}. It is also clear that V_{s} is generated by members of A. From these results it follows in all cases that U_{s}^{j} is generated by members of A ∪ B^{j}. Next we show that every member c of C^{j} is in U_{s}^{j}. There is nothing to prove in cases with 0 ≤ j < |s| − p, because in those cases C^{j} is

empty.

From now on, suppose that |s| − p ≤ j < |s|. By Lemma 11.4

im(w_{|s|−p,p−q}^{∨} − id) ⊂ V_{s} ⊂ U_{s}^{j},

so it suffices to show that w_{|s|−p,p−q}^{∨}c ∈ U_{s}^{j}. We do this by considering the various terms of

w_{|s|−p,p−q}^{∨}c = (∂_{|s|−p+1}^{∨})^{p−q}(ϵ_{|s|−p}^{∨})^{p−q}c

+ (∂_{|s|−p+1}^{∨})^{p−q}(∂_{0}^{∨})^{|s|−p}(ϵ_{0}^{∨})^{|s|−q}c + (∂_{0}^{∨})^{|s|−p}(ϵ_{0}^{∨})^{|s|−p}c (see Proposition 10.4).

If |s| − p ≤ j < |s| − q then it follows from the inductive hypothesis that the first term is zero or is in (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{|s|−p}; if |s| − q ≤ j < |s| then it similarly follows from the inductive hypothesis that the first term is zero or is in (∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j−p+q}; in any case we see that the first term is in U_{s}^{j}.

For all j with |s| − p ≤ j < |s| the second and third terms are zero or are in (∂_{0}^{∨})^{|s|−p}U_{(p)}^{j−|s|+p}, so they are also in U_{s}^{j}.

This completes the proof.

We deduce that the morphisms ψ_{i,s}^{∨} act in the required way.

11.6. Proposition. The morphisms ψ_{i,s}^{∨} are such that

(ψ_{i,s}^{∨} − id)Δ(|s|) ⊂ U_{s} (0 < i < |s|).

Proof. We use induction on the number of terms in s.

Suppose that s = (s′, q, p) and 0 < i < |s| − p. By Proposition 10.5,

im(ψ_{i,s}^{∨} − id) ⊂ (∂_{|s|−p+1}^{∨})^{p−q} im(ψ_{i,s′} − id) + im(w_{|s|−p,p−q}^{∨} − id).

By the inductive hypothesis,

(∂_{|s|−p+1}^{∨})^{p−q} im(ψ_{i,s′} − id) ⊂ (∂_{|s|−p+1}^{∨})^{p−q}U_{s′} ⊂ U_{s};

by Lemma 11.4,

im(w_{|s|−p,p−q}^{∨} − id) ⊂ V_{s} ⊂ U_{s}.

Therefore im(ψ_{i,s}^{∨} − id) ⊂ U_{s}.

Now suppose that s = (s′, q, p) and i = |s| − p. We can apply a similar argument, using Propositions 10.6 and 10.7.

Finally suppose that |s| − p < i < |s|. By Notation 9.8 and Proposition 10.2,

ψ_{i,s}^{∨} − id = ϕ_{i,1}^{∨} − id = ∂_{i+1}^{∨}(ϵ_{i−1}^{∨} − ϵ_{i}^{∨}).

Because of Lemma 11.3, it suffices to show that ∂_{i+1}^{∨}(ϵ_{i−1}^{∨} − ϵ_{i}^{∨})a is in U_{s} when a is a basis element including i + 1. If a is a basis element of that form not including i, then ∂_{i+1}^{∨}(ϵ_{i−1}^{∨} − ϵ_{i}^{∨})a = 0; if a is a basis element of that form including i, then ∂_{i+1}^{∨}(ϵ_{i−1}^{∨} − ϵ_{i}^{∨})a is a basis element including i − 1 and i but not i + 1. In view of Lemma 11.5, this suffices to show that ∂_{i+1}^{∨}(ϵ_{i−1}^{∨} − ϵ_{i}^{∨})a is in U_{s} in all cases, and this completes the proof.

