Equivalence of categories
Last fall I held a talk about functors, natural transformations and equivalences of categories. This talk was part two of five in a student seminar on introductory category theory. There was mostly second year students attending but also a couple more experienced students. To make the talk a bit interesting for them as well I said that an equivalence of categories is the correct notion of “sameness” of categories, and not isomorphisms due to the fact that categories naturally lie in a $2$-category. An isomorphism of categories would be the correct notion of sameness if the category of categories had only trivial $2$-categorical structure, and we didn’t have to worry about higher morphisms. In this post I want to look at this statement and show that it is true. ...