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

Abstract

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. These equivalences enable us to obtain more examples of groupoids.

Keywords

• crossed module
• crossed square
• groupoid
• cat$^{bf {1}}-$group
• cat$^{bf {2}}-$group