On Algebraic Models for Homotopy 3-types

We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2. Introduction Crossed modules defined by Whitehead, [23], are algebraic models of connected (weak homotopy) 2-types. Crossed squares as introduced by Loday and GuinWalery, [22], model connected 3-types. Crossed n-cubes model connected (n + 1)types, (cf. [21]). Conduché, [10], gave an alternative model for connected 3-types in terms of crossed modules of groups of length 2 which he calls ‘2-crossed module’. Conduché also constructed (in a letter to Brown in 1984) a 2-crossed module from a crossed square. Baues, [3], gave the notion of quadratic module which is a 2-crossed module with additional ‘nilpotency’ conditions. A quadratic module is thus a ‘nilpotent’ algebraic model of connected 3-types. Another algebraic model of connected 3-types is ‘braided regular crossed module’ introduced by Brown and Gilbert (cf. [5]). These notions are then related to simplicial groups. Conduché has shown that the category of simplicial groups with Moore complex of length 2 is equivalent to that of 2-crossed modules. Baues gives a construction of a quadratic module from a simplicial group in [3]. Berger, [4], gave a link between 2-crossed modules and double loop spaces. Some light on the 2-crossed module structure was also shed by Mutlu and Porter, [20], who suggested ways of generalising Conduché’s construction to higher n-types. Also Carrasco-Cegarra, [9], gives a generalisation of the Dold-Kan theorem to an equivalence between simplicial groups and a non-Abelian chain complex with a lot of extra structure, generalising 2-crossed modules. The present article aims to show some relations among algebraic models of connected 3-types. Thus the main points of this paper are: (i) to give a complete description of the passage from a crossed square to a 2crossed module by using the ‘Artin-Mazur’ codiagonal functor and prove directly a 2-crossed module structure; (ii) to give a functor from 2-crossed modules to quadratic modules based on Baues’s work (cf. [3]); Received October 26, 2005, revised February 3, 2005; published on February 19, 2006. 2000 Mathematics Subject Classification: 18D35 18G30 18G50 18G55.