The homotopy categorical crossed module of a CW-complex

On the Homotopy Type and the Fundamental Crossed Complex of the Skeletal Filtration of a CW-Complex

We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks D, when the later are given their natural CW-decomposition with unique cells of order 0, (n − 1) and n; a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the Colimit Theorem for the Fundamental Crosse...

On the Second Homotopy Module of Two-dimensional Cw Complexes

Let A' be a connected 2-dimensional CW complex. This note reproves from a very simple point of view two classical theorems of H. Hopf relating the homology of the fundamental group it = irx(X) of X and the Hurewicz map on ir2(X). This point of view also allows the dual theorems to be proved. If w is a finite group, a new interpretation is given for tf,(7r; Z) (/' = 2, 3) in terms of v2(X).

On Categorical Crossed Modules

The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group. We also prove a suitable 2-dimensional version of the kernelcokernel lemma for a diagram of categorical crossed modules. We th...

Categorical homotopy theory

Categorical Homotopy Theory

This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves wi...