Item 5b (011K) — The Clowder Project

11.6.1 Faithful Functors

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

Definition 11.6.1.1.1 ▶ Faithful Functors

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is faithful if, for each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the action on morphisms

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

of $F$ at $\webleft (A,B\webright )$ is injective.

Proposition 11.6.1.1.2 ▶ Properties of Faithful Functors

Let $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ be functors.

1.
Interaction With Composition. If $F$ and $G$ are faithful, then so is $G\circ F$.
2.
Interaction With Postcomposition. The following conditions are equivalent:
1. (a)
  The functor $F\colon \mathcal{C}\to \mathcal{D}$ is faithful.
2. (b)
  For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
  
  \[ F_{*} \colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright ) \to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]
  
  is faithful.
3. (c)
  The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 13: Types of Morphisms in Bicategories, Definition 13.1.1.1.1.
3.
Interaction With Precomposition I. Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
1. (a)
  If $F$ is faithful, then the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  can fail to be faithful.
2. (b)
  Conversely, if the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  is faithful, then $F$ can fