(2-dep,$Σ$)-categories are not generalised categories with families

Abstract

The notion of a generalised category with families, introduced by Coraglia and Emmenegger is one of the most general notions introduced to capture categorically the notion of dependent typing. It is shown that these generalised categories with families are biequivalent to comprehension categories. We will show that the notion of a (2-dep,$Σ$)-category, an extension of the notion of a (dep,$Σ$)-category, introduced by Petrakis, is not biequivalent to generalised categories with families, but instead is equivalent to a direct generalisation of that notion.