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2-crossed module

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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 11 (that is, at most stuff in dimensions 00 and 11) will be the internal nerve of a strict 22-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 00, 11, and 22; 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 22-nerve construction.

In other words, a 22-crossed module is the Moore complex of a 22-truncated simplicial group.

Definition

A 22-crossed module is a normal complex of groups

L 2M 1N,L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,

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

{,}:M×ML\{ - ,- \} : M\times M \to L

such that

  1. the action of NN on itself is by conjugation, and 2\partial_2 and 1\partial_1 are NN-equivariant;

  2. for all m 0,m 1Mm_0,m_1 \in M,

    2{m 0,m 1}= 1m 0m 1.m 0m 1 1m 0 1;\partial_2\{m_0,m_1\} = \,^{\partial_1 m_0}m_1 . m_0m_1^{-1}m_0^{-1};
  3. if 0, 0L\ell_0,\ell_0 \in L, then

    { 2 0, 2}=[ 1, 0];\{\partial_2\ell_0,\partial_2\ell\} = [\ell_1,\ell_0];
  4. if L\ell \in L and mMm\in M, then

    {m,}{,m}= m. 1;\{m,\partial \ell\}\{\partial \ell,m\} = \,^{\partial m}\ell.\ell^{-1};
  5. for all m 0,m 1,m 2Mm_0,m_1,m_2 \in M,

    • {m 0,m 1m 2}={m 0,m 1}{{m 0,m 2},(m 0m 1m 0 1)}{m 0,m 2}\{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}= m 0{m 1,m 2}{m 0,m 1m 2m 1 1}\{m_0m_1,m_2\} = \,^{\partial m_0}\{m_1,m_2\}\{m_0,m_1m_2m_1^{-1}\};
  6. if nNn\in N and m 0,m 1Mm_0,m_1 \in M, then

    n{m 0,m 1}={ nm 0, nm 1}. \,^{n} \{m_0,m_1\} = \{ \,^{n}m_0, \,^{n}m_1\}.

The pairing {,}:M×ML\{ - ,- \} : M\times M \to L is often called the Peiffer lifting of the 22-crossed module.

Remarks

  • In a 22-crossed module as above the structure 2:LM\partial_2: L \to M is a crossed module, but 1:MN\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 LL 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 π 0\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 2δG 1 G_2 \stackrel{\delta }{\to}{G_1} gives a 2-crossed module, L 2M 1N,L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N, by setting L=1L = 1, the trivial group, and, of course, M=G 2M = G_2, N=G 1N = G_1. Conversely any 2-crossed module having trivial top dimensional group (L=1L=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 2M 1NL\stackrel{\partial_2}{\longrightarrow} M \stackrel{\partial_1}{\longrightarrow}N

is a 2-crossed module, then Im 2Im\, \partial_2 is a normal subgroup of MM, and then there is an obvious induced crossed module structure on

1:MIm 2N.\partial_1 : \frac{M}{Im\, \partial_2} \to N.

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

11C 3 3C 2 2C 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 3L = C_3, M=C 2M = C_2 and N=C 1N = 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\{m_1,m_2\} = 1 for all m 1m_1, m 2Mm_2 \in M, axiom 3 then shows that LL 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]Crs_{2]} of crossed complexes of length 2 is equivalent to the full subcategory of 2CMod2-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 LL generated by the Peiffer lifting, …. is that all?)

From simplicial groups to 2-crossed modules

If GG is a simplicial group then

𝒩G 2d 0(𝒩G 3)𝒩G 1𝒩G 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

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

be a crossed square then NN acts on MM via PP, so nm:= ν(n)m{}^n m := {}^{\nu(n)}m, and so we can form MNM\rtimes N and the sequence

L((λ) 1,λ)MNμνPL\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: (μ,ν)(m,n)=μ(m)ν(n)(\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

References:

  • H. J. Baues: Combinatorial homotopy and 44-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 nn-types. Extracta Math. 9 (1994), no. 1, 42–47.

  • Daniel Conduché: Modules croisés généralisés de longueur 22. Proceedings of the Luminy conference on algebraic KK-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.

Revised on February 26, 2013 11:43:07 by Tim Porter (95.147.236.196)