Subsection 9.6.8 (0148): Isomorphisms of Categories

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Table of Contents
Part 3: Category Theory
Chapter 9: Categories
Section 9.6: Conditions on Functors
Subsection 9.6.8: Isomorphisms of Categories (cite)

9.6.8 Isomorphisms of Categories

Definition 9.6.8.1.1 ▶ Isomorphisms of Categories

An isomorphism of categories is a pair of functors

\begin{align*} F & \colon \mathcal{C}\to \mathcal{D},\\ G & \colon \mathcal{D}\to \mathcal{C} \end{align*}

such that we have

\begin{align*} G\circ F & = \text{id}_{\mathcal{C}},\\ F\circ G & = \text{id}_{\mathcal{D}}. \end{align*}

Example 9.6.8.1.2 ▶ Equivalent But Non-Isomorphic Categories

Categories can be equivalent but non-isomorphic. For example, the category consisting of two isomorphic objects is equivalent to $\mathsf{pt}$, but not isomorphic to it.

Proposition 9.6.8.1.3 ▶ Properties of Isomorphisms of Categories

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

Characterisations. If $\mathcal{C}$ and $\mathcal{D}$ are small, then the following conditions are equivalent:
1. The functor $F$ is an isomorphism of categories.
2. The functor $F$ is fully faithful and bijective on objects.
3. For each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
  \[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  is an isomorphism of categories.
4. For each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
  \[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]
  
  is an isomorphism of categories.