nLab
crossed complex

Idea

A crossed complex (of groupoids) is a nonabelian and many object generalization of a chain complex of abelian groups.

Crossed complexes were defined by Blakers in 1948 (following a suggestion of Eilenberg) and developed by Whitehead in 1949 and 1950 (but these authors used different terminology). They were applied by Huebschmann to group cohomology in 1980. They were further developed in series of articles by Ronnie Brown and collaborators in the context of nonabelian algebraic topology, and partly because they were found equivalent to form of (strict) cubical $\omega$-groupoid with connections. This equivalence enabled a number of new results, including van Kampen type theorems and monoidal closed structures for crossed complexes.

Crossed complexes are an equivalent way to encode the information contained in strict ω-groupoids: the groups appearing in the crossed complex in degree $n\ge 2$ are the source-fibers of the collection of $n$-morphisms of the $\omega$-groupoid.

See also homotopy n-type.

One way to think of a crossed complex is as a chain complex in which the bottom part is a crossed module and the rest is a chain complex of modules over the fundamental group of the crossed complex (that is its cokernel). This is easy to think of in the case where there is a single object (crossed complex of groups), and it is a simple step to extend to the many object case.

Examples

If $X_{*}$ is a filtered space, then there is a crossed complex $\Pi X_{*}$ which in dimension 0 is $X_0$, in dimension 1 is the fundamental groupoid ${\pi }_1(X_1,X_0)$ and in dimension $n>1$ is the family of relative homotopy groups $\{{\pi }_n(X_n,X_{n-1},p):p\in X_0\}$. This gives a functor $\Pi$ from filtered spaces to crossed complexes, which may be used to construct the generalisation of the Dold-Kan correspondence, which in this case goes between crossed complexes and simplicial T-complexes.

An important special case of the above is when the filtered space is a CW-complex and the filtration is by skeleta. Particularly useful instances of this are the $n$-cubes and $n$-simplices, with their CW-filtration. We obtain $\Pi (I^n)$ and $\Pi ({\Delta }^n)$. These are used to define cubical and simplicial nerves of a crossed complex and these in turn define the Dold-Kan correspondence mentioned above. For instance if $C$ is a crossed complex, then its simplicial nerve is the simplicial set with $\mathrm{Ner}(C{)}_n=\mathrm{Crs}(\Pi ({\Delta }^n),C)$ in dimension $n$.

Remarks

• In low degrees crossed complexes are the following:

  • A crossed complex of length 0 is a set.
  • A crossed complex of length 1 is a groupoid.
  • A crossed complex of length 2 is a crossed module, which is equivalent to a strict kind of 2-group by the Brown-Spencer Theorem, (R. Brown and C.B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296 – 302.)

A survey of the use of crossed complexes is in R. Brown Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems, to appear in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier, 2008/2009. (available as math.AT/0212274 v7).

Tim: One of many terminological problems that arise in this stuff is whether ‘length’ of a chain complex or crossed complex refers to the number of transitions/arrows or the number of nodes /groups or whatever?

(Alex Nelson): I think that the “length” refers to the number of morphisms. I suspect this since a length 0 chain complex is a set, which makes intuitive sense if it’s just a single object without any morphisms…but I may be (and probably am) wrong.

A discussion of the kind of homotopy types modelled by crossed complexes, namely a linear model, and a conjecture as to how to extend this, is in homotopy n-type.

Crossed complexes as Moore complexes

Crossed complexes (of groups) correspond to group T-complexes. Any group $T$-complex is a simplicial group and in the entry for them it is mentioned that a simplicial group has a group $T$-complex structure if and only if $𝒩G\cap D$ is the trivial graded subgroup, where $D=(D_n{)}_{n\ge 1}$ is the graded subgroup of $G$ generated by the degenerate elements. If $G$ is such a group $T$-complex then its Moore complex has a natural structure of a crossed complex. In general the obstruction to a given simplicial group to have Moore complex which is a crossed complex is exactly that graded subgroup, $𝒩G\cap D$. (The Whitehead products for $G$ live in this graded subgroup, so this provides one way of showing that the homotopy types representable by crossed complexes have trivial Whitehead products.)

Conversely, for any crossed complex, $C$, there is a simplicial group, $K(C)$, constructed using an analogue of the inverse in the Dold-Kan correspondence, which is a group $T$-complex and whose Moore complex is isomorphic to $C$.

The generalisation to general crossed complexes (of groupoids) and simplicially enriched groupoids if quite easy to do. We will usually state results below for the group case, leaving the general case to the ‘reader’.

From simplicial group(oid)s to crossed complexes

It is fairly clear that crossed complexes / group(oid) $T$-complexes correspond to simplicial group(oid)s in which certain equations hold. It therefore is reasonable that they are equivalent to a variety / reflective subcategory in the category, $\mathrm{SSet}\mathrm{Grpd}$, of simplicially enriched groupoids. (The discussion in the entry on group T-complex is relevant here.)

There is a functor $C(-)$ from simplicial groups to crossed complexes given by

$$C(G{)}_{n+1}=\frac{𝒩G_n}{(𝒩G_n\cap D_n)d_0(𝒩G_{n+1}\cap D_{n+1})},$${C}(G)_{n+1} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},

in higher dimensions with at its ‘bottom end’, the crossed module,

$$\frac{𝒩G_1}{d_0(𝒩G_2\cap D_2)}\to 𝒩G_0$$\frac{\mathcal{N}G_1}{d_0(\mathcal{N}G_2\cap D_2)} \to \mathcal{N}G_0

with $\partial$ induced from the boundary in the Moore complex.

The category of crossed complexes form a variety in the category of all hypercrossed complexes. Alternatively, groupoid T-complexes (the groupoid version of group T-complex) form a variety in the category of all simplicial groups.