  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.
