Item 4 (018X) — 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 $\mathsf{Fun}(\mathcal{C},\mathcal{D})$² where

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

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

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

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

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

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 $F,G,H\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(\mathcal{C},\mathcal{D}))$, the composition map

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

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

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

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 $\mathsf{Fun}(\mathcal{C},\mathcal{D})$.
²Further Notation: Also written $\mathcal{D}^{\mathcal{C}}$ and $[\mathcal{C},\mathcal{D}]$.

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 $\mathcal{C},\mathcal{D},(\mathcal{C},\mathcal{D})\mapsto \mathsf{Fun}(\mathcal{C},\mathcal{D})$ define functors

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

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

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

natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$.

4.
2-Adjointness. We have 2-adjunctions
witnessed by isomorphisms of categories

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

natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats}_{\mathsf{2}})$.

Interaction With Punctual Categories. We have a canonical isomorphism of categories

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

natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$.

Objectwise Computation of Co/Limits. Let

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

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

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

naturally in $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$.

Interaction With Co/Completeness. If $\mathcal{E}$ is co/complete, then so is $\mathsf{Fun}(\mathcal{C},\mathcal{E})$.

Monomorphisms and Epimorphisms. Let $\alpha \colon F\Longrightarrow G$ be a morphism of $\mathsf{Fun}(\mathcal{C},\mathcal{D})$. The following conditions are equivalent:

(a)
The natural transformation

\[ \alpha \colon F \Longrightarrow G \]

is a monomorphism (resp. epimorphism) in $\mathsf{Fun}(\mathcal{C},\mathcal{D})$.
(b)
For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the morphism

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

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