%0 Journal Article %T Crossed squares, crossed modules over groupoids and cat$^{\bf {1-2}}-$groupoids %J Categories and General Algebraic Structures with Applications %I Shahid Beheshti University %Z 2345-5853 %A Temel, Sedat %D 2020 %\ 07/01/2020 %V 13 %N 1 %P 125-142 %! Crossed squares, crossed modules over groupoids and cat$^{\bf {1-2}}-$groupoids %K crossed module %K crossed square %K groupoid %K cat$^{bf {1}}-$group %K cat$^{bf {2}}-$group %R 10.29252/cgasa.13.1.125 %X 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. %U https://cgasa.sbu.ac.ir/article_87511_b595acc78c30c8580458ad2fc0331d2f.pdf