Problem solving #2

Textbook

Basic Category Theory by Tom Leinster

Exercise 2.1.12

This problem asks what we can say about adjunctions between discrete categories. The answer is that if we have an adjunction between discrete categories, then the categories are isomorphic and the adjoint functors are isomorphisms.

Let \(A\) and \(B\) be discrete categories with \(F : A \to B\) and \(G: B \to A\) such that \(F \dashv G\).

Pick any \(a \in \textsf{obj}(A)\) and let \(F(a) = b \in \textsf{obj}(B)\). Now consider \(\textsf{Hom}(F(a), b) = \{\textsf{id}_b\}\). Since \(F \dashv G\), we must have \(\textsf{Hom}(F(a), b) \cong \textsf{Hom}(a, G(b))\). Since \(A\) is discrete, we must have \(G(b) = a\) (otherwise the hom-set is empty). The same holds in the other direction. This implies that \(F(a) = b\) iff \(G(b) = a\), so the two functors are inverses and the categories are isomorphic. \(\blacksquare\)