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 by a Peiffer lifting function , such that, on writing ,
each , and are -modules and the for , together with the codomain restriction of , are -module homomorphisms;
the -module structure on is the action induced from the -action on for which the action of 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 , the corresponding category.
Any 2-crossed module clearly gives a 2-crossed complex (with trivial ‘tail’).
From simplicial groups to 2-crossed complexes. If is a simplicial group, then
has the structure of a 2-crossed complex, where is the Moore complex of , is the subgroup of generated by the degenerate elements, and, for ,
is the -dimensional term of the crossed complex, , associated to the simplicial group 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 has the structure of a 2-crossed complex if and only if for each , 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 , 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 , for al ) is isomorphic to a crossed complex in this sense.
This functor, from to , has a left adjoint which is the identity on the subcategory of with trivial Peiffer liftings, so is equivalent to a reflective subcategory of
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