theory, and change of base for groupoids and multiple

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The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into an exact sequence.

In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2-groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kan-fibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...

Dedicated to the memory of Frank Adams

In this note we generalise Hopfs formula for the second homology of a group G in terms of a free presentation R>- • F^ » G. We prove: THEOREM 1. Let Rlt...,Rn be normal subgroups of a group F such that F/Y\iiiinRi = G, and for each proper subset A of <«> = {1,...,«} the groups Hr(F/Y[ieA R,) are trivial for r = 2ifA = 0, and for r = \A \ + 1 and \A \ + 2 if A # 0 (for example, the groups F/\\ieARi are free for A-£(n)). Then there is an isomorphism-l)l /»£<«> ieA ilA s tne Here \A \ denotes the order of A, and ^6^1 * subgroup of F generated by trivial subgroup). Also is understood to mean F. Thus for n = 2 the formula reads the subgroups Ri with ieA (in particular, I~[ie0^i * S ^ e f]ie0Rt H3(G) « {R, n R2 n [F, F]}/{[F, R, 0^) [Rlt R2]}. Note that for any group G and n ^ 1, such an F and Rt can be found: let F l (G) be the free group on G; define inductively Ft = F^-^G)), and set F = F n (G); for 1 ^ / ^ n let e(: F n (G)-> F n ~\G) denote the canonical homorphisms induced by applying F"' * to the standard 'augmentation ' map F^F 1 " 1 ^))->F l ~\G) (where /ro((7) = Gyf and set ^ ( = Kerfii An alternative method, analogous to methods in [4,5], is best illustrated for n = 2. Choose any surjections F{^G with F { free, / = 1,2. Let P be the pullback of these surjections and choose a surjection F- » P with F free. Let Rf be the kernel of the composite F-> P-> Ft. In general, one constructs inductively an «-cube of groups F such that, for A c <«>: (i) /r js free if ^ ^ <«>, (ii) /r js G for ^ = <«>, and (iii) the morphism FA-»limB=)/4FB is surjective. Such an /i-cube might be called afibrant n-presentation of G. Again, suppose G = F/HKwhere //and #are normal subgroups of Fsuch that F, F/H and F/KSLTQ free. For example, we might be given a presentation <A ^ £/, V) of

This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.

Tamsamani’s weak n-groupoids are known to model n-types. In this paper we show that every Tamsamani weak n-groupoid representing a connected n-type is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak n-groupoids and cat n−1-groups as models of connected n-types.

We explore the relations among quadratic modules, 2-crossed modules, crossed squares and simplicial groups with Moore complex of length 2.

Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)-types through cat n-groups. 1.