nLab 2-crossed module

Idea

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.

Definition

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

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

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

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

such that

1. the action of $N$ on itself is by conjugation, and $\partial_2$ and $\partial_1$ are $N$-equivariant;

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

   $$\partial_2\{m_0,m_1\} = \,^{\partial_1 m_0}m_1 . m_0m_1^{-1}m_0^{-1};$$

3. if $\ell_0,\ell_0 \in L$, then

   $$\{\partial_2\ell_0,\partial_2\ell\} = [\ell_1,\ell_0];$$

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

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

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

   • $\{m_0,m_1m_2\} = \{m_0,m_1\}\{ \partial \{m_0,m_2\},(m_0m_1m_0^{-1})\}\{m_0,m_2\}$;
   • $\{m_0m_1,m_2\} = \,^{\partial m_0}\{m_1,m_2\}\{m_0,m_1m_2m_1^{-1}\}$;

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

   $$\,^{n} \{m_0,m_1\} = \{ \,^{n}m_0, \,^{n}m_1\}.$$

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’.

• 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.

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

$$L\stackrel{\partial_2}{\longrightarrow} M \stackrel{\partial_1}{\longrightarrow}N$$

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

$$\partial_1 : \frac{M}{Im\, \partial_2} \to N.$$

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

$$\ldots \to 1 \to 1 \to C_3\stackrel{\partial_3}{\longrightarrow} C_2 \stackrel{\partial_2}{\longrightarrow}C_1,$$

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 $Crs_{2]}$ 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?)

From simplicial groups to 2-crossed modules

If $G$ is a simplicial group then

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

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

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

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

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

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

Related concepts

• group

• 2-group, crossed module, differential crossed module

• 3-group, 2-crossed module / crossed square, differential 2-crossed module

• n-group

• ∞-group, simplicial group, crossed complex, hypercrossed complex

References:

• 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. (EuDML)

• P. Carrasco and T. Porter, Coproduct of 2-crossed modules. Applications to a definition of a tensor product for 2-crossed complexes, Collectanea Mathematica, DOI:10.1007/s13348-015-0156-9.

• Daniel Conduché, Modules croisés généralisés de longueur $2$, in:

  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, The fundamental 2-crossed complex of a reduced CW-complex, Homology Homotopy Appl. 13(2): 129-157 (2011) (web pdf.)

• Graham Ellis, Crossed squares and combinatorial homotopy, Math. Z. 214 (1993), no. 1, 93–110.

See also:

• Masaki Fukuda, Tommy Shu: An Equivalence of Categories between 3-Crossed Modules and Gray 4-Groups [arXiv:2609.24034]