Algebraic Models of 3‐Types and Automorphism Structures for Crossed Modules

Abstract

ii) g* x) =x~xgx, for all g, x e G and / e Aut G. We see that Aut G is naturally considered as part of a crossed module: that is, a group homomorphism d: M-+P together with an action of P on M satisfying CM1: d(m p )=p-1 d(m)p, CM2: mo {m) = m~lm o m, for all m 0 , meM andp eP.Crossed modules were introduced by J. H. C. Whitehead [16] and among the standard examples are the inclusion M<-+P of a normal subgroup M of P, the zero homomorphism M->P when M is a P-module, and any surjection Af-»P with central kernel. There is also an important topological example: if F-»£-» # is a fibration sequence of pointed spaces, then the induced homomorphism jiiF-> it X E of fundamental groups is naturally a crossed module. Now groups are algebraic models of 1-types: that is, there is a classifying space functor B: (groups)-• (OV-complexes) such that for any group G the space BG satisfies JZ X BG = G and JtjBG = 0 for j > 1, and further any pointed, connected CW-complex X with jtjX = 0 for j > 1 is of the homotopy type of Bn x X.Crossed modules are algebraic models of 2-types. There is a classifying space functor B: (crossed modules)-» (ClV-complexes)

Actions and automorphisms of crossed modules

“…This 2-crossed module is one of the automorphism structures of (T^G,9) derived by BROWN and GILBERT in [2] from a monoidal closed structure on the category of crossed modules (over groupoids).…”

Braided regular crossed modules bifibered over regular groupoids

“…We recall some basic definitions from [4]. A groupoid C is a small category in which every morphism is an isomorphism.…”

Homotopy Theory of Higher Categories

“…This observation, which I learned from G. Maltsiniotis and A. Bruguières, had been used by many people to argue that strict n-categories do not contain sufficient information to model homotopy n-types, as soon as n ≥ 3. See for example Brown [56], with Gilbert [57] and with Higgins [58] [59]; Grothendieck's discussion of this in various places in [108], and the paper of Berger [29].…”