The internal structure of categories

Chapter 2
The internal structure of categories

In the previous chapter, our discussion of categories focused on their external theory: relating them via functors and equivalences, and constructing them via products or functor categories. Something we have mostly ignored so far is the internal structure that categories are expected to have: beyond objects, they should also have morphisms, and there should be a unital and associative composition law.

Rather than directly axiomatizing such internal structure, we will instead gain access to it by ‘probing’ a category under consideration with various test categories. We begin with morphisms and commutative diagrams in Section 2.1. The Segal axiom in Section 2.2 then provides composition of morphisms, and the Rezk axiom in Section 2.3 relates isomorphisms to identifications of objects.

Finally, we relate the primitive animae of our theory to the structural notion of a groupoid, a category in which every morphism is an isomorphism. The groupoid recognition axiom identifies these two points of entry. This also explains why the collection of objects \(C^\simeq \) introduced in the previous chapter is called the groupoid core of \(C\): it remembers the objects of \(C\) and their isomorphisms, while forgetting the noninvertible morphisms.