Theory and Applications of Categories, Vol. 3, No. 7, pp. 160–181.

CROSSED SQUARES AND 2-CROSSED MODULES OF COMMUTATIVE ALGEBRAS

ZEKERİYA ARVASİ

Transmitted by J. L. Loday

ABSTRACT. In this paper, we construct a neat description of the passage from crossed squares of commutative algebras to 2-crossed modules analogous to that given by Conduché in the group case. We also give an analogue, for commutative algebra, of T.Porter’s [13] simplicial groups to n-cubes of groups which implies an inverse functor to Conduché’s one.

Introduction

Simplicial commutative algebras play an important role in homological algebras, homotopy theory and algebraic K-theory. In each theory, the internal structure has been studied relatively little. The present article intends to study the 3-types of simplicial algebras.

Crossed modules were initially defined by Whitehead (cf. [14]) as a model for 2-types. Conduché has defined a 2-crossed module as a model for 3-types. His unpublished work determines that there exists an equivalence (up to homotopy) between the category of crossed squares of groups and that of 2-crossed modules of groups. This construction was used by T.Porter in the work of n-Type of simplicial groups and crossed n-cubes, [13]. This result is true for the commutative algebra case.

In [3], we showed how to go from a simplicial algebra to a 2-crossed module of algebras and back to a truncated form of the simplicial algebra, and the link between simplicial algebras and crossed squares is explicitly given. It is natural for us to ask if the passage from a crossed square to a simplicial algebra and hence a 2-crossed module can be performed.

The main points of this paper are thus:

i) to construct a functor from simplicial commutative algebras to crossed n-cubes analogous to that given by T.Porter in the group case.

ii) to give a full description of the passage from a crossed square to a 2-crossed module of commutative algebras by using the Artin-Mazur “codiagonal” functor.

iii) to prove directly 2-crossed modules in a neat way.

Received by the editors 1997 May 20 and, in revised form, 1997 August 24.

Published on 1997 October 5

1991 Mathematics Subject Classification : 18G30,18G55.

Key words and phrases: Simplicial Algebras, Crossed n-Cubes, Crossed Squares, 2-Crossed Modules.

\begin{matrix}c \\ ◯ \end{matrix} Zekeri̇ya Arvasi̇ 1997. Permission to copy for private use granted.

Acknowledgements. The author wishes to thank Prof. T.Porter for useful conversations and the referee for helpful comments.

1. Preliminaries

1.1. Simplicial algebras. Let k be a commutative ring with identity. We will use the term commutative algebra to mean a commutative algebra over k. The category of commutative algebras will be denoted by CA.

A simplicial (commutative) algebra E consists of a family of algebras {E_{n}} together with face and degeneracy maps d_{i} = d_{i}^{n} : E_{n} → E_{n−1}, 0 ≤ i ≤ n, (n ≠ 0) and s_{i} = s_{i}^{n} : E_{n} → E_{n+1}, 0 ≤ i ≤ n, satisfying the usual simplicial identities given in Illusie [10] for example. It can be completely described as a functor E: Δ^{op} →CommAlg_{k} where Δ is the category of finite ordinals [n] = {0 < 1 < ⋯ < n} and increasing maps. We will denote the category of simplicial commutative algebras by SimpAlg. We have for each k ≥ 0 a subcategory Δ_{≤k} determined by the objects [j] of Δ with j ≤ k. A k-truncated simplicial commutative algebra is a functor from (Δ_{≤k}^{op}) to CommAlg.

Consider the product Δ × Δ whose objects are pairs ([p], [q]) and whose maps are pairs of weakly increasing maps. A functor E.,.: (Δ × Δ)^{op} → CommAlg_{k} is called a bisimplicial algebra with value in CommAlg. To give E.,. is equivalent with giving for each (p, q) algebra E_{p,q} and homomorphisms

d_{i}^{h} : E_{p,q} → E_{p−1,q}
s_{i}^{h} : E_{p,q} → E_{p+1,q} i : 0, …, p
d_{j}^{v} : E_{p,q} → E_{p,q−1}
s_{j}^{v} : E_{p,q} → E_{p,q+1} j : 0, …, q

such that the maps d_{i}^{h}, s_{i}^{h} commute with d_{j}^{v}, s_{j}^{v} and d_{i}^{h}, s_{i}^{h} (resp. d_{j}^{v}, s_{j}^{v}) satisfy the usual simplicial identities. Here d_{i}^{h}, s_{i}^{h} denote the horizontal operators and d_{j}^{v}, s_{j}^{v} denote the vertical operators.

By an ideal chain complex of algebras, (X, d) we mean one in which each Imd_{i+1} is an ideal of X_{i}. Given any ideal chain complex (X, d) of algebras and an integer n the truncation, t_{n]}X, of X at level n will be defined by

(t_{n]}X)_{i} = \{ \begin{matrix}X_{i} & if i < n \\ X_{i}/Imd_{n+1} & if i = n \\ 0 & if i > n. \end{matrix}

The differential d of t_{n]}X is that of X for i < n, d_{n} is induced by the nth differential of X and all other are zero.

1.2. The Moore complex of simplicial algebra. We recall that given a simplicial algebra E, the Moore complex (NE, ∂) of E is the chain complex defined by

(NE)_{n} = ⋂_{i=0}^{n−1} Kerd_{i}^{n}

with ∂_{n} : NE_{n} → NE_{n−1} induced by d_{n}^{n} by restriction.

The n^{th} homotopy module π_{n}(E) of E is the n^{th} homology of the Moore complex of E, i.e.,

π_{n}(E) ≅ H_{n}(NE, ∂) = ⋂_{i=0}^{n} Kerd_{i}^{n}/d_{n+1}^{n+1}( ⋂_{i=0}^{n} Kerd_{i}^{n+1}).

We say that the Moore complex NE of a simplicial algebra is of length k if NE_{n} = 0 for all n ≥ k + 1 so that a Moore complex is of length k also of length r for r ≥ k.

A simplicial map f : E → E′ is called a n- equivalence if it induces isomorphisms

π_{n}(E) ≅ π_{n}(E′) for n ≥ 0.

Two simplicial algebras E and E′ are said to be have the same n-type if there is a chain of n-equivalences linking them. A simplicial commutative algebra E is an n-type if π_{i}(E) = 0 for i > n.

2. 2-crossed modules from simplicial algebras

Crossed modules techniques give a very efficient way of handling informations about on a homotopy type. They correspond to a 2-type (see [3]). As mentioned the above, Conduché, [5], in 1984 described the notion of 2-crossed module as a model for 3-types.

Throughout this paper we denote an action of r ∈ R on c ∈ C by r · c.

