A simplicial group whose Moore complex has length (that is, at most stuff in dimensions and ) will be the internal nerve of a strict -group and the Moore complex will be the corresponding crossed module. What if we have a simplicial group whose Moore complex has at most stuff in dimensions , , and ; can we describe its structure in some similar way? Yes, and Conduché provided a neat description of the structure involved. From the structure one can rebuild a simplicial group, a type of internal -nerve construction.
In other words, a -crossed module is the Moore complex of a -truncated simplicial group.
A -crossed module is a normal complex of groups
together with an action of on all three groups and a mapping
the action of on itself is by conjugation, and and are -equivariant;
for all ,
if , then
if and , then
for all ,
if and , then
The pairing is often called the Peiffer lifting of the -crossed module.
In a -crossed module as above the structure is a crossed module, but may not be one, as the Peiffer identity need not hold. The Peiffer commutator?, which measures the failure of that identity, may not be trivial, but it will be a boundary element and the Peiffer lifting gives a structured way of getting an element in that maps down to it.
It is sometimes useful to consider a crossed module as being a crossed complex of length 1 (i.e. on possibly non-trivial morphism only). Likewise one can consider a 2-crossed module as a special case of a 2-crossed complex. Such a gadget is intuitively a 2-crossed module with a ‘tail’, which is a chain complex of modules over the of the base 2-crossed module, much as a crossed complex is a crossed module together with a ‘tail’.
A quadratic module, as developed by H.-J. Baues, is a special case of a 2-crossed module, satisfying nilpotency conditions at the level of the underlying pre-crossed module (which is a close to being a crossed module as possible.) The fundamental quadratic module of a CW-complex yields an equivalence of categories between the category of pointed 3-types and the category of quadratic modules.
A functorial fundamental 2-crossed module of a CW-complex can also be defined, by using Graham Ellis fundamental crossed square of a CW-complex; this is explained in the article of João Faria Martins, below. We can also define this fundamental 2-crossed module of a CW-complex, by using Kan’s fundamental simplicial group of a CW-complex, and by applying the usual reflection from simplicial groups to simplicial groups of Moore complex of lenght two, known to be equivalent to 2-crossed modules.
The homotopy theory of 2-crossed modules can be addressed by noting that 2-crossed modules, inducing a reflective subcategory of the category of simplicial groups, inherit a natural Quillen model structure, as explored in the article of Cabello and Garzon below. A version very close to the usual homotopy theory of crossed complexes was developed in the article of Joao Faria Martins below in a parallel way to the homotopy theory of quadratic module?s and quadratic complex?es as introduced by H. J. Baues.
Any crossed module, gives a 2-crossed module, by setting , the trivial group, and, of course, , . Conversely any 2-crossed module having trivial top dimensional group () ‘is’ a crossed module. This gives an inclusion of the category of crossed modules into that of 2-crossed modules, as a reflective subcategory.
The reflection is given by noting that, if
is a 2-crossed module, then is a normal subgroup of , and then there is an obvious induced crossed module structure on
But we can do better than this. More generally, let
be a truncated crossed complex (of groups) in which all higher dimensional terms are trivial, then taking , and , with trivial Peiffer lifting, gives one a 2-crossed complex. Conversely suppose we have a 2-crossed module with trivial Peiffer lifting: for all , , axiom 3 then shows that is an Abelian group, and similarly the other axioms can be analysed to show that the result is a truncated crossed complex.
The category of crossed complexes of length 2 is equivalent to the full subcategory of given by those 2-crossed modules with trivial Peiffer lifting.
Of course, the resulting ‘inclusion’ has a left adjoint, which is quite fun to check out! (You kill off the subgroup of generated by the Peiffer lifting, …. is that all?)
If is a simplicial group then
is a 2-crossed module. (You are invited to find the Peiffer lifting!)
Both crossed squares and 2-crossed modules model all connected homotopy 3-types so one naturally asks how to pass from one description to the other. Going from crossed squares to 2-crossed modules is easy, so will be given here (going back is harder).
be a crossed square then acts on via , so , and so we can form and the sequence
is then a 2-crossed complex.
(And, yes, these are actually group homomorphisms: , the product of the two elements! Try it!)
The full result and an explanation of what is going on here is given in
H. J. Baues: Combinatorial homotopy and -dimensional complexes. With a preface by Ronald Brown. de Gruyter Expositions in Mathematics, 2. Walter de Gruyter \& Co., Berlin, 1991.
Daniel Conduché: Modules croisés généralisés de longueur . Proceedings of the Luminy conference on algebraic -theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 155–178.
Graham Ellis: Crossed squares and combinatorial homotopy. Math. Z. 214 (1993), no. 1, 93–110.