Item 8b (0193) — The Clowder Project

11.10.1 Functor Categories

Let $\mathcal{C}$ be a category and $\mathcal{D}$ be a small category.

Definition 11.10.1.1.1 ▶ Functor Categories

The category of functors from $\mathcal{C}$ to $\mathcal{D}$¹ is the category $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$² where

•
Objects. The objects of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ are functors from $\mathcal{C}$ to $\mathcal{D}$.
•
Morphisms. For each $F,G\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright )$, we have

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}\webleft (F,G\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Nat}}\webleft (F,G\webright ). \]
•
Identities. For each $F\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright )$, the unit map

\[ \mathbb {1}^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F} \colon \mathrm{pt}\to \operatorname {\mathrm{Nat}}\webleft (F,F\webright ) \]

of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ at $F$ is given by

\[ \operatorname {\mathrm{id}}^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{F}, \]

where $\operatorname {\mathrm{id}}_{F}\colon F\Longrightarrow F$ is the identity natural transformation of $F$ of Example 11.9.3.1.1.
•
Composition. For each $F,G,H\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright )$, the composition map

\[ \circ ^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F,G,H} \colon \operatorname {\mathrm{Nat}}\webleft (G,H\webright ) \times \operatorname {\mathrm{Nat}}\webleft (F,G\webright ) \to \operatorname {\mathrm{Nat}}\webleft (F,H\webright ) \]

of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ at $\webleft (F,G,H\webright )$ is given by

\[ \beta \mathbin {{\circ }^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F,G,H}}\alpha \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta \circ \alpha , \]

where $\beta \circ \alpha $ is the vertical composition of $\alpha $ and $\beta $ of Item 1 of Proposition 11.9.4.1.2.

¹Further Terminology: Also called the functor category $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
²Further Notation: Also written $\mathcal{D}^{\mathcal{C}}$ and $\webleft [\mathcal{C},\mathcal{D}\webright ]$.

Proposition 11.10.1.1.2 ▶ Properties of Functor Categories

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

1.
Functoriality. The assignments $\mathcal{C},\mathcal{D},\webleft (\mathcal{C},\mathcal{D}\webright )\mapsto \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ define functors

\[ \begin{array}{ccc} \mathsf{Fun}\webleft (\mathcal{C},-\webright )\colon \mkern -15mu & \mathsf{Cats} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats},\\ \mathsf{Fun}\webleft (-,\mathcal{D}\webright )\colon \mkern -15mu & \mathsf{Cats}\mathrlap {{}^{\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats},\\ \mathsf{Fun}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Cats}^{\mathsf{op}}\times \mathsf{Cats} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}. \end{array} \]
2.
2-Functoriality. The assignments $\mathcal{C},\mathcal{D},\webleft (\mathcal{C},\mathcal{D}\webright )\mapsto \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ define 2-functors

\[ \begin{array}{ccc} \mathsf{Fun}\webleft (\mathcal{C},-\webright )\colon \mkern -15mu & \mathsf{Cats}_{\mathsf{2}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}_{\mathsf{2}},\\ \mathsf{Fun}\webleft (-,\mathcal{D}\webright )\colon \mkern -15mu & \mathsf{Cats}_{\mathsf{2}}^{\mathrlap {\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}_{\mathsf{2}},\\ \mathsf{Fun}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Cats}_{\mathsf{2}}^{\mathsf{op}}\times \mathsf{Cats}_{\mathsf{2}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}_{\mathsf{2}}. \end{array} \]
3.
Adjointness. We have adjunctions
witnessed by bijections of sets

\begin{align*} \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{D},\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathsf{Fun}\webleft (\mathcal{D},\mathcal{E}\webright )\webright ), \end{align*}

natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$.
4.
2-Adjointness. We have 2-adjunctions
witnessed by isomorphisms of categories

\begin{align*} \mathsf{Fun}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \mathsf{Fun}\webleft (\mathcal{D},\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )\webright ),\\ \mathsf{Fun}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \mathsf{Fun}\webleft (\mathcal{C},\mathsf{Fun}\webleft (\mathcal{D},\mathcal{E}\webright )\webright ), \end{align*}

natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}_{\mathsf{2}}\webright )$.
5.
Interaction With Punctual Categories. We have a canonical isomorphism of categories

\[ \mathsf{Fun}\webleft (\mathsf{pt},\mathcal{C}\webright ) \cong \mathcal{C}, \]

natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$.
6.
Objectwise Computation of Co/Limits. Let

\[ D \colon \mathcal{I} \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright ) \]

be a diagram in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$. We have isomorphisms

\begin{align*} \operatorname*{\operatorname {\mathrm{lim}}}\webleft (D\webright )_{A} & \cong \operatorname*{\operatorname {\mathrm{lim}}}_{i\in \mathcal{I}}\webleft (D_{i}\webleft (A\webright )\webright ),\\ \operatorname*{\operatorname {\mathrm{colim}}}\webleft (D\webright )_{A} & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{i\in \mathcal{I}}\webleft (D_{i}\webleft (A\webright )\webright ), \end{align*}

naturally in $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.
7.
Interaction With Co/Completeness. If $\mathcal{E}$ is co/complete, then so is $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )$.
8.
Monomorphisms and Epimorphisms. Let $\alpha \colon F\Longrightarrow G$ be a morphism of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$. The following conditions are equivalent:
1. (a)
  The natural transformation
  
  \[ \alpha \colon F \Longrightarrow G \]
  
  is a monomorphism (resp. epimorphism) in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.

(b)
For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the morphism

\[ \alpha _{A} \colon F_{A} \to G_{A} \]

is a monomorphism (resp. epimorphism) in $\mathcal{D}$.