A crossed module is an algebra morphism ∂ : C → R with an action of R on C satisfying ∂(r · c) = r∂c and ∂(c) · c′ = cc′ for all c, c′ ∈ M, r ∈ R. The last condition is called the Peiffer identity. We will denote such a crossed module by (C, R, ∂). A morphism of crossed modules from (C, R, ∂) to (C′, R′, ∂′) is a pair of k-algebra morphisms, ϕ : C → C′, ψ : R → R′ such that ϕ(r · c) = ψ(r)ϕ(c). We thus get a category XMod of crossed modules.

Examples of crossed modules: (i) any ideal I in R gives an inclusion map, inc: I → R which is a crossed module. Conversely given an arbitrary crossed module ∂ : C → R, one can easily sees that the Peiffer identity implies that ∂C is an ideal in R. (ii) Given any R-module, M, one can give M an algebra structure by giving it the zero multiplication. With this structure the zero morphism 0 : M → R is a crossed module.

We recall from Grandjeán and Vale [8] the definition of 2-crossed module:

A 2-crossed module of k-algebras consists of a complex of C_{0}-algebras

[figure: C_{2}; ∂_{2}; C_{1}; ∂_{1}; C_{0}]

with ∂_{2}, ∂_{1} morphisms of C_{0}-algebras, where the algebra C_{0} acts on itself by multiplication, such that

C_{2}  \overline{∂_{2} ^{}}C_{1}

is a crossed module in which C_{1} acts on C_{2}, (we require thus that for all x ∈ C_{2}, y ∈ C_{1} and z ∈ C_{0} that (xy)z = x(yz)), further, there is a C_{0}-bilinear function giving

{ ⊗ } : C_{1} ⊗_{C_{0}} C_{1} ⟶ C_{2},

called a Peiffer lifting, which satisfies the following axioms:

\begin{aligned}P L1 : & ∂_{2}\{y_{0} ⊗ y_{1}\} & = & y_{0}y_{1} − y_{0} · ∂_{1}(y_{1}), \\ P L2 : & \{∂_{2}(x_{1}) ⊗ ∂_{2}(x_{2})\} & = & x_{1}x_{2}, \\ P L3 & \{y_{0} ⊗ y_{1}y_{2}\} & = & \{y_{0}y_{1} ⊗ y_{2}\} + ∂_{1}y_{2} · \{y_{0} ⊗ y_{1}\}, \\ P L4 : & \{y ⊗ ∂_{2}(x)\} + \{∂_{2}(x) ⊗ y\} & = & ∂_{1}(y) · x, \\ P L5 : & \{y_{0} ⊗ y_{1}\} · z & = & \{y_{0} · z ⊗ y_{1}\} = \{y_{0} ⊗ y_{1} · z\}, \end{aligned}

for all x, x_{1}, x_{2} ∈ C_{2}, y, y_{0}, y_{1}, y_{2} ∈ C_{1} and z ∈ C_{0}.

A morphism of 2-crossed modules of algebras can be defined in a obvious way. We thus define the category of 2-crossed module denoting it by X_{2}Mod.

We denote the category of simplicial algebras with Moore complex of length n by SimpAlg_{≤n} in the following.

In [3], we studied the truncated simplicial algebras and saw what properties that has. Later, we turned to a simplicial algebra E which is 2-truncated, i.e., its Moore complex

look like:

⋯ ⟶ 0 ⟶ NE_{2} ⟶^{∂_{2}} NE_{1} ⟶^{∂_{1}} NE_{0}

and we showed the following result:

2.1. Theorem. The category X_{2}Mod of 2-crossed modules is equivalent to the category SimpAlg_{≤2} of simplicial algebras with Moore complex of length 2.

That is, this result can be pictured by the following diagram:

SimpAlg_{≤2} [figure: C^{(2)}; \underline{K^{(2)}}] X_{2}Mod
[figure: t_{2]}]
Tr_{2}SimpAlg.^{}

where Tr_{2}SimpAlg denotes the category of 2-truncated simplicial algebras.

3. Cat^{2}-algebras and crossed squares

We will recall the definition (due to Ellis [7]) of a crossed square of commutative algebras. A crossed square of commutative algebras is a commutative diagram

[figure: L; λ; M; N; λ′; ν; R; μ]

together with actions of R on L, M and N. There are thus commutative actions of M on L and N via μ, and N acts on L and M via ν and a function h : M × N → L, the h-map. These data have to satisfy the following axioms: for all m, m′ ∈ M, n, n′ ∈ N, r ∈ R, l ∈ L, k ∈ k;

1. each of the maps λ, λ′, μ, ν′ and the composite μλ = νλ′ are crossed modules,

2. the maps λ, λ′ preserve the action of R,

3. kh(m, n) = h(km, n) = h(m, kn),

4. h(m + m′, n) = h(m, n) + h(m′, n),

5. h(m, n + n′) = h(m, n) + h(m, n′),

6. r · h(m, n) = h(r · m, n) = h(m, r · n),

7. λh(m, n) = m · n,

8. λ′h(m, n) = n · m,

9. h(m, λ′l) = m · l,

10. h(λl, n) = n · l.

In the simplest example of crossed squares (cf.[3]) μ and ν are ideal inclusions and L = M ∩ N, with h being the multiplication map. We also mentioned that if

[figure: M ∩ N; M; N; E]

was a simplicial crossed square constructed from a simplicial algebra E and two simplicial ideals M and N then applying π_{0} to the square gives a crossed square and that up to isomorphism all crossed squares arise his way, see [3].

Although when first introduced by Loday and Walery [9], the notion of crossed squares of groups was not linked to that of cat^{2}-groups, it was in this form that Loday gave their generalisation to an n-fold structure, cat^{n}-groups (see [11]).

We recall for low dimension Ellis equivalence between the category of crossed n-cubes and that of cat^{n}-algebras is given by Ellis (cf. [7]) it was not given an explicit description in low dimensions. We now examine this for low dimensions.

Recall from [12] that a cat^{1}-alg is a triple, (E, s, t), where E is an k-algebra and s, t are endomorphisms of E satisfying the following conditions:

(i) st = t and ts = s (ii) KersKert = 0. It was shown in [12] that setting C =Kers, B =Ims and ∂ = t | C, then the action of

B on C within E makes

∂ : C ⟶ B

into a crossed module. Conversely if ∂ : C → B is a crossed module, then setting E = C ⋊ B and letting s, t be defined by

s(c, b) = (0, b)

and

t(c, b) = (0, ∂(c) + b)

for c ∈ C, b ∈ B, then (E, s, t) is a cat^{1}-algebra.

