Crossed complexes are a useful extension of crossed modules allowing not only the encoding of an algebraic model for the homotopy 2-type, but also information on the ‘complex of chains on the universal cover?’. The category of crossed complexes is a monoidal closed category equivalent to various types of strict infinity-groupoid.
To model the homotopy 3-type of a space, we can use either a 2-crossed module or a crossed square (or various other algebraic models to be added some time in the future). A crossed complex is a ‘hybrid’, part crossed module but with a ‘tail’ which is a chain complex. What would be the ‘hybrid’ between a 2-crossed module and a chain complex? Are there examples that are easily constructed? What sort of information do they encode? Are they easy to analyse, understand, … and useful?
A 2-crossed complex is a normal complex of groups
together with a 2-crossed module structure given on $C_2\to C_1\to C_0$ by a Peiffer lifting function $\{ -,-\} : C_1\times C_1 \to C_2$, such that, on writing $\pi = Coker(C_1\to C_0)$,
each $C_n$, $n\geq 3$ and $Ker \,\partial_2$ are $\pi$-modules and the $\partial_n$ for $n\geq 4$, together with the codomain restriction of $\partial_3$, are $\pi$-module homomorphisms;
the $\pi$-module structure on $Ker \partial_2$ is the action induced from the $C_0$-action on $C_2$ for which the action of $\partial_1 C_1$ is trivial.
A 2-crossed complex morphism is defined in the obvious way, being compatible with all the actions, the pairings and Peiffer liftings. We will denote by $2 Crs$, the corresponding category.
Any 2-crossed module clearly gives a 2-crossed complex (with trivial ‘tail’).
From simplicial groups to 2-crossed complexes. If $G$ is a simplicial group, then
has the structure of a 2-crossed complex, where $\mathcal{N}G$ is the Moore complex of $G$, $D_n$ is the subgroup of $G_n$ generated by the degenerate elements, and, for $n\gt2$,
is the $n$-dimensional term of the crossed complex, $C(G)$, associated to the simplicial group $G$ as in the entry crossed complex (in the section From simplicial group(oid)s to crossed complexes.)
(There is an obvious extension of the group based definition above to a groupoid based one, and of this construction to one which takes as input a simplicially enriched groupoid.)
The Moore complex of a simplicial group $G$ has the structure of a 2-crossed complex if and only if for each $n\gt 2$, $\mathcal{N}G_n\cap D_n$ is trivial. This means that the axioms of a group T-complex are almost satisfied, but not necessarily in dimension 2.
A quadratic chain complex as defined by H.J. Baues is a special case of a 2-crossed complex, satisfying additional (pre-crossed module) nilpotency condition at the level of the underlying pre-crossed module. (In fact the category of quadratic chain complexes is a reflective subcategory of the category of 2-crossed complexes.) In Baues’ book referenced below, there is the construction of the fundamental quadratic chain complex of a pointed CW-complex. The reflection (or cotruncation) of this to the category of quadratic modules (i.e. 3-truncated quadratic chain complexes) faithfully represents the homotopy 3-type of a CW-space (at the level of spaces and maps between them).
Graham Ellis defined the fundamental squared complex of a CW-complex from triad homotopy groups and generalised Whitehead products, and showed how Baues fundamental quadratic chain complex of a CW-complex can be obtained from it. A homotopy 2-crossed complex of a CW-complex can also be defined is the same way, see the work of João Faria Martins below.
Any crossed complex can be given the structure of a 2-crossed complex simply by defining a trivial Peiffer lifting, $\{-,-\}$. As the Peiffer lifting covers the Peiffer commutators in $C_1$, and these are trivial (since the bottom of the crossed complex is a crossed module), this trivial Peiffer lifting works and gives a 2-crossed complex structure. This defines a functor from the category of crossed complexes to that of 2-crossed complexes.
Any 2-crossed complex which has a Peiffer lifting that is trivial $\{x,y\} = 1$, for al $x,y \in C_1$) is isomorphic to a crossed complex in this sense.
This functor, from $Crs$ to $2-Crs$, has a left adjoint which is the identity on the subcategory of $2-Crs$ with trivial Peiffer liftings, so $Crs$ is equivalent to a reflective subcategory of $2-Crs$
See the Crossed Menagerie, chapter 4.
H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes , de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).
Graham Ellis, Crossed squares and combinatorial homotopy. Mathematische Zeitschrift Volume 214, Number 1, 93-110, DOI: 10.1007/BF02572393
João Faria Martins, Homotopies of 2-crossed complexes and the homotopy category of pointed 3-types (web pdf)