Groupoids and crossed objects in algebraic topology

Crossed squares, crossed modules over groupoids and cat$^{bf {1-2}}-$groupoids

The aim of this paper is to introduce the notion of cat$^{bf {1}}-$groupoids which are the groupoid version of cat$^{bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}-$groupoids, and then we show their categories are equivalent....

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.

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

Algebraic Topology Seminar

Grothendieck Toposes are often considered as generalized spaces; indeed, every space gives rise to a topos of sheaves, and various invariants and constructions from (algebraic) topology can be generalized to the level of toposes. In this talk, I will introduce a newly discovered invariant called the isotropy group of a topos and illustrate by considering special cases such as continuous group a...