For a cat^{2}-algebra, we again have an algebra, E, but this time with two independent cat^{1}-algebra structures on it. Explicitly:

A cat^{2}-algebra is a 5-tuple (E, s_{1}, t_{1}, s_{2}, t_{2}), where (E, s_{i}, t_{i}), i = 1, 2, are cat^{1}-algebras and

s_{i}s_{j} = s_{j}s_{i}, t_{i}t_{j} = t_{j}t_{i}, s_{i}t_{j} = t_{j}s_{i}

for i, j = 1, 2, i ≠ j. 3.1. Proposition. There is an equivalence of categories between the category of cat^{2}- algebras and that of crossed squares. Proof. The cat^{1}-algebra (E, s_{1}, t_{1}) will give us a crossed module

∂ : C ⟶ B

with C =Kers, B =Ims and ∂ = t | C, but as the two cat^{1}-algebra structures are independent, (E, s_{2}, t_{2}) restricts to give cat^{1}-algebra structures on C and B and makes ∂ a morphism of cat^{1}-algebras. We thus get a morphism of crossed modules

Kers_{1} ∩ Kers_{2} \overline{}Ims_{1} ∩ Kers_{2}

[figure: Kers_{2} ∩ Ims_{1}; Ims_{1} ∩ Ims_{2}]

where each morphism is a crossed module for the natural action, i.e. multiplication in E. It remains to produce an h-map, but it is given by the multiplication within E since if x ∈Kers_{2}∩Ims_{1} and y ∈Ims_{2}∩Kers_{1} then xy ∈Kers_{1}∩Kers_{2}. It is easy to check the crossed squares axioms.

Conversely, if

[figure: L; M; N; R]

is a crossed square, then we can think of it as a morphism of crossed modules

L  M

\overline{}

[figure: N; R.]

Using the equivalence between crossed modules and cat^{1}-algebras this gives a morphism

∂ : (L ⋊ N, s, t) ⟶ (M ⋊ R, s′, t′)

of cat^{1}-algebras. There is an action of (m, r) ∈ M ⋊ R on (l, n) ∈ L ⋊ N given by

(m, r) · (l, n) = (r · l + ∂m · l + h(m, n), r · n + mn).

Using this action, we thus form its associated cat^{1}-algebra with big algebra (L ⋊ N) ⋊ (M ⋊ R) and induced endomorphisms s_{1}, t_{1}, s_{2}, t_{2}.

4. Crossed n-cubes and simplicial algebras

Crossed n-cubes in algebraic settings such as commutative algebras, Jordan algebras, Lie algebras have been defined by Ellis [7].

A crossed n-cube of commutative algebras is a family of commutative algebras, M_{A} for A ⊆< n >= {1, ..., n} together with homomorphisms μ_{i} : M_{A} → M_{A−{i}} for i ∈< n > and for A, B ⊆< n >, functions

h : M_{A} × M_{B} ⟶ M_{A∪B}

such that for all k ∈ k, a, a′ ∈ M_{A}, b, b′ ∈ M_{B}, c ∈ M_{C}, i, j ∈< n > and A ⊆ B

\begin{aligned}1) & μ_{i}a & = & a if i ∉ A \\ 2) & μ_{i}μ_{j}a & = & μ_{j}μ_{i}a \\ 3) & μ_{i}h(a, b) & = & h(μ_{i}a, μ_{i}b) \\ 4) & h(a, b) & = & h(μ_{i}a, b) = h(a, μ_{i}b) if i ∈ A ∩ B \\ 5) & h(a, a′) & = & aa′ \\ 6) & h(a, b) & = & h(b, a) \\ 7) & h(a + a′, b) & = & h(a, b) + h(a′, b) \\ 8) & h(a, b + b′) & = & h(a, b) + h(a, b′) \\ 9) & k · h(a, b) & = & h(k · a, b) = h(a, k · b) \\ 10) & h(h(a, b), c) & = & h(a, h(b, c)) = h(b, h(b, c)). \end{aligned}

A morphism of crossed n-cubes is defined in the obvious way: It is a family of commutative algebra homomorphisms, for A ⊆< n >, f_{A} : M_{A} ⟶ M′_{A} commuting with the μ_{i}’s and h’s. We thus obtain a category of crossed n-cubes denoted by Crs^{n}.

Examples. (a) For n = 1, a crossed 1-cube is the same as a crossed module. (b) For n = 2, one has a crossed square:

[figure: M_{<2>}; μ_{2}; M_{{1}}; M_{{2}}; μ_{1}; μ_{2}; M_{∅}.; μ_{1}]

Each μ_{i} is a crossed module as is μ_{1}μ_{2}. The h-functions give actions and a pairing

h : M_{{1}} × M_{{2}} ⟶ M_{<2>}.

The maps μ_{2} (or μ_{1}) also define a map of crossed modules. In fact a crossed square can be thought of as a crossed module in the category of crossed modules.

(c) Let E be a simplicial algebra. Then the following diagram

[figure: NE_{2}/∂_{3}NE_{3}; δ; NE_{1}; \overline{NE}_{1}; δ′; ∂; E_{1}; ∂′]

is a crossed square. Here NE_{1} =Kerd_{0}^{1} and \overline{NE}_{1} =Kerd_{1}^{1}.

Since E_{1} acts on NE_{2}/∂_{3}NE_{3}, \overline{NE}_{1} and NE_{1}, there are actions of \overline{NE}_{1} on NE_{2}/∂_{3}NE_{3} and NE_{1} via ∂, and NE_{1} acts on NE_{2}/∂_{3}NE_{3} and \overline{NE}_{1} via ∂′. As ∂ and ∂′ are inclusions, all actions can be given by multiplication. The h-map is

\begin{matrix}NE_{1} × \overline{NE}_{1} & ⟶ & NE_{2}/∂_{3}NE_{3} \\ (x, \overline{y}) & ⟼ & h(x, \overline{y}) = s_{1}x(s_{1}y − s_{0}y) + ∂_{3}NE_{3}, \end{matrix}

which is bilinear. Here x and y are in NE_{1} as there is a natural bijection between NE_{1} and \overline{NE}_{1} (by lemma 2.3.1 in [2]). The element ȳ is the image of y under this. The reader is referred to the author’s thesis, [2], for verifying of the crossed square axioms.

This last example effectively introduces the functor

M(−, 2) : SimpAlg \overline{}Crs^{2}.

This is the 2-dimensional case of a general construction to which we turn next.

By an ideal (n + 1)-ad will be meant an algebra with n selected ideals (possibly with repeats), (R, I_{1}, …, I_{n}).