11.7. Proposition. The morphisms ψ_{i,s}^{∨} are such that

ψ_{i,s}^{∨}U_{s}^{j} ⊂ U_{s}^{j} (0 < i < j < |s|).

Proof. We use induction on the number of terms in s.

Suppose first that s = (p). By Notation 9.8 and Proposition 10.2

ψ_{i,s}^{∨} = ϕ_{i,1}^{∨} = ∂_{i+1}^{∨}(ϵ_{i−1}^{∨} − ϵ_{i}^{∨}) + id .

According to Notation 11.1, the chain complex U_{s}^{j} is generated by the basis elements with at least two terms less than or equal to j and with no term equal to j + 1. The result now follows from a simple computation.

Now suppose that s = (s′, q, p), and let p′ be the last term of s′. Recall from Notation 11.1 that U_{s}^{j} is a sum of constituents which may have one of the following forms:

(∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j′}, (∂_{0}^{∨})^{|s−p}U_{(p)}^{j′′}, V_{s}.

In almost all cases it follows straightforwardly from Propositions 10.5–10.9 and the inductive hypothesis that ψ_{i,s}^{∨} maps the constituents of U_{s}^{j} into U_{s}^{j}. The exceptional cases are

ψ_{|s|−p,s}^{∨}(∂_{0}^{∨})^{|s|−p}U_{(p)}^{j−|s|+p} (q > 0, |s| − p < j < |s|)

and

ψ_{|s|−q+1,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q}U_{s′}^{j−p+q} (|s| − q + 1 < j < |s|).

We deal with these cases as follows.

In the first case let c be a generator for U_{(p)}^{j−|s|+p}, so that c is a basis element with at least two terms less than or equal to j − |s| + p and with no term equal to j − |s| + p + 1. By Proposition 10.7,

[ψ_{|s|−p,s}^{∨} − id](∂_{0}^{∨})^{|s|−p}c = (∂_{|s|−p+1}^{∨})^{p−q}c′,

where

c′ = ∂_{|s|−p+1}^{∨}[ϵ_{|s|−p−1}^{∨} − ϵ_{|s|−p}^{∨}](ϵ_{|s|−p+1}^{∨})^{p−q}(∂_{0}^{∨})^{|s|−p}c.

If |s| − p < j ≤ |s| − q then c′ = 0; if |s| − q < j < |s| then c′ is a linear combination of basis elements with at least two terms in the set

{ |s| − p − 1, |s| − p, …, j − p + q }

and with no term equal to j − p + q + 1, so that c′ ∈ U_{s′}^{j−p+q} by Lemma 11.5. In all cases it follows that ψ_{|s|−p}^{∨} ∈ U_{s}^{j}.

It remains to show that

ψ_{|s|−q+1,s}^{∨}(∂_{s|−p+1}^{∨})^{p−q}U_{s′}^{j−p+q} ⊂ U_{s}^{j} (|s| − q + 1 < j < |s|).

Because of Lemma 11.5 it suffices to show that ψ_{|s|−q+1,s}^{∨}(∂_{s|−p+1}^{∨})^{p−q}c is in U_{s}^{j} when c is a basis element in U_{s′}^{j−p+q} with two terms less than or equal to |s| − p, or with two terms less than or equal to j − p + q and with no term j − p + q + 1. By Lemma 11.4

im(w_{|s|−p,p−q}^{∨} − id) ⊂ V_{s} ⊂ U_{s}^{j};

it therefore suffices to show that w_{|s|−p,p−q}^{∨}ψ_{|s|−q+1,s}^{∨}(∂_{s|−p+1}^{∨})^{p−q}c is in U_{s}^{j} for each such basis element c. By Proposition 10.8

w_{|s|−p,p−q}^{∨}ψ_{|s|−q+1,s}^{∨}(∂_{|s|−p+1}^{∨})^{p−q}c = (∂_{|s|−p+1}^{∨})^{p−q}ψ_{|s|−p+1,s′}^{∨}c

+ (∂_{0}^{∨})^{|s|−p}[ϕ_{p−q+1,1}^{∨}(∂_{1}^{∨})^{p−q} − (∂_{1}^{∨})^{p−q}ϕ_{1,1}^{∨}](ϵ_{0}^{∨})^{|s|−p}c.

