Item 2 (0140) — The Clowder Project

11.6.7 Equivalences of Categories

Definition 11.6.7.1.1 ▶ Equivalences of Categories

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.

1.
An equivalence of categories between $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors

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

together with natural isomorphisms

\begin{align*} \eta & \colon \operatorname {\mathrm{id}}_{\mathcal{C}} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}G\circ F,\\ \epsilon & \colon F\circ G \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\operatorname {\mathrm{id}}_{\mathcal{D}}. \end{align*}
2.
An adjoint equivalence of categories between $\mathcal{C}$ and $\mathcal{D}$ is an equivalence $\webleft (F,G,\eta ,\epsilon \webright )$ between $\mathcal{C}$ and $\mathcal{D}$ which is also an adjunction.

Proposition 11.6.7.1.2 ▶ Properties of Equivalences of Categories

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

1.
Characterisations. If $\mathcal{C}$ and $\mathcal{D}$ are small¹, then the following conditions are equivalent:²
1. (a)
  The functor $F$ is an equivalence of categories.
2. (b)
  The functor $F$ is fully faithful and essentially surjective.
3. (c)
  The induced functor
  
  \[ \left.F\right\vert _{\mathsf{Sk}\webleft (\mathcal{C}\webright )}\colon \mathsf{Sk}\webleft (\mathcal{C}\webright )\to \mathsf{Sk}\webleft (\mathcal{D}\webright ) \]
  
  is an isomorphism of categories.
4. (d)
  For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
  
  \[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  is an equivalence of categories.
5. (e)
  For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
  
  \[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]
  
  is an equivalence of categories.

2.
Two-Out-of-Three. Let
be a diagram in $\mathsf{Cats}$. If two out of the three functors among $F$, $G$, and $G\circ F$ are equivalences of categories, then so is the third.

Stability Under Composition. Let

be a diagram in $\mathsf{Cats}$. If $\webleft (F,G\webright )$ and $\webleft (F',G'\webright )$ are equivalences of categories, then so is their composite $\webleft (F'\circ F,G'\circ G\webright )$.

Equivalences vs.Adjoint Equivalences. Every equivalence of categories can be promoted to an adjoint equivalence.³

Interaction With Groupoids. If $\mathcal{C}$ and $\mathcal{D}$ are groupoids, then the following conditions are equivalent:

(a)
The functor $F$ is an equivalence of groupoids.
(b)
The following conditions are satisfied:
1. (i)
  The functor $F$ induces a bijection
  
  \[ \pi _{0}\webleft (F\webright )\colon \pi _{0}\webleft (\mathcal{C}\webright )\to \pi _{0}\webleft (\mathcal{D}\webright ) \]
  
  of sets.
2. (ii)
  For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the induced map
  
  \[ F_{x,x}\colon \mathrm{Aut}_{\mathcal{C}}\webleft (A\webright )\to \mathrm{Aut}_{\mathcal{D}}\webleft (F_{A}\webright ) \]
  
  is an isomorphism of groups.

¹Otherwise there will be size issues. One can also work with large categories and universes, or require $F$ to be constructively essentially surjective; see [kilian, Equivalence of categories and axiom of choice].
²In ZFC, the equivalence between Item 1a and Item 1b is equivalent to the axiom of choice; see [user30818, Category and the axiom of choice]. In Univalent Foundations, this is true without requiring neither the axiom of choice nor the law of excluded middle.
³More precisely, we can promote an equivalence of categories $\webleft (F,G,\eta ,\epsilon \webright )$ to adjoint equivalences $\webleft (F,G,\eta ',\epsilon \webright )$ and $\webleft (F,G,\eta ,\epsilon '\webright )$.

Proof of Proposition 11.6.7.1.2.

Item 1: Characterisations

We claim that Item 1a, Item 1b, Item 1c, Item 1d, and Item 1e are indeed equivalent:

1.
Item 1a$\implies $Item 1b: Clear.
2.
Item 1b$\implies $Item 1a: Since $F$ is essentially surjective and $\mathcal{C}$ and $\mathcal{D}$ are small, we can choose, using the axiom of choice, for each $B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{D}\webright )$, an object $j_{B}$ of $\mathcal{C}$ and an isomorphism $i_{B}\colon B\to F_{j_{B}}$ of $\mathcal{D}$.

Since $F$ is fully faithful, we can extend the assignment $B\mapsto j_{B}$ to a unique functor $j\colon \mathcal{D}\to \mathcal{C}$ such that the isomorphisms $i_{B}\colon B\to F_{j_{B}}$ assemble into a natural isomorphism $\eta \colon \operatorname {\mathrm{id}}_{\mathcal{D}}\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}F\circ j$, with a similar natural isomorphism $\epsilon \colon \operatorname {\mathrm{id}}_{\mathcal{C}}\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}j\circ F$. Hence $F$ is an equivalence.
3.
Item 1a$\implies $Item 1c: This follows from Item 4 of Proposition 11.1.3.1.3.
4.
Item 1c$\implies $Item 1a: Omitted.
5.
Item 1a, Item 1d, and Item 1e Are Equivalent: This follows from .