Let R be an algebra with ideals I_{1}, …, I_{n} of R. Let

M_{A} = ⋂{I_{i} : i ∈ A} and M_{∅} = R

with A ⊆< n > . If i ∈< n >, then M_{A} is an ideal of M_{A−{i}.} Define μ_{i} : M_{A} ⟶ M_{A−{i}} to be the inclusion. If A, B ⊆< n >, then M_{A∪B} = M_{A} ∩ M_{B}, let

\begin{aligned}h : & M_{A} × M_{B} & ⟶ & M_{A∪B} \\ & (a, b) & ⟼ & ab \end{aligned}

as M_{A}M_{B} ⊆ M_{A} ∩ M_{B}, where a ∈ M_{A}, b ∈ M_{B}. Then {M_{A} : A ⊆< n >, μ_{i}, h} is a crossed n-cube, called the inclusion crossed n-cube given by the ideal (n + 1)-ad of commutative algebras (R; I_{1}, …, I_{n}).

4.1. Proposition. Let (E; I_{1}, …, I_{n}) be a simplicial ideal (n + 1)-ad of algebras and define for A ⊆< n >

M_{A} = π_{0}( ⋂_{i∈A} I_{i})

with homomorphisms μ_{i} : M_{A} → M_{A−{i}} and h-maps induced by the corresponding maps in the simplicial inclusion crossed n-cube, constructed by applying the previous example to each level. Then {M_{A} : A ⊆< n >, μ_{i}, h} is a crossed n-cube.

The proof is straightforward and so has been omitted.

Up to isomorphism, all crossed n-cubes arise in this way. In fact any crossed n-cube can be realised (up to isomorphism) as a π_{0} of a simplicial inclusion crossed n-cube coming from a simplicial ideal (n + 1)-ad in which π_{0} is a non-trivial homotopy module.

In 1991, T. Porter described a functor, [13], from the category of simplicial groups to that of crossed n-cubes of groups, based on ideas of Loday. Here, we adapt that description to give an obvious analogue of this functor for the commutative algebra case. The functor here constructed is defined using the décalage functor studied by Illusie [10] and Duskin [6] and is a π_{0}-image of a functor taking values in a category of simplicial ideal (n + 1)-ads.

The décalage functor forgets the last face operators at each level of a simplicial algebra E and moves everything down one level. It is denoted by Dec. Thus

(DecE)_{n} = E_{n+1}.

The last degeneracy of E yields a contraction of Dec^{1}E as an augmented simplicial algebra,

Dec^{1}E ≃ K(E_{0}, 0),

by an explicit natural homotopy equivalence (cf. [6]). The last face map will be denoted

δ_{0} : Dec^{1}E ⟶ E.

Iterating the Dec construction gives an augmented bisimplicial algebra

[figure: [... Dec^{3}E; Dec^{2}E; δ_{0}; δ_{1}; Dec^{1}E]]

which in expanded form is the total décalage of E:

[figure: [... Dec^{3}E; Dec^{2}E; δ_{0}; δ_{1}; Dec^{1}E]; δ_{0}; E.]

(see [6] or [10] for details). The maps from Dec^{i}E to Dec^{i−1}E coming from the i last face maps will be labelled δ_{0}, …, δ_{i−1} so that δ_{0} = d_{last}, δ_{1} = d_{last but one} and so on.

For a simplicial algebra E and a given n, we write M(E, n) for the crossed n-cube, arising from the functor

M(−, n) : SimpAlg ⟶ Crs^{n},

which is given by π_{0}(DecE; Kerδ_{0}, …, Kerδ_{n−1}). The following propositions and their proofs aim to explore this compressed definition, giving an ‘elementary’ specification of the construction.

4.2. Proposition. If E be a simplicial algebra, then the crossed n-cube M(E,n) is determined by:

(i) for A ⊆< n >,

M(E, n)_{A} = \frac{⋂_{j∈A} Kerd_{j−1}^{n}}{d_{n+1}^{n+1}(Kerd_{0}^{n+1} ∩ \{⋂_{j∈A} Kerd_{j}^{n+1}\});}

(ii) the inclusion

⋂_{j∈A} Kerd_{j−1}^{n} ⟶ ⋂_{j∈A−{i}} Kerd_{j−1}^{n}

induces the morphism

μ_{i} : M(E, n)_{A} ⟶ M(E, n)_{A−{i}};

(iii) the functions, for A, B ⊆< n >,

h : M(E, n)_{A} × M(E, n)_{B} ⟶ M(E, n)_{A∪B}

given by

h(x̄, ȳ) = \overline{xy},

where an element of M(E, n)_{A} is denoted by x̄ with x ∈ ⋂_{j∈A}Kerd_{j−1}^{n}.

Proof. First some explanation of the definition of M(−, n) as

π_{0}(DecE; Kerδ_{0}, …, Kerδ_{n−1}).

For each simplicial algebra, E, we start by looking at the canonical augmentation map δ_{0} : Dec^{1}E ⟶ E, which has kernel the simplicial algebra, Kerd_{last} mentioned above. Then take the simplicial inclusion crossed module Kerδ_{0} ⟶ Dec^{1}E to be M(E, 1) defining thus a functor

M( , 1) : SimpAlg ⟶ Simp(IncCrs^{1}).

Then it is easy to show that

π_{0}(Kerδ_{0}) ⟶ π_{0}(Dec^{1}E)

is precisely M(E, 1). The higher order analogues M( , n) are as follows:

For each simplicial algebra, E, there is a functorial short exact sequence

[figure: Kerδ_{0}; Dec^{1}E; δ_{0}; E.]

This corresponds to the 0-skeleton of the total décalage of E,

[figure: [... Dec^{3}E; Dec^{2}E; δ_{0}; δ_{1}; Dec^{1}E]; δ_{0}; E.]

For n = 2, the 1-skeleton of that total décalage gives a commutative diagram

Dec^{2}E \overline{δ_{0}_{}}Dec^{1}E

[figure: Dec^{1}E; δ_{1}; δ_{0}; E.; δ_{0}]

Here δ_{1} is d_{n−1}^{n} in dimension n whilst δ_{0} is d_{n}^{n}. Forming the square of kernels gives

Kerδ_{0} ∩ Kerδ_{1}  \overline{}Kerδ_{1}

[figure: Kerδ_{0}; Dec^{2}E.]

Again, π_{0} of this gives M(E, 2). In general, we use the (n − 1)-skeleton of the total décalage to form an n-cube and thus a simplicial inclusion crossed n-cube corresponding to the simplicial ideal (n + 1)-ad

(Dec^{n}E; Kerδ_{n−1}, …, Kerδ_{0}).

This simplicial inclusion n-cube will be denoted by M(E, n), and its associated crossed n-cube by

π_{0}(M(E, n)) = M(E, n).

The result now follows by direct calculation on examining the construction of π_{0} as the zeroth homology of the Moore complex of each term in the inclusion crossed n-cube, M(E, n).

4.3. Lemma. If E is a simplicial algebra with A ⊆< n >, A ≠< n >, then

d_{n}( ⋂_{i∈A} Kerd_{i}^{n}) = ⋂_{i∈A} Kerd_{i−1}^{n−1}.

Proof. If i ∈ A, then

d_{n}( ⋂_{i∈A} Kerd_{i}^{n}) ⊆ ⋂_{i∈A} Kerd_{i−1}^{n−1},

since d_{i−1}d_{n} = d_{n−1}d_{i−1}.

Conversely, we suppose that x is an element in ⋂_{i∈A} Kerd_{i−1}^{n−1} and consider the element

y = s_{n}x − s_{n−1}x + … + (−1)^{n−k}s_{k}x = ∑_{i=0}^{n−k}(−1)^{i+1}s_{i+k}x,

where k is the first integer in < n > \A. Then d_{n}y = x and d_{i}y = 0for all i ∈ A and hence y ∈ ⋂_{i∈A} Kerd_{i}^{n} implies x ∈ d_{n}(⋂_{i∈A} Kerd_{i}^{n}) as required.

This lemma gives the following proposition:

4.4. Proposition. If E is a simplicial algebra, then i) for A ⊆< n >, A ≠< n >,

M(E, n)_{A} ≅ ⋂_{i∈A} Kerd_{i−1}^{n−1}

so that in particular, M(E, n)_{∅} ≅ E_{n−1}; in every case the isomorphism is induced by d_{n}, ii) if A ≠< n > and i ∈< n >,

μ_{i} : M(E, n)_{A} ⟶ M(E, n)_{A\{i}}

is the inclusion of an ideal,

iii) for j ∈< n >,

μ_{j} : M(E, n)_{<n>} ⟶ ⋂_{i≠j} Kerd_{i}^{n+1}

is induced by d_{n}.

Proof. By proposition 4.2 and the previous lemma, one can obtain, for A ≠< n >,

M(E, n)_{A} = \frac{⋂_{i∈A} Kerd_{i−1}^{n}}{d_{n+1}(Kerd_{0}^{n+1}∩\{⋂_{i∈A} Kerd_{i}^{n+1}\})}

= \frac{⋂_{i∈A} Kerd_{i−1}^{n}}{Kerd_{0}^{n}∩(⋂_{i∈A} Kerd_{i−1}^{n}).}

The epimorphism d_{n} : E_{n} → E_{n−1}, which is split, d_{n}s_{n−1} = id, can be restricted to an epimorphism

⋂_{i∈A} Kerd_{i−1}^{n} ⟶ ⋂_{i∈A} Kerd_{i−1}^{n−1},

by lemma 4.3 It follows then that

Ker( ⋂_{i∈A} Kerd_{i−1}^{n} ⟶^{d_{n}} ⋂_{i∈A} Kerd_{i−1}^{n−1}) = Kerd_{0}^{n} ∩ ( ⋂_{i∈A} Kerd_{i−1}^{n}).

which completes the proof of (i). (ii) and (iii) are now consequences.

Expanding this out for low values of n gives:

1) For n = 0, M(E, 0) = E_{0}/d_{1}(Kerd_{0}) ≅ π_{0}(E) = H_{0}(E).

2) For n = 1, M(E, n) is the crossed module μ_{1} : Kerd_{0}^{1}/d_{2}^{2}(NE_{2}) → E_{1}/d_{2}^{2}(Kerd_{0}^{2}). Since d_{2}^{2}(NE_{2}) =Kerd_{0}^{1}Kerd_{1}^{1}, this implies

μ : NE_{1}/Kerd_{0}^{1}Kerd_{1}^{1} ⟶ E_{0}.

3) For n = 2, M(E, n) is

Kerd_{0}^{2} ∩ Kerd_{1}^{2}/d_{3}^{3}(Kerd_{0}^{3} ∩ Kerd_{1}^{3} ∩ Kerd_{2}^{3}) \overline{μ_{2}_{}}Kerd_{0}^{2}/d_{3}^{3}(Kerd_{0}^{3} ∩ Kerd_{1}^{3})

[figure: Kerd_{1}^{2}/d_{3}^{3}(Ker d_{0}^{3} ∩ Kerd_{2}^{3}); μ_{1}; μ_{2}; E_{2}/d_{3}^{3}(Kerd_{0}^{3}).; μ_{1}]

By proposition 4.4, this is isomorphic to

[figure: NE_{2}/d_{3}^{3}(NE_{3}); μ_{2}; Kerd_{0}^{1}; Kerd_{1}^{1}; μ_{1}; μ_{2}; E_{1}; μ_{1}]

Here the h-map is

h : Kerd_{0}^{1} × Kerd_{1}^{1} ⟶ NE_{2}/d_{3}^{3}(NE_{3})

given by

h(x, y) = s_{1}x(s_{1}y − s_{0}y) + ∂_{3}NE_{3}.

where x, y ∈ NE_{1}.

5. 2-crossed modules from crossed squares

Conduché constructed (in a letter to R.Brown in 1984) the 2-crossed module from a crossed square

[figure: L; λ; M; N; λ′; ν; R; μ]

as

[figure: L; (−λ, λ′); M ⋊ N; μ + ν; R.]

This construction can be briefly described as follows: apply the nerve in the both directions so as to get a bisimplicial algebra, then apply either the diagonal or the Artin-Mazur ‘codiagonal’ functor to get to a simplicial algebra and take the Moore complex.

We saw above the category of crossed modules is equivalent to that of cat^{1}-algebras. Ellis (cf. [7]) generalised this equivalence to the dimension n, i.e., the equivalence between crossed n-cubes and cat^{n}-algebras.

We form the associated cat^{2}-algebra. This is

[figure: (L ⋊ N) ⋊ (M ⋊ R); s; t; N ⋊ R; M ⋊ R; s′; t′; s_{M}; t_{M}; R.; s_{N}; t_{N}]

The source and target maps are defined as follows:

\begin{matrix}s((l, n), (m, r)) = (n, r) & t((l, n), (m, r)) = (λ′l + μ(m)n, μ(m) + r) \\ s′((l, n), (m, r)) = (m, r) & t′((l, n), (m, r)) = (λl + m, ν(n) + r) \\ s_{M}(m, r) = r & t_{M}(m, r) = μ(m) + r \\ s_{N}(n, r) = r & t_{N}(n, r) = ν(n) + r \end{matrix}

for l ∈ L, m ∈ M, n ∈ N and r ∈ R.

We now take the binerve, that is the nerves in the both directions of the cat^{2}-algebra constructed. This is a bisimplicial algebra. The first few entries in the bisimplicial array are given below (where A^{B} := A ⋊ B)

[figure: ⋯; (L^{N}) ⋊ ((L^{N}) ⋊ (M^{R})); N ⋊ (N ⋊ R)]

[figure: ((L^{L}) ⋊ N) ⋊ ((M^{M}) ⋊ R); (L^{N}) ⋊ (M^{R}); N ⋊ R]

[figure: M ⋊ (M ⋊ R); M ⋊ R; R]

Some reduction has already been done. For example, the double semi-direct product represents the algebra of pairs of elements ((m_{1}, r_{1}), (m_{2}, r_{2})) ∈ M ⋊ R where μ(m_{1}) + r_{1} = r_{2}. This is the algebra M ⋊ (M ⋊ R) where the action is M ⋊ R on M given by (m, r) · m′ = (μm + r)m′.

We will recall the ‘Artin-Mazur’ codiagonal functor ∇ (cf. [1]) from bisimplicial algebras to simplicial algebras.

Let E.,. be a bisimplicial algebra. Put

E_{(n)} = ⨁_{p+q=n} E_{p,q}

and define ∇_{n} ⊂ E_{(n)} as follows: An element (x_{0}, …, x_{n}) of E_{(n)} with x_{p} ∈ E_{p,n−p} is in ∇_{n} if and only if

d_{0}^{v}x_{p} = d_{p+1}^{h}x_{p+1} for p = 0, …, n − 1.

Next, define the faces and degeneracies

\begin{matrix}D_{j} : ∇_{n} → ∇_{n−1} \\ S_{j} : ∇_{n} → ∇_{n+1} & j = 0, …, n \end{matrix}

by

\begin{aligned}D_{j}(x) & = & (d_{j}^{v}x_{0}, d_{j−1}^{v}x_{1}, …, d_{1}^{v}x_{j−1}, d_{j}^{h}x_{j+1}, d_{j}^{h}x_{j+2}, …, d_{j}^{h}x_{n}) \\ S_{j}(x) & = & (s_{j}^{v}x_{0}, s_{j−1}^{v}x_{1}, …, s_{0}^{v}x_{j−1}, s_{j}^{h}x_{j}, s_{j}^{h}x_{j+1}, …, s_{j}^{h}x_{n}). \end{aligned}

Thus ∇(E., .) = {∇_{n} : D_{j}, S_{j}} is a simplicial algebra.

We now examine this construction in low dimensions: Example: For n = 0, E_{(0)} = E_{0,0}. For n = 1, we have

∇_{1} ⊂ E_{(1)} = E_{1,0} ⊕ E_{0,1}

where

∇_{1} = {(e_{1,0}, e_{0,1}) : d_{0}^{h}e_{1,0} = d_{1}^{v}e_{0,1}}

together with the homomorphisms

D_{0}^{1}(e_{1,0}, e_{0,1}) = (d_{0}^{v}e_{1,0}, d_{0}^{h}e_{0,1})
D_{1}^{1}(e_{1,0}, e_{0,1}) = (d_{1}^{v}e_{1,0}, d_{1}^{h}e_{0,1})
S_{0}^{0}(e_{0,0}) = (s_{0}^{v}e_{0,0}, s_{0}^{h}e_{0,0}).

For n = 2, we have

∇_{2} ⊂ E_{(2)} = ⨁_{p+q=2} E_{p,q} = E_{2,0} ⊕ E_{1,1} ⊕ E_{0,2}

where

∇_{2} = {(e_{2,0}, e_{1,1}, e_{0,2}) : d_{0}^{h}e_{2,0} = d_{1}^{v}e_{1,1}, d_{0}^{h}e_{1,1} = d_{2}^{v}e_{0,2}} .

We use the ‘Artin-Mazur’ codiagonal functor to obtain a simplicial algebra A (of some complexity);

The base algebra is still A_{0} ≅ R. However the algebra of 1-simplices is subset of

E_{1,0} ⊕ E_{0,1} = (M ⋊ R) ⊕ (N ⋊ R),

consisting of elements ((m, r), (n, r′)) where μ(m) + r = r′, i.e.,

A_{1} = {((m, r), (n, r′)) : μ(m) + r = r′} .

Let us see what the composite of two such elements

((m_{1}, μ(m_{1}) + r_{1}), (n_{1}, r_{1}))and ((m_{2}, μ(m_{2}) + r_{2})), (m_{2}, r_{2}))

becomes:

((m_{1}, μ(m_{1}) + r_{1}), (n_{1}, r_{1})) ∘ ((m_{2}, μ(m_{2}) + r_{2})), (m_{2}, r_{2}))
= (m_{1}, μ(m_{1}) + r_{1}) · (m_{2}, μ(m_{2}) + r_{2}) ∘ ((n_{1}, r_{1}) · (m_{2}, r_{2}))by the inter-change law
= ((m_{1}m_{2} + m_{1}(μ(m_{2}) + r_{2}) + m_{2}(μ(m_{1}) + r_{1}), (μ(m_{1}) + r_{1})(μ(m_{2}) + r_{2})),
n_{1}n_{2} + n_{1}r_{2} + n_{1}r_{1}, r_{1}r_{2}).

This means that the subalgebra A_{1} is M ⋊ (N ⋊ R) where the action of N ⋊ R on M via R is the obvious one. Thus

A_{1} ≅ M ⋊ (N ⋊ R) = {(m, n, r) : m ∈ M, n ∈ N, r ∈ R} .

We next turned to the algebra of 2-simplices which is the subset A_{2} of

E_{2,0} ⊕ E_{1,1} ⊕ E_{0,2} = (M ⋊ (M ⋊ R)) ⊕ ((L ⋊ N) ⋊ (M ⋊ R)) ⊕ (N ⋊ (N ⋊ R)),

whose elements ((m_{2}, m_{1}, r), (l, n, m, r′), (n_{2}, n_{1}, r′′)) are such that (m_{2}, μm_{1} +r) = (m, r′) and (λ′l + n, μm + r′) = (n_{1}, r′′).

By similar calculation given above, the composition of two such elements

((m_{2}, m_{1}, r), ((l, n), (m_{2}, μm_{1} + r)), (n_{2}, (λ′l + n, μm + r′)))

and

((m′_{2}, m′_{1}, r_{1}), ((l′, n′), (m′_{2}, μm′_{1} + r_{1})), (n′_{2}, (λ′l′ + n′, μm′ + r′_{1})))

becomes such an element (l, m, n, r) ∈ L ⋊ (M ⋊ (N ⋊ R)) where l ∈ L, m ∈ M, n ∈ N and r ∈ R. This is the algebra L ⋊ (M ⋊ (N ⋊ R)) where the action of M ⋊ (N ⋊ R) on L is the obvious one. Thus

A_{2} ≅ L ⋊ (M ⋊ (N ⋊ R)).

Therefore we can get a 2-truncated simplicial algebra A^{(2)} that looks like

[figure: A^{(2)} :; L ⋊ (M ⋊ (N ⋊ R)); d_{0}^{2}, d_{1}^{2}, d_{2}^{2}; \underline{s_{0}^{1}, s_{1}^{1}}; M ⋊ (N ⋊ R); d_{0}^{1}, d_{1}^{1}; \underline{s_{0}^{0}}; R.]

with the faces and degeneracies

d_{0}^{1}(m, n, r) = r
d_{1}^{1}(m, n, r) = μ(m) + r′ where r′ = ν(n) + r
s_{0}^{0}(r) = (0, n, r)

and

d_{0}^{2}(l, m, n, r) = (m, n, r)
d_{1}^{2}(l, m, n, r) = (m, n, r)
d_{2}^{2}(l, m, n, r) = (−λ(l) + m, λ′(l) + n, μ(m) + r′); where r′ = ν(n) + r
s_{0}^{1}(m, n, r) = (0, m, n, r)
s_{1}^{1}(m, n, r) = (0, m, n, r).

Loday,[11], defined a mapping cone of a complex which is analogous to the construction of the Moore complex of a simplicial group. One can easily give an obvious description for commutative algebras.

We next give a mapping cone of a crossed square of algebras as follows:

5.1. Proposition. The Moore complex of the simplicial algebra A^{(2)} is the mapping cone, in the sense of Loday, of the crossed square. Furthermore, this mapping cone complex has a structure of 2-crossed module of algebras.

Proof. Given the 2-truncated simplicial algebra A^{(2)} described the above and look at its Moore complex:

Of course NA_{0} = A_{0} ≅ R. The second term of the Moore complex is NA_{1} =Kerd_{0}. The d_{0} map is d_{0} on N ⋊ R, that is it picks out the point r, i.e., d_{0}(n, r) = r. Thus the kernel of d_{0} is the subalgebra of M ⋊ (N ⋊ R) when r = 0. This is the algebra M ⋊ N = NA_{1} =Kerd_{0}. The map d_{1} |_{Kerd_{0}} is d_{1}(m, n, 0) = μ(m) + ν(n). This is d_{1} on M ⋊ R, which is μ(m) + r′ where r′ = ν(n) + r, and of course r = 0.

It is not hard to see that the kernel of d_{0} and d_{1} is the subalgebra of L⋊(M ⋊(N ⋊R)) where m = 0, n = 0, r = 0. This is the algebra L. In other words NA_{2} = L. The map d_{2} |_{Kerd_{0}∩Kerd_{1}} is

d_{2}(l, 0, 0, 0) = (−λl, λ′l, 0),

This is ∂_{2} =d_{2} on L, which is, ∂_{2}l = (−λl, λ′l).

Next we will see that ∂_{1}∂_{2} = 0,

∂_{1}∂_{2}(l) = ∂_{1}(−λl, λ′l)
= μ(−λl) + νλ′l
= −μλ(l) + μλ(l) by μλ = νλ′
= 0.

Thus, if given a crossed square

[figure: L; λ; M; N; λ′; ν; R; μ]

then its mapping cone is

[figure: L; ∂_{2}; M ⋊ N; ∂_{1}; R]

where ∂_{1}(m, n) = μ(m) + ν(n) and ∂_{2}l = (−λl, λ′l).

Now we will show the second part of the proof of result:

The semi-direct product M ⋊ N can be formed by making N act on M via R,

n · m = ν(n)m

where the R-action is the given one. The fact that ∂_{2} and ∂_{1} are algebra homomorphisms:

\begin{aligned}(m, n)(m′, n′) & = & (n · m′ + n′ · m + mm′, nn′) \\ & = & (ν(n)m′ + ν(n′)m + mm′, nn′) \end{aligned}

Thus, (i)

∂_{1}((m, n)(m′, n′)) = ∂_{1}(ν(n)m′ + ν(n′)m + mm′, nn′)
= μ(ν(n)m′ + ν(n′)m + mm′) + ν(nn′)
= ν(n)μ(m′) + ν(n′)μ(m) + μ(m)μ(m′) + ν(n)ν(n′)
= (μ(m) + ν(n))(μ(m′) + ν(n′))
= ∂_{1}(m, n)∂_{1}(m′, n′)

Theory and Applications of Categories, Vol. 3, No. 7 178

and similarly of course

∂_{1}((m, n) + (m′, n′)) = ∂_{1}(m + m′, n + n′)
= μ(m + m′) + ν(n + n′)
= (μ(m) + ν(n)) + (μ(m′) + ν(n′))
= ∂_{1}(m, n) + ∂_{1}(m′, n′)

(ii) if l_{1}, l_{2} ∈ L, then

\begin{aligned}∂_{2}(l_{1}l_{2}) & = & (−λ(l_{1}l_{2}), λ′(l_{1}l_{2})) \\ & = & (−(λl_{1}λl_{2}), λ′(l_{1})λ′(l_{2})) \end{aligned}

whilst

\begin{aligned}∂_{2}(l_{1})∂_{2}(l_{2}) & = & (−λl_{1}, λ′l_{1})(−λl_{2}, λ′l_{2}) \\ & = & ((λ′l_{1}) · (−λl_{2}) + (λ′l_{2}) · (−λl_{1}) + (−λl_{1})(−λl_{2}), λ′(l_{1})λ′(l_{2})) \end{aligned}

but νλ′ = μλ, so the first coordinates are equal. Similarly

∂_{2}(l_{1} + l_{2}) = (−λ(l_{1} + l_{2}), λ′(l_{1} + l_{2}))
= (−λl_{1}, λ′l_{1}) + (−λl_{2}, λ′l_{2}).
= ∂_{2}(l_{1}) + ∂_{2}(l_{2}).

These elementary calculations are useful as they pave the way for calculation of the Peiffer multiplication of x = (m, n) and y = (c, a) in the above complex:

⟨x, y⟩ = ∂_{1}x · y − xy
= (μ(m) + ν(n)) · (c, a) − (m, n)(c, a)
= ((μ(m) + ν(n)) · c, (μ(m) + ν(n)) · a) − (n · c + a · m + mc, na)
= (a · m, (μ(m) + ν(n)) · a − na) since any dependence on cvanishes
= (ν(a)m, μ(m) · a + ν(n) · a − na) by definition of action
= (ν(a)m, μ(m)a + na − na) since ν is a crossed module
= (ν(a)m, μ(m)a).

Thus the Peiffer lifting

{ ⊗ } : (M ⋊ N) ⊗ (M ⋊ N) ⟶ L

for this structure is given by

{x ⊗ y} = h(m, na).

It is immediate that this works:

∂_{2} {x ⊗ y} = (−λh(m, na), λ′h(m, na))
= ((−m) · na, (na) · m)
= (−ν(na)m, μ(m)(na))

by the axioms of a crossed square.

We will not check all the axioms for a 2-crossed module for this structure, but will note the proof for two or three of them as it illustrates the connection between the h-map and the Peiffer lifting.

PL2: {∂_{2}(l_{0}) ⊗ ∂_{1}(l_{1})} = l_{0}l_{1}.

As ∂_{2}l = (−λl, λ′l), this needs the calculation of

h(−λl_{0}, λ′(l_{0}l_{1}))

but the crossed square axiom h(λl, n) = n · l, and h(m, λ′l) = m · l, together with the fact that the map λ : L → M is a crossed module give:

{∂_{2}(l_{0}) ⊗ ∂_{1}(l_{1})} = {(−λl_{0}, λ′l_{0}) ⊗ (−λl_{1}, λ′l_{1})}
= h(−λl_{0}, λ′(l_{0}l_{1}))
= −λl_{0} · (l_{0}l_{1})
= l_{0}l_{1}

PL3: {y_{0} ⊗ y_{1}y_{2}} = {y_{0} ⊗ y_{1}y_{2}} + ∂_{1}y_{2} · {y_{0} ⊗ y_{1}y_{2}} , where y_{0} = (m_{0}, n_{0}), y_{1} = (m_{1}, n_{1}) and y_{2} = (m_{2}, n_{2}) ∈ M ⋊ N

{y_{0} ⊗ y_{1}y_{2}} = {(m_{0}, n_{0}) ⊗ (m_{1}, n_{1})(m_{2}, n_{2})}
= {(m_{0}, n_{0}) ⊗ (n_{1} · m_{2} + n_{2} · m_{1} + m_{1}m_{1}, n_{1}n_{2})}
= h(m_{0}, n_{0}n_{1}n_{2})

whilst

{y_{0} ⊗ y_{1}y_{2}} + ∂_{1}y_{2} · {y_{0} ⊗ y_{1}y_{2}} = {(m_{0}, n_{0})(m_{1}, n_{1}) ⊗ (m_{2}, n_{2})}
+∂_{1}y_{2} · {(m_{0}, n_{0}) ⊗ (m_{1}, n_{1})(m_{2}, n_{2})}
= h(n_{0} · m_{1} + n_{1} · m_{0}, n_{0}n_{1}(n_{2})) + h(m_{0}, ∂_{1}y_{2} · (n_{0}n_{1}))

and this simplifies as expected to give the correct result.

PL4:

{x ⊗ ∂_{2}l} + {∂_{2}l ⊗ x} = {(m, n) ⊗ (−λl, λ′l)} + {(−λl, λ′l) ⊗ (m, n)}
= h(m, nλ′l) + h(−λl, λ′(l)n)
= h(m, λ′(n · l)) + h(−λl, λ′(n · l)) by λ′ cros. mod.
= (μ(m) + ν(n)) · l
= ∂_{1}(m, n) · l
= ∂_{1}x · l

We thus have two ways of going from simplicial algebras to 2-crossed modules

(i) directly to get

\frac{NE_{2}}{∂_{2}NE_{3}} ⟶ NE_{1} ⟶ NE_{0},

(ii) indirectly via the square axiom M(E, 2) and then by the above construction to get

\frac{NE_{2}}{∂_{2}NE_{3}} ⟶ Kerd_{0} ⋊ Kerd_{1} ⟶ E_{1},

and they clearly give the same homotopy type. More precisely E_{1} decomposes as Kerd_{1} ⋊ s_{0}E_{0} and the Kerd_{0} factor in the middle term of (ii) maps down to that in this decomposition by the identity map. Thus d_{0} induces a quotient map from (ii) to (i) with kernel isomorphic to

0 ⟶ Kerd_{0} ⟶^{=} Kerd_{0}

which is thus contractible.

Remark. The situation in this paper can be summarised in the following diagram:

SimpAlg_{≤2}
[figure: M(−, 2)]
[figure: Crs^{2}; ( )_{2}; X_{2}Mod,; K_{(2)}; C_{(2)}]

where ( )_{2} is given by Proposition 5.1. The diagram is commutative, linking the constructions of section 4 and 5 with those given here.

References

[1] M. Artin and B. Mazur. On the Van Kampen theorem, Topology, 5, 179-189 (1966).

[2] Z. Arvasi. Applications in commutative algebra of the Moore complex of a simplicial algebra, Ph.D. Thesis, U.C.N.W. Maths Preprint, 94.09.

[3] Z. Arvasi and T. Porter. Freeness conditions for 2-crossed modules of commutative algebras, Applied Categorical Structures, (to appear).

[4] R. Brown and J-L. Loday. Van Kampen theorems for diagram of spaces, Topology 26 (1987) 311-335.

[5] D. Conduché. Modules croisés généralisés de longueur 2, J. Pure Applied Algebra, 34 (1984), 155-178.

[6] J. Duskin. Simplicial methods and the interpretation of triple cohomology, Memoirs A.M.S. Vol. 3 163 (1975).

[7] G.J. Ellis. Higher dimensional crossed modules of algebras. J. Pure and Appl. Algebra 52 (1988) 277-282.

[8] A.R. Grandjeán and M.J. Vale. 2-Modulos cruzados en la cohomologia de André-Quillen. Memorias de la Real Academia de Ciencias, 22 (1986), 1-28.

[9] D. Guin-Walery and J.L. Loday. Obstructions á l’excision en K-theories algébrique. In: Friedlander, E.M.,Stein, M.R.(eds.) Evanston conf. on algebraic Ktheory 1980. (Lect. Notes Math., vol. 854, pp. 179-216) Berlin Heidelberg New York: Spinger 1981.

[10] L. Illusie. Complex cotangent et déformations, I, Lectures Notes in Math. 239 (1971); II, 283 (1972).

[11] J.L. Loday. Spaces with finitely many non-trivial homotopy groups. J. Pure and Appl. Algebra 24 (1982) 179-202.

[12] T. Porter. Homology of commutative algebras and an invariant of Simis and Vasconceles. J.Algebra 99 No:2 (1987).

[13] T. Porter. n-Type of simplicial groups and crossed n-cubes. Topology 32 5-24 (1993).

[14] J.H.C. Whitehead. Combinatorial homotopy. Bull. Amer. Math. Soc 55 (1949) 453-496.

Department of Mathematics Faculty of Science Osmangazi University 26480, Eskişehir Turkey

Email: zarvasi@mail.ogu.edu.tr

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Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.math.mcgill.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università della Calabria: carboni@unical.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: mthfwl@ubvms.cc.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@gar.union.edu Robert Paré, Dalhousie University: pare@cs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca