Item 2 (00ZZ) — The Clowder Project

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

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

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

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

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

satisfying the following conditions:

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

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

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

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

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 $F\webleft (A\webright )$.

2.
Given a functor

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

we also write $F^{A}$ for $F\webleft (A\webright )$.

Given a functor

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

we also write $F_{A,B}$ for $F\webleft (A,B\webright )$.

Given a functor

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

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

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

Notation 11.5.1.1.3 ▶ Additional Notation for Functors

Following the notation $[\mspace {-3mu}[x\mapsto f\webleft (x\webright )]\mspace {-3mu}]$ 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

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

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

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

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

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

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

Proof of Example 11.5.1.1.4.

Preservation of Identities

We have $\operatorname {\mathrm{id}}_{\mathcal{C}}\webleft (\operatorname {\mathrm{id}}_{A}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{A}$ for each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ 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}}\webleft (g\circ f\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\mathcal{C}}\webleft (g\webright )\circ \operatorname {\mathrm{id}}_{\mathcal{C}}\webleft (f\webright ). \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 $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, we have

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

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

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

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

Proof of Definition 11.5.1.1.5.

Preservation of Identities

For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, 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 $\webleft (g,f\webright )$ 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 $F\webleft (f\webright )$ 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\webleft (f\webright )^{-1}\circ F\webleft (f\webright ) & = F\webleft (f^{-1}\circ f\webright )\\ & = F\webleft (\operatorname {\mathrm{id}}_{A}\webright )\\ & = \operatorname {\mathrm{id}}_{F\webleft (A\webright )} \end{align*}

and

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

showing $F\webleft (f\webright )$ to be an isomorphism.

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