and
nonabelian homological algebra
A 2-crossed module encodes a semistrict 3-group – a Gray-group – in generalization of how a crossed module encodes a strict 2-group.
A simplicial group whose Moore complex has length $1$ (that is, at most stuff in dimensions $0$ and $1$) will be the internal nerve of a strict $2$-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 $0$, $1$, and $2$; 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 $2$-nerve construction.
In other words, a $2$-crossed module is the Moore complex of a $2$-truncated simplicial group.
A $2$-crossed module is a normal complex of groups
together with an action of $N$ on all three groups and a mapping
such that
the action of $N$ on itself is by conjugation, and $\partial_2$ and $\partial_1$ are $N$-equivariant;
for all $m_0,m_1 \in M$,
if $\ell_0,\ell_0 \in L$, then
if $\ell \in L$ and $m\in M$, then
for all $m_0,m_1,m_2 \in M$,
if $n\in N$ and $m_0,m_1 \in M$, then
The pairing $\{ - ,- \} : M\times M \to L$ is often called the Peiffer lifting of the $2$-crossed module.
In a $2$-crossed module as above the structure $\partial_2: L \to M$ is a crossed module, but $\partial_1: M\to N$ 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 $L$ 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 $\pi_0$ 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, $G_2 \stackrel{\delta }{\to}{G_1}$ gives a 2-crossed module, $L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,$ by setting $L = 1$, the trivial group, and, of course, $M = G_2$, $N = G_1$. Conversely any 2-crossed module having trivial top dimensional group ($L=1$) ‘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 $Im\, \partial_2$ is a normal subgroup of $M$, 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 $L = C_3$, $M = C_2$ and $N = C_1$, with trivial Peiffer lifting, gives one a 2-crossed complex. Conversely suppose we have a 2-crossed module with trivial Peiffer lifting: $\{m_1,m_2\} = 1$ for all $m_1$, $m_2 \in M$, axiom 3 then shows that $L$ is an Abelian group, and similarly the other axioms can be analysed to show that the result is a truncated crossed complex.
This gives:
The category $Crs_{2]}$ of crossed complexes of length 2 is equivalent to the full subcategory of $2-CMod$ 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 $L$ generated by the Peiffer lifting, …. is that all?)
If $G$ 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).
Let
be a crossed square then $N$ acts on $M$ via $P$, so ${}^n m := {}^{\nu(n)}m$, and so we can form $M\rtimes N$ and the sequence
is then a 2-crossed complex.
(And, yes, these are actually group homomorphisms: $(\mu,\nu)(m,n) = \mu(m)\nu(n)$, the product of the two elements! Try it!)
The full result and an explanation of what is going on here is given in
3-group, 2-crossed module / crossed square, differential 2-crossed module
∞-group, simplicial group, crossed complex, hypercrossed complex
H. J. Baues: Combinatorial homotopy and $4$-dimensional complexes. With a preface by Ronald Brown. de Gruyter Expositions in Mathematics, 2. Walter de Gruyter \& Co., Berlin, 1991.
Julia G. Cabello?, Antonio R. Garzón: Quillen’s theory for algebraic models of $n$-types. Extracta Math. 9 (1994), no. 1, 42–47.
Daniel Conduché: Modules croisés généralisés de longueur $2$. Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 155–178.
Joao Faria Martins Homotopy of 2-crossed complexes and the homotopy category of pointed 3-types (wed pdf)
Graham Ellis: Crossed squares and combinatorial homotopy. Math. Z. 214 (1993), no. 1, 93–110.