nLab
crossed n-cube

Idea

  • A crossed module is the ‘algebraic core’ of a cat 1\cat^1-group, in as much as within a cat 1\cat^1-group we can find a crossed module in a simple way from which the whole of the cat 1\cat^1-group in all its glory can be reconstructed.

  • A crossed square is similarly the ‘algebraic core’ of a cat 2\cat^2group.

  • A crossed nn-cube should be the ‘algebraic core’ of a cat n\cat^n-group.

A crossed square in the Guin-Valery Loday specification has quite a few axioms, many more than those for a crossed module. Does that mean that crossed nn-cubes will be defined with a very large number of axioms (perhaps dependent on nn)? No (unless you think that 11 is large!)

Finally a particular example of a crossed square is given by a group with two normal subgroups, and their intersection together with the commutator map. It would be appropriate if a group together with nn normal subgroups, all the intersections and all the commutator maps, formed an example of a crossed nn-cube. It does. The axioms below govern the hh-maps, by using analogues of the relationships between commutator maps in such a nn-cube of inclusions of intersections.

The result (definition due to Graham Ellis and Richard Steiner, reference below) is a set of 11 axioms. That is all, and these objects model all homotopy (n+1)(n+1)-types.

Definition

We denote by n\langle n \rangle the set {1,2,,n}\{1,2,\ldots, n\}.

A crossed nn-cube, MM, is a family of groups, {M A:An}\{M_A: A \subseteq \langle n \rangle\}, together with homomorphisms, μ i:M AM A{i}\mu_i :M_A \rightarrow M_{A-\{i\}}, for in,Ani \in \langle n \rangle , A \subseteq \langle n \rangle, and functions, h:M A×M BM ABh:M_{A} \times M_{B} \rightarrow M_{A\cup B}, for A,BnA, B \subseteq \langle n \rangle, such that if ab^{a}b denotes h(a,b)bh(a,b)b for aM Aa \in M_{A} and bM Bb \in M_{B} with ABA \subseteq B, then for a,a M A,b,b M B,cM Ca, a^\prime \in M_{A},\, b, b^\prime \in M_{B},\, c \in M_{C} and i,jni, j \in \langle n \rangle, the following axioms hold: 1. μ ia=a\mu_i a = a if aAa \notin A 1. μ iμ ja=μ jμ ia\mu_i\mu_j a = \mu_j\mu_i a 1. μ ih(a,b)=h(μ ia,μ ib)\mu_i h(a,b) = h(\mu_i a,\mu_i b) 1. h(a,b)=h(μ ia,b)=h(a,μ ib)h(a,b) = h(\mu_i a,b) = h(a,\mu_i b) if iABi \in A \cap B 1. h(a,a )=[a,a ]h(a,a^\prime ) = [a,a^\prime ] 1. h(a,b)=h(b,a) 1h(a,b) = h(b,a)^{-1} 1. h(a,b)=1h(a,b) = 1 if a=1a = 1 or b=1b = 1 1. h(aa ,b)= ah(a ,b)h(a,b)h(aa^\prime ,b) = {}^{a}h(a^\prime ,b)h(a,b) 1. h(a,bb )=h(a,b) bh(a,b )h(a,bb^\prime ) = h(a,b){ }^b h(a,b^\prime ) 1. ah(h(a 1,b),c) ch(h(c 1,a),b) bh(h(b 1,c),a)=1{ }^{a}h(h(a^{-1},b),c)^{c}h(h(c^{-1},a),b)^{b}h(h(b^{-1},c),a) = 1 1. ah(b,c)=h( ab, ac){ }^{a}h(b,c) = h(^{a}b,^{a} c) if ABCA \subseteq B \cap C.

A morphism of crossed nn-cubes

f:{M A}{M A }f :\{M_{A}\} \rightarrow \{M^{\prime}_{A}\}

is a family of homomorphisms, {f A:M AM A |An}\{f_{A}: M_{A} \rightarrow M^{\prime}_A \,|\, A \subseteq \langle n \rangle\}, which commute with the maps, μ i\mu_i, and the functions, hh.

This gives us a category, Crs nCrs^{n} , equivalent to that of cat n\cat^n-groups.

Homotopical example

The fundamental crossed nn-cube of groups functor Π\Pi ' is defined from nn-cubes of pointed spaces to crossed nn-cubes of groups: ΠX *\Pi 'X_{*} is simply the crossed nn-cube of groups equivalent to the catn^n-group ΠX *\Pi X_{*}. It is easier to identify Π\Pi ' in classical terms in the case X *X_{*} is the nn-cube of spaces arising from a pointed (n+1)(n + 1)-ad 𝒳=(X;X 1,,X n)\mathcal{X} = (X;X_1,\ldots ,X_n). That is, let X n=XX_{ \langle n \rangle } = X and for AA properly contained in n\langle n \rangle let X A= i¬AX iX_A = \bigcap _{i \not\in A} X_i. Then M=Π𝒳M = \Pi '\mathcal{X} is given as follows (Ellis and Steiner, 1987): M =π 1(X )M_{\emptyset} = \pi_1(X_\emptyset ); if A=i 1,,i rA = {i_1,\ldots ,i_r}, in the right order, then MM is the homotopy (r+1)(r + 1)-ad group π r+1(X A;X AX i 1,,X AX i r)\pi _{r+1}(X _A;X_A \cap X_{i_1} ,\ldots ,X_A \cap X_{i_r} ); the maps μ\mu are given by the usual boundary maps; the hh-functions are given by generalised Whitehead products. Note that whereas these separate elements of structure had all been considered previously, the aim of this theory is to consider the whole structure, despite its apparent complications. The equivalence of categories is a convincing reason for supposing that the axioms for a crossed nn-cube of groups are a complete axiomatisation of this homotopical structure, as was not previously known.

Algebraic example

Let us put a bit of flesh on the example given in the introduction. Suppose that GG is a group and N 1,N 2,..,N nN_1, N_2, .., N_n are nn-normal subgroups of GG (there may be repeats).

Now define for AnA \subseteq \langle n \rangle, M A={N iiA}M_A = \bigcap\{N_i\mid i \in A\}, and for AM AA\in M_A, bM Bb\in M_B, h(a,b)=[a,b]h(a,b) = [a,b], the commutator of aa and bb in GG. The result is a crossed nn-cube (sometimes called the inclusion crossed nn-cube determined by the normal nn-ad of subgroups).

Simplicial example

If instead of a space you start with a simplicial group GG, as model for a connected homotopy type, then there is a crossed nn-cube generalising the crossed square given in terms of the Moore complex.

References

  • G. J. Ellis and R. Steiner, Higher dimensional crossed modules and the homotopy groups of (n + 1)-ads, J. Pure Appl. Alg., 46, (1987), 117–136.
Revised on October 20, 2013 06:00:20 by Tim Porter (95.147.236.247)