The first of the terms on the right hand side is in U_{s}^{j} by the inductive hypothesis. If c has two terms less than or equal to |s| − p, then the second term on the right hand side is zero. If c has two terms less than or equal to j − p + q and has no term j − p + q + 1, then the second term on the right hand side is a linear combination of basis elements with at least two terms in the set

{ |s| − p, |s| − p + 1, …, j }

and with no term j + 1 and is in U_{s}^{j} by Lemma 11.5. This completes the proof.

11.8. Proposition. The morphisms ψ_{j,s}^{∨} are such that

ψ_{j,s}^{∨}U_{s}^{j} ⊂ U_{s}^{j−1} (0 < j < |s|).

Proof. This is similar. Again we use induction on the number of terms in s.

Suppose that s = (p). By definition, U_{s}^{j} is generated by the basis elements with at least two terms less than or equal to j and with no term j + 1. If a is such a basis element then

ψ_{j,s}^{∨}a = ϕ_{j,1}^{∨}a = ∂_{j+1}^{∨}(ϵ_{j−1}^{∨} − ϵ_{j}^{∨})a + a,

and this is a linear combination of basis elements with at least two terms less than or equal to j − 1 and with no term j. Therefore ψ_{j,s}^{∨}U_{s}^{j} ⊂ U_{s}^{j−1}.

Now suppose that s = (s′, q, p). In almost all cases it follows from Propositions 10.5– 10.9 and the inductive hypothesis that ψ_{j,s}^{∨} maps each constituent of U_{s}^{j} into U_{s}^{j−1}. The only difficulty is to show that

ψ_{|s|−q+1,s}^{∨}(∂_{s|−p+1}^{∨})^{p−q}U_{s′}^{|s|−p+1} ⊂ U_{s}^{|s|−q}

in the case q > 1. By Lemma 11.5, it suffices to show that

ψ_{|s|−q+1,s}^{∨}(∂_{s|−p+1}^{∨})^{p−q}c ∈ U_{s}^{|s|−q}

when c is a basis element in U_{s′}^{|s|−p+1} with two terms less than or equal to |s| − p, or with two terms less than or equal to |s| − p + 1 and with no term |s| − p + 2. As in the proof of Proposition 11.7 it suffices to show that

(∂_{|s|−p+1}^{∨})^{p−q}ψ_{|s|−p+1,s′}^{∨}c + (∂_{0}^{∨})^{|s|−p}[ϕ_{p−q+1,1}^{∨}(∂_{1}^{∨})^{p−q} − (∂_{1}^{∨})^{p−q}ϕ_{1,1}^{∨}](ϵ_{0}^{∨})^{|s|−p}c

is in U_{s}^{|s|−q}.

The first of these terms is in U_{s}^{|s|−q} by the inductive hypothesis. If c has two terms less than or equal to |s|−p, then the second term is zero. If c has two terms less than or equal to |s| − p + 1 and has no term |s| − p + 2, then the second term is a linear combination of basis elements with terms |s| − p and |s| − q and with no term |s| − q + 1, and is therefore in U_{s}^{|s|−q} by Lemma 11.5. This completes the proof.

It follows from Propositions 11.2 and 11.6–11.8 that Ψ_{s}^{∨} is idempotent with kernel U_{s}. Recall from Theorem 6.11 that if X is a set with complicial identities then X_{|s|} ≅ Hom[λΔ(|s|), X]. We draw the following conclusions.

11.9. Proposition. Let s be an up-down vector and let X be a set with complicial identities. Then Ψ_{s} is an idempotent operation on X_{|s|}. There is a natural bijection

Ψ_{s}X_{|s|} ≅ Hom[λS_{s}, X],

where

S_{s} = Δ(|s|)/U_{s},

and the inclusion of Ψ_{s}X_{|s|} in

X_{|s|} ≅ Hom[λΔ(|s|), X]

is induced by the quotient homomorphism Δ(|s|) → S_{s}. The image Ψ_{s}X_{|s|} is the subset of X_{|s|} consisting of the elements x such that

ψ_{1,s}x = ⋯ = ψ_{|s|−1,s}x = x.

