nLab
2-crossed module

Idea

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.

Definition

A $2$-crossed module is a normal complex of groups

$$L\stackrel{{\partial }_2}{\to }M\stackrel{{\partial }_1}{\to }N,$$L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,

together with an action of $N$ on all three groups and a mapping

$$\{-,-\}:M\times M\to L$$\{ - ,- \} : M\times M \to L

such that

1. the action of $N$ on itself is by conjugation, and ${\partial }_2$ and ${\partial }_1$ are $N$-equivariant;

2. for all $m_0,m_1\in M$,

   $${\partial }_2\{m_0,m_1\}={}^{{\partial }_1{m}_0}{m}_1.m_0{m}_1^{-1}{m}_0^{-1};$$\partial_2\{m_0,m_1\} = \,^{\partial_1 m_0}m_1 . m_0m_1^{-1}m_0^{-1};

3. if ${\ell }_0,{\ell }_0\in L$, then

   $$\{{\partial }_2{\ell }_0,{\partial }_2\ell \}=[{\ell }_1,{\ell }_0];$$\{\partial_2\ell_0,\partial_2\ell\} = [\ell_1,\ell_0];

4. if $\ell \in L$ and $m\in M$, then

   $$\{m,\partial \ell \}\{\partial \ell ,m\}={}^{\partial m}\ell .{\ell }^{-1};$$\{m,\partial \ell\}\{\partial \ell,m\} = \,^{\partial m}\ell.\ell^{-1};

5. for all $m_0,m_1,m_2\in M$,

   • $\{m_0,m_1{m}_2\}=\{m_0,m_1\}\{\partial \{m_0,m_2\},(m_0{m}_1{m}_0^{-1})\}\{m_0,m_2\}$;
   • $\{m_0{m}_1,m_2\}={}^{\partial m_0}\{m_1,m_2\}\{m_0,m_1{m}_2{m}_1^{-1}\}$;

6. if $n\in N$ and $m_0,m_1\in M$, then

   $${}^n\{m_0,m_1\}=\{{}^n{m}_0,{}^n{m}_1\}.$$\,^{n} \{m_0,m_1\} = \{ \,^{n}m_0, \,^{n}m_1\}.

The pairing $\{-,-\}:M\times M\to L$ is often called the Peiffer lifting of the $2$-crossed module.

Remarks

• 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’. The homotopy theory of these is little developed as yet, but, in part, is related to the theory of quadratic module?s and quadratic complex?es as introduced by Baues.

Examples

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

$$L\stackrel{{\partial }_2}{⟶}M\stackrel{{\partial }_1}{⟶}N$$L\stackrel{\partial_2}{\longrightarrow} M \stackrel{\partial_1}{\longrightarrow}N

is a 2-crossed module, then $\mathrm{Im}{\partial }_2$ is a normal subgroup of $M$, and then there is an obvious induced crossed module structure on

$${\partial }_1:\frac{M}{\mathrm{Im}{\partial }_2}\to N.$$\partial_1 : \frac{M}{Im\, \partial_2} \to N.

But we can do better than this. More generally, let

$$\dots \to 1\to 1\to C_3\stackrel{{\partial }_3}{⟶}{C}_2\stackrel{{\partial }_2}{⟶}{C}_1,$$\ldots \to 1 \to 1 \to C_3\stackrel{\partial_3}{\longrightarrow} C_2 \stackrel{\partial_2}{\longrightarrow}C_1,

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:

Proposition

The category ${\mathrm{Crs}}_{2]}$ of crossed complexes of length 2 is equivalent to the full subcategory of $2-\mathrm{CMod}$ 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?)

From simplicial groups to 2-crossed modules

If $G$ is a simplicial group then

$$\frac{𝒩G_2}{d_0(𝒩G_3)}\to 𝒩G_1\to 𝒩G_0,$$\frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,

is a 2-crossed module. (You are invited to find the Peiffer lifting!)

From crossed squares to 2-crossed modules

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

$$\begin{array}{ccccc}& L& {\to }^{\lambda }& M& \\ {\lambda }^{\prime }& \downarrow & & \downarrow & \mu \\ & N& {\to }_{\nu }& P& \\ \end{array}$$\array{& L & {\to}^\lambda & M & \\ \lambda^\prime & \downarrow &&\downarrow & \mu\\ &N & {\to}_{\nu}& P & \\ }

be a crossed square then $N$ acts on $M$ via $P$, so ${}^{n}m:={}^{\nu (n)}m$, and so we can form $M⋊N$ and the sequence

$$L\stackrel{((\lambda \prime {)}^{-1},\lambda )}{⟶}M⋊N\stackrel{\mu \nu }{⟶}P$$L\stackrel{((\lambda')^{-1},\lambda)}{\longrightarrow}M\rtimes N\stackrel{\mu\nu}{\longrightarrow}P

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

• D. Conduché, Simplicial Crossed Modules and Mapping Cones, Georgian Math. J., 10, (2003), 623–636