CAT$^1$-HYPERGROUPS AND PULLBACK CAT$^1$-HYPERGROUPS

Abstract

Loday's 1-cat group definition plays very powerful role in makingsome new applications to crossed module due to Whitehead. Thereare many applications of cat$^1$-groups such ascat$^1$-polygroups and pullback cat$^1$-polygroups. The importance of hypergroups come from theproperties of hypergroups such that hypergroups in the sense ofMarty do not have identity element, inverse element and they aregeneralization of the well known groups. In this paper, we introduce the concept of cat$^1$-hypergroups, their examples and some related properties.Also, we investigate pullback cat$^1$-hypergroups and properties such as: every cat$^1$-group is a cat$^1$-hypergroup; construction of a cat$^1$-group from a crossed module of hypergroups and vice versa. Finally, we present the definition of pullback cat$^1$-hypergroups and some of their properties.