Proof. We need only prove the final statement. To do this we first observe that if 0 < i < |s| then

im(ψ_{i,s}^{∨} − id) ⊂ U_{s} = ker Ψ_{s}^{∨}

by Proposition 11.6, hence ψ_{i,s}Ψ_{s} = Ψ_{s}. We then recall from Notation 9.12 that Ψ_{s} is an iterated composite of the operations ψ_{i,s}. The result follows.

12. The pull-back property

Let X be a set with complicial identities. According to Proposition 11.9 there is a functor S_{s} ↦ Ψ_{s}X_{|s|} from simple chain complexes to sets. According to Theorem 8.12 (see also Notations 8.3 and 8.9) the image of an s-simple square is given by

\xymatrix{Ψ_{s}X_{|s|} \ar[d]_{∂_{|s|−p+1}^{p−q}} \ar[r]^{∂_{0}^{|s|−p}} & Ψ_{(p)}X_{p} \ar[d]^{∂_{1}^{p−q}} \\ Ψ_{s′}X_{|s′|} \ar[r]_{∂_{0}^{|s|−p}} & Ψ_{(q)}X_{q}.}

In this section we show that the functor yields an ω-category; that is, we show that the images of simple squares are pull-back squares (see Proposition 5.8).

Recall from Proposition 11.9 that the image of Ψ_{s} is the intersection of the fixed point sets of the operations ψ_{i,s}. We begin the proof by making the following observations.

12.1. Proposition. Let s be an up-down vector with last term p and let k be an integer with |s| − p ≤ k < |s|. Then

ψ_{k+1,s}x = ψ_{k+2,s}x = ⋯ = ψ_{|s|−1,s}x = x

if and only if

∂_{k+2}x = ϵ_{k}∂_{k+1}^{2}x, ∂_{k+3}x = ϵ_{k}^{2}∂_{k+1}^{3}x, …, ∂_{|s|}x = ϵ_{k}^{|s|−k−1}∂_{k+1}^{|s|−k}x.

Proof. For k < i < |s| we have ψ_{i,s} = ϕ_{i,1} (Notation 9.8), hence, by Proposition 9.3,

ψ_{i,s}x = x ⟺ ∂_{i+1}x ∈ im ϵ_{i−1}.

Note also that ∂_{i}ϵ_{i−1} = id and ∂_{i}∂_{i+1} = ∂_{i}∂_{i}, hence

∂_{i+1}x ∈ im ϵ_{i−1} ⟺ ∂_{i+1}x = ϵ_{i−1}∂_{i}∂_{i+1}x ⟺ ∂_{i+1}x = ϵ_{i−1}∂_{i}∂_{i}x.

It follows from this that if ψ_{i,s}x = x for all i with k < i < |s| then

∂_{k+2}x = ϵ_{k}∂_{k+1}∂_{k+1}x = ϵ_{k}∂_{k+1}^{2}x,
∂_{k+3}x = ϵ_{k+1}∂_{k+2}∂_{k+2}x = ϵ_{k+1}ϵ_{k}∂_{k+2}∂_{k+1}^{2}x = ϵ_{k}^{2}∂_{k+1}^{3}x,
…,
∂_{|s|}x = ϵ_{k}^{|s|−k−1}∂_{k+1}^{|s|−k}x.

Conversely, if ∂_{i+1}x = ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x for all i with k < i < |s| then ∂_{i+1}x ∈ im ϵ_{i−1} for all i with k < i < |s|, hence ψ_{i,s}x = x for all i with k < i < |s|. This completes the proof.

12.2. Proposition. Let x be an n-dimensional element in the image of an operation w_{k,l}. If k < i ≤ k + l then

[∂_{i+1}x = ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x] ⟺ [∂_{i−k+1}∂_{0}^{k}x = ϵ_{0}^{i−k}∂_{1}^{i−k+1}∂_{0}^{k}x].

If k + l < i < n then

[∂_{i+1}x = ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x] ⟺ [∂_{i−l+1}∂_{k+1}^{l}x = ϵ_{k}^{i−k−l}∂_{k+1}^{i−k−l+1}∂_{k+1}^{l}x

and ∂_{i−k+1}∂_{0}^{k}x = ϵ_{0}^{i−k}∂_{1}^{i−k+1}∂_{0}^{k}x].

Proof. Recall from Proposition 10.4 that

w_{k,l}^{∨} = (∂_{k+1}^{∨})^{l}(ϵ_{k}^{∨})^{l} − (∂_{k+1}^{∨})^{l}(∂_{0}^{∨})^{k}(ϵ_{0}^{∨})^{k+l} + (∂_{0}^{∨})^{k}(ϵ_{0}^{∨})^{k}.

Direct computations show that

∂_{i+1}w_{k,l} = \begin{cases}w_{k,l−1}∂_{i+1} & (k < i < k + l), \\ w_{k,l}∂_{i+1} & (k + l ≤ i < n) \end{cases}

and

ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}w_{k,l} = \begin{cases}w_{k,l−1}ϵ_{k}^{i−k}∂_{k+1}^{i−k+1} & (k < i < k + l), \\ w_{k,l}ϵ_{k}^{i−k}∂_{k+1}^{i−k+1} & (k + l ≤ i < n). \end{cases}

Recall from Notation 9.6 and Proposition 9.7 that

w_{k,l}x = ∂_{k+1}^{l}x ∧_{k,l} ∂_{0}^{k}x, ∂_{k+1}^{l}w_{k,l}x = ∂_{k+1}^{l}x, ∂_{0}^{k}w_{k,l}x = ∂_{0}^{k}x.

For x ∈ im w_{k,l} and for k < i < k + l it follows that ∂_{i+1}x = ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x if and only if

∂_{k+1}^{l−1}∂_{i+1}x = ∂_{k+1}^{l−1}ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x, ∂_{0}^{k}∂_{i+1}x = ∂_{0}^{k}ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x;

for k + l ≤ i < n it follows that ∂_{i+1}x = ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x if and only if

∂_{k+1}^{l}∂_{i+1}x = ∂_{k+1}^{l}ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x, ∂_{0}^{k}∂_{i+1}x = ∂_{0}^{k}ϵ_{k}^{i−k}∂_{k+1}^{i−k+1}x.

This gives the result. (In cases with k < i ≤ k + l the first condition is omitted from the statement of the proposition because it is satisfied automatically).

12.3. Proposition. Let s = (s′, q, p) be an up-down vector with more than one term, let X be a set with complicial identities, let x be a member of X_{|s|} such that w_{|s|−p,p−q}x = x, and let y = ∂_{|s|−p+1}^{p−q}x, z = ∂_{0}^{|s|−p}x. Then x ∈ Ψ_{s}X_{|s|} if and only if y ∈ Ψ_{s′}X_{|s′|} and z ∈ Ψ_{(p)}X_{p}.

Proof. According to Proposition 11.9 we must show that

ψ_{1,s}x = ⋯ = ψ_{|s|−1,s}x = x

if and only if

ψ_{1,s′}y = ⋯ = ψ_{|s′|−1,s′}y = y, ψ_{1,(p)}z = ⋯ = ψ_{p−1,(p)}z = ∂_{0}^{|s|−p}z.

We will consider ψ_{i,s}x for i > |s| − p, then for i < |s| − p, then for i = |s| − p.

Let p′ be the last term in s′, and recall that |s| − p > |s′| − p′. It follows from Propositions 12.1 and 12.2 that

ψ_{|s|−p+1,s}x = ⋯ = ψ_{|s|−1,s}x = x

if and only if

ψ_{|s|−p+1,s′}y = ⋯ = ψ_{|s′|−1,s′}y = y, ψ_{1,(p)}z = ⋯ = ψ_{p−1,(p)}z = ∂_{0}^{|s|−p}z.

Recall that x = y ∧_{|s|−p,p−q} z (Notation 9.6). For 0 < i < |s| − p we have ψ_{i,s}x = ψ_{i,s′}y ∧_{|s|−p,p−q} z (Notation 9.11), hence ψ_{i,s}x = x if and only if ψ_{i,s′}y = y.

In the case q = 0 we have ψ_{|s|−p,s}x = w_{|s|−p,p−q}x = x by hypothesis (see Notation 9.9). This completes the proof in the case q = 0.

From now on suppose that q > 0. It suffices to show that

ψ_{|s|−p,s}x = x ⟺ ψ_{|s|−p,s′}y = y.

Equivalently, since |s| − p > |s′| − p′, it suffices to show that

w_{|s|−p,p−q}ϕ_{|s|−p,p−q+1}x = x ⟺ ϕ_{|s|−p,1}y = y

(see Notations 9.9 and 9.8).

To do this, suppose first that w_{|s|−p,p−q}ϕ_{|s|−p,p−q+1}x = x. Then

∂_{|s|−p+1}y = ∂_{|s|−p+1}^{p−q+1}x
= ∂_{|s|−p+1}^{p−q}w_{|s|−p,p−q}ϕ_{|s|−p,p−q+1}x
= ∂_{|s|−p+1}^{p−q}ϕ_{|s|−p,p−q+1}x

by Proposition 9.7, hence ∂_{|s|−p+1}y ∈ im ϵ_{|s|−p−1} (Proposition 9.2), hence ϕ_{|s|−p,1}y = y (Proposition 9.3).

Conversely, suppose that ϕ_{|s|−p,1}y = y. By Proposition 9.3

∂_{|s|−p+1}^{p−q+1}x = ∂_{|s|−p+1}y ∈ im ϵ_{|s|−p−1},

hence

w_{|s|−p,p−q}ϕ_{|s|−p,p−q+1}x = w_{|s|−p,p−q}x = x.

This completes the proof.

12.4. Proposition. If s = (s′, q, p) and if X is a set with complicial identities, then the square

\xymatrix{Ψ_{s}X_{|s|} \ar[d]_{∂_{|s|−p+1}^{p−q}} \ar[r]^{∂_{0}^{|s|−p}} & Ψ_{(p)}X_{p} \ar[d]^{∂_{1}^{p−q}} \\ Ψ_{s′}X_{|s′|} \ar[r]_{∂_{0}^{|s|−p}} & Ψ_{(q)}X_{q}.}

is a pull-back square of sets.

Proof. By Proposition 9.7 there is a pull-back square

\xymatrix{w_{|s|−p,p−q}X_{|s|} \ar[d]_{∂_{|s|−p+1}^{p−q}} \ar[r]^{∂_{0}^{|s|−p}} & X_{p} \ar[d]^{∂_{1}^{p−q}} \\ X_{|s′|} \ar[r]_{∂_{0}^{|s|−p}} & X_{q}}

with w_{|s|−p,p−q} idempotent. By Proposition 11.9 and Notation 9.9, if x ∈ Ψ_{s}X_{|s|} then

w_{|s|−p,p−q}x = w_{|s|−p,p−q}ψ_{|s|−p,s}x = ψ_{|s|−p,s}x = x.

The result now follows from Proposition 12.3.

For a set with complicial identities X it now follows from Proposition 11.9 that the functor

S ↦ Hom[λS, X]

takes simple squares of chain complexes to pull-back squares of sets. By Proposition 5.8 this determines an ω-category functorially in X. We will use the following notation.

12.5. Notation. Let β be the functor from sets with complicial identities to ω-categories such that

Hom[νS, βX] = Hom[λS, X]

for every simple chain complex S.

13. The equivalence

We have constructed functors α and β between ω-categories and sets with complicial identities (see Notations 7.3 and 12.5). We will now show that these functors are inverse equivalences.

In particular we must show that αβX ≅ X for every set with complicial identities X. We will do this by showing that

Ψ_{(m)}(αβX)_{m} ≅ Ψ_{(m)}X_{m} (m ≥ 0).

We must therefore show that a set with complicial identities is determined up to isomorphism by the images of the operations Ψ_{(m)}. We will do this by induction on m: for m > 0 we will show that an m-dimensional element c is equivalent to the family consisting of the image Ψ_{(m)}c and of its faces.

Recall that Ψ_{(m)} is a composite of the operations ψ_{i,(m)}, where

ψ_{i,(m)}x = ϕ_{i,1}x = ∂_{i+1}(ϵ_{i−1}∂_{i+1}x ∧_{i} x)

(see Notations 9.12, 9.8 and 9.1). We begin by considering an individual operation ϕ_{i,1}.

13.1. Proposition. Let X be a set with complicial identities, let m and i be integers with 0 < i < m, and let T be the set of triples (a, y, z) in ϕ_{i,1}X_{m} × X_{m−1}^{2} such that

∂_{i−1}y = ∂_{i}z, ∂_{i−1}a = ∂_{i}(y ∧_{i−1} z), ∂_{i+1}a = ϵ_{i−1}∂_{i}y.

Then there is a bijection f: X_{m} → T given by

f(c) = (ϕ_{i,1}c, ∂_{i+1}c, ∂_{i−1}c).

Proof. It follows from the axioms (Definition 3.1) that the formula for f defines a function whose image is contained in T. We will show that there is an inverse function g: T → X_{m} given by

g(a, y, z) = ∂_{i}[a ∧_{i−1} (y ∧_{i−1} z)];

the wedges in this formula exist because

∂_{i−1}y = ∂_{i}z, ∂_{i−1}a = ∂_{i}(y ∧_{i−1} z).

First we show that gf(c) = c for c ∈ X_{m}. Let

A = ϵ_{i−1}∂_{i+1}c ∧_{i} c = (ϵ_{i−1}∂_{i}∂_{i+1}c ∧_{i−1} ∂_{i+1}c) ∧_{i} c,

so that ϕ_{i,1}c = ∂_{i+1}A. By Definition 3.1(6),

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

therefore

gf(c) = ∂_{i}[ϕ_{i,1}c ∧_{i−1} (∂_{i+1}c ∧_{i−1} ∂_{i−1}c]
= ∂_{i}[∂_{i+1}A ∧_{i−1} (∂_{i+1}c ∧_{i−1} ∂_{i−1}c)]
= ∂_{i}A
= c.

Conversely we will show that fg(a, y, z) = (a, y, z) for (a, y, z) in T. Let

B = a ∧_{i−1} (y ∧_{i−1} z),

so that g(a, y, z) = ∂_{i}B. Then

∂_{i+1}g(a, y, z) = ∂_{i+1}∂_{i}B
= ∂_{i}∂_{i+2}B
= ∂_{i}[∂_{i+1}a ∧_{i−1} ∂_{i+1}(y ∧_{i−1} z)]
= ∂_{i}(ϵ_{i−1}∂_{i}y ∧_{i−1} y)
= ∂_{i}ϵ_{i−1}y
= y,

and

∂_{i−1}g(a, y, z) = ∂_{i−1}∂_{i}B = ∂_{i−1}∂_{i−1}B = ∂_{i−1}(y ∧_{i−1} z) = z.

We also deduce that

ϵ_{i−1}∂_{i+1}g(a, y, z) = ϵ_{i−1}y = ϵ_{i−1}∂_{i}y ∧_{i−1} y = ∂_{i+1}a ∧_{i−1} y.

By Definition 3.1(5),

B = (∂_{i+1}a ∧_{i−1} y) ∧_{i} ∂_{i}B,

hence

ϕ_{i,1}g(a, y, z) = ∂_{i+1}[ϵ_{i−1}∂_{i+1}g(a, y, z) ∧_{i} g(a, y, z)]
= ∂_{i+1}[(∂_{i+1}a ∧_{i−1} y) ∧_{i} ∂_{i}B]
= ∂_{i+1}B
= a.

Therefore fg(a, y, z) = (a, y, z).

This completes the proof.

According to this proposition, if X is a set with complicial identities and if 0 < i < m, then an m-dimensional member c of X can be recovered from the image ϕ_{i,1}c and the faces ∂_{i+1}c, ∂_{i−1}c. The triples (ϕ_{i,1}c, ∂_{i+1}c, ∂_{i−1}c) that can occur are those permitted by the formulae

∂_{i+1}ϕ_{i,1}c = ϵ_{i−1}∂_{i}∂_{i+1}c, ∂_{i−1}ϕ_{i,1}c = ∂_{i}(∂_{i+1}c ∧_{i} ∂_{i−1}c).

We extend this as follows.

13.2. Proposition. Let X be a set with complicial identities. Then X_{0} = Ψ_{(0)}X_{0}. For m > 0 the function on X_{m} given by

c ↦ (Ψ_{(m)}c, ∂_{0}c, …, ∂_{m}c)

is injective. The image consists of the (m + 2)-tuples

(a, u_{0}, …, u_{m}) ∈ Ψ_{(m)}X_{(m)} × X_{m−1} × … × X_{m−1}

such that

∂_{i}a = F_{i}(u_{0}, …, u_{m}) (0 < i < m),

where F_{i} is the operation such that

∂_{i}Ψ_{(m)}c = F_{i}(∂_{0}c, …, ∂_{m}c)

for all c.

Proof. Recall from Notations 9.12 and 9.8 that Ψ_{(m)} is a composite of the operations ϕ_{i,1} (an empty composite in the case m = 0). The result therefore follows from Proposition 13.2.

13.3. Theorem. The categories of ω-categories and sets with complicial identities are equivalent under the functors α and β.

Proof. Let C be an ω-category and let X be a set with complicial identities. We will construct natural isomorphisms

βαC ≅ C, αβX ≅ X.

Let m be a nonnegative integer. By Proposition 5.4 and Definition 2.5, S_{(m)} is a free ω-category on one m-dimensional generator. By Theorem 6.11 λΔ(m) is a free set with complicial identities on one m-dimensional generator. It follows that

C_{m} ≅ Hom[νS_{(m)}, C], X_{m} ≅ Hom[λΔ(m), X];

recall also from Proposition 11.9 that Hom[λS_{(m)}, X] ≅ Ψ_{(m)}X_{m}. It is convenient to write

Ψ_{(m)} Hom[λΔ(m), X] = { x ∈ Hom[λΔ(m), X] : x(λΨ_{(m)}^{∨}) = x },

so that

Hom[λS_{(m)}, X] ≅ Ψ_{(m)} Hom[λΔ(m), X].

Analogously we will write

Ψ_{(m)} Hom[νΔ(m), C] = { x ∈ Hom[νΔ(m), C] : x(νΨ_{(m)}^{∨}) = x },

so that

Hom[νS_{(m)}, C] ≅ Ψ_{(m)} Hom[νΔ(m), C].

It follows from Notations 7.3 and 12.5 that

(βαC)_{m} ≅ Hom[νS_{(m)}, βαC]
≅ Hom[λS_{(m)}, αC]
≅ Ψ_{(m)} Hom[λΔ(m), αC]
≅ Ψ_{(m)} Hom[νΔ(m), C]
≅ Hom[νS_{(m)}, C]
≅ C_{m},

hence βαC ≅ C. Analogously

Ψ_{(m)}(αβX)_{m} ≅ Hom[λS_{(m)}, αβX]
≅ Ψ_{(m)} Hom[λΔ(m), αβX]
≅ Ψ_{(m)} Hom[νΔ(m), βX]
≅ Hom[νS_{(m)}, βX]
≅ Hom[λS_{(m)}, X]
≅ Ψ_{(m)}X_{m},

hence αβX ≅ X by Proposition 13.2. This completes the proof.

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] M. Makkai and M. Zawadowski, Duality for simple ω-categories and disks, Theory Appl. Categ. 8 (2001), 114–243.

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

[4] R. Steiner, Simple omega-categories and chain complexes, Homology Homotopy Appl. 9 (2007), no. 1, 451–465.

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

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

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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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