Subsection 11.5.1 (00ZV): Foundations

11.5.1 Foundations

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

Definition 11.5.1.1.1 ▶ Functors

A functor $F\colon \mathcal{C}\to \mathcal{D}$ from $\mathcal{C}$ to $\mathcal{D}$¹ consists of:

1.
Action on Objects. A map of sets

\[ F \colon \operatorname {\mathrm{Obj}}(\mathcal{C}) \to \operatorname {\mathrm{Obj}}(\mathcal{D}), \]

called the action on objects of $F$.
2.
Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, a map

\[ F_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B) \to \operatorname {\mathrm{Hom}}_{\mathcal{D}}(F(A),F(B)), \]

called the action on morphisms of $F$ at $(A,B)$².

satisfying the following conditions:

1.
Preservation of Identities. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the diagram
commutes, i.e. we have

\[ F(\operatorname {\mathrm{id}}_{A}) = \operatorname {\mathrm{id}}_{F(A)}. \]
2.
Preservation of Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the diagram
commutes, i.e. for each composable pair $(g,f)$ of morphisms of $\mathcal{C}$, we have

\[ F(g\circ f) = F(g)\circ F(f). \]

¹Further Terminology: Also called a covariant functor.
²Further Terminology: Also called action on $\operatorname {\mathrm{Hom}}$-sets of $F$ at $(A,B)$.

Notation 11.5.1.1.2 ▶ Subscript and Superscript Notation for Functors

Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and write $\mathcal{C}^{\mathsf{op}}$ for the opposite category of $\mathcal{C}$ of , .

1.
Given a functor

\[ F\colon \mathcal{C}\to \mathcal{D}, \]

we also write $F_{A}$ for $F(A)$.
2.
Given a functor

\[ F\colon \mathcal{C}^{\mathsf{op}}\to \mathcal{D}, \]

we also write $F^{A}$ for $F(A)$.
3.
Given a functor

\[ F\colon \mathcal{C}\times \mathcal{C}\to \mathcal{D}, \]

we also write $F_{A,B}$ for $F(A,B)$.
4.
Given a functor

\[ F\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathcal{D}, \]

we also write $F^{A}_{B}$ for $F(A,B)$.

We employ a similar notation for morphisms, writing e.g. $F_{f}$ for $F(f)$ given a functor $F\colon \mathcal{C}\to \mathcal{D}$.

Notation 11.5.1.1.3 ▶ Additional Notation for Functors

Following the notation $[\mspace {-3mu}[x\mapsto f(x)]\mspace {-3mu}]$ for a function $f\colon X\to Y$ introduced in Chapter 3: Sets, Notation 3.1.1.1.2, we will sometimes denote a functor $F\colon \mathcal{C}\to \mathcal{D}$ by

\[ F\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[A\mapsto F(A)]\mspace {-3mu}], \]

specially when the action on morphisms of $F$ is clear from its action on objects.

Example 11.5.1.1.4 ▶ Identity Functors

The identity functor of a category $\mathcal{C}$ is the functor $\operatorname {\mathrm{id}}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ where

1.
Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have

\[ \operatorname {\mathrm{id}}_{\mathcal{C}}(A) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A. \]
2.
Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms

\[ (\operatorname {\mathrm{id}}_{\mathcal{C}})_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B) \to \underbrace{\operatorname {\mathrm{Hom}}_{\mathcal{C}}(\operatorname {\mathrm{id}}_{\mathcal{C}}(A),\operatorname {\mathrm{id}}_{\mathcal{C}}(B))}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)} \]

of $\operatorname {\mathrm{id}}_{\mathcal{C}}$ at $(A,B)$ is defined by

\[ (\operatorname {\mathrm{id}}_{\mathcal{C}})_{A,B} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)}. \]

Proof of Example 11.5.1.1.4.

Preservation of Identities

We have $\operatorname {\mathrm{id}}_{\mathcal{C}}(\operatorname {\mathrm{id}}_{A})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{A}$ for each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$ by definition.

Preservation of Compositions

For each composable pair $A\xrightarrow {f}B\xrightarrow {g}B$ of morphisms of $\mathcal{C}$, we have

\begin{align*} \operatorname {\mathrm{id}}_{\mathcal{C}}(g\circ f) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\mathcal{C}}(g)\circ \operatorname {\mathrm{id}}_{\mathcal{C}}(f). \end{align*}

This finishes the proof.

Definition 11.5.1.1.5 ▶ Composition of Functors

The composition of two functors $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ is the functor $G\circ F$ where

•
Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have

\[ [G\circ F](A) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G(F(A)). \]
•
Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms

\[ (G\circ F)_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B) \to \operatorname {\mathrm{Hom}}_{\mathcal{E}}(G_{F_{A}},G_{F_{B}}) \]

of $G\circ F$ at $(A,B)$ is defined by

\[ [G\circ F](f) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G(F(f)). \]

Proof of Definition 11.5.1.1.5.

Preservation of Identities

For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have

\begin{align*} G_{F_{\operatorname {\mathrm{id}}_{A}}} & = G_{\operatorname {\mathrm{id}}_{F_{A}}}\tag {functoriality of $F$}\\ & = \operatorname {\mathrm{id}}_{G_{F_{A}}}.\tag {functoriality of $G$} \end{align*}

Preservation of Composition

For each composable pair $(g,f)$ of morphisms of $\mathcal{C}$, we have

\begin{align*} G_{F_{g\circ f}} & = G_{F_{g}\circ F_{f}}\tag {functoriality of $F$}\\ & = G_{F_{g}}\circ G_{F_{f}}.\tag {functoriality of $G$} \end{align*}

This finishes the proof.

Proposition 11.5.1.1.6 ▶ Elementary Properties of Functors

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

1.
Preservation of Isomorphisms. If $f$ is an isomorphism in $\mathcal{C}$, then $F(f)$ is an isomorphism in $\mathcal{D}$.¹

¹When the converse holds, we call $F$ conservative, see Definition 11.6.4.1.1.

Proof of Proposition 11.5.1.1.6.

Item 1: Preservation of Isomorphisms

Indeed, we have

\begin{align*} F(f)^{-1}\circ F(f) & = F(f^{-1}\circ f)\\ & = F(\operatorname {\mathrm{id}}_{A})\\ & = \operatorname {\mathrm{id}}_{F(A)} \end{align*}

and

\begin{align*} F(f)\circ F(f)^{-1} & = F(f\circ f^{-1})\\ & = F(\operatorname {\mathrm{id}}_{B})\\ & = \operatorname {\mathrm{id}}_{F(B)}, \end{align*}

showing $F(f)$ to be an isomorphism.

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