B. Davvaz, M. Alp

Loday's 1-cat group definition plays very powerful role in making some new applications to crossed module due to Whitehead. There are many applications of cat$^1$-groups such as cat$^1$-polygroups and pullback cat$^1$-polygroups. The importance of hypergroups come from the properties of hypergroups such that hypergroups in the sense of Marty do not have identity element, inverse element and they are generalization 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.