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 ...

Homotopy 3-types can be modelled algebraically by Tamsamani’s weak 3-groupoids as well as, in the path connected case, by cat 2-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using Tamsamani’s model. Both are then related to Gray 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 introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geometric) simplicial complexes as given by Brown, [2], or Spanier, [13], reveals the join operation to be a basic geometric construction. It is used in the development of several areas of geometric topology (cf. Hudson, [8]) whilst also being applied to the basic properties of polyhedra relating to homology. The theories of geometric and abstract simplicial complexes run in a largely parallel way and when describing the theory, expositions often choose which aspect -- abstract combinatorial or geometric -- to emphasise at each step. Historically in algebraic topology geometric simplicial complexes, as tools, were largely replaced by CW complexes whilst the combinatorial abstract complex became pa...

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.

In this paper we give an elementary derivation of a 2-groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2-groupoids and cat 1 --groupoids.

We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial

Crossed squares and 2-crossed modules