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

Crossed complexes and higher 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 ω-groupoids 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 in non simply connected situations.

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.

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...

Nonabelian Algebraic Topology : filtered spaces , crossed complexes ,

Groupoids and crossed objects in algebraic topology

This is an introductory survey of the passage from groups to groupoids and their higher dimensional versions, with most emphasis on calculations with crossed modules and the construction and use of homotopy double groupoids.