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

Abstract. The aim of this paper is to introduce the notion of cat1−groupoids
which are the groupoid version of cat1−groups and to prove the categorical
equivalence between crossed modules over groupoids and cat1−groupoids. In
section 4 we introduce the notions of crossed squares over groupoids and of
cat2−groupoids, and then we show their categories are equivalent. These
equivalences enable us to obtain more examples of groupoids.

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

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