Group Objects and Internal Categories

TL;DR

This paper introduces the concept of group objects and internal categories within algebraic structures, highlighting their equivalences and foundational aspects in category theory.

Contribution

It provides an elementary introduction to internal categories and group objects, clarifying their relationships and applications in algebra and category theory.

Findings

Group objects in Grp are abelian groups

Internal categories in Grp are equivalent to group objects in Cat

Internal categories correspond to crossed modules of groups

Abstract

Algebraic structures such as monoids, groups, and categories can be formulated within a category using commutative diagrams. In many common categories these reduce to familiar cases. In particular, group objects in Grp are abelian groups, while internal categories in Grp are equivalent both to group objects in Cat and to crossed modules of groups. In this exposition we give an elementary introduction to some of the key concepts in this area.