Crossed complexes and higher homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

We outline the main features of the definitions and applications of crossed complexes and cubical ωgroupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types.

Nonabelian Algebraic Topology : filtered spaces , crossed complexes , cubical higher homotopy groupoids

O ct 2 00 4 A homotopy double groupoid of a Hausdorff space II : a van Kampen theorem

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems...

A Homotopy Double Groupoid of a Hausdorff Space Ii: a Van Kampen Theorem

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems...

Tensor Products and Homotopies for Ω-groupoids and Crossed Complexes∗

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product...