Groupoids and crossed objects in algebraic topology

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.

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

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....

Nonabelian Algebraic Topology : filtered spaces , crossed complexes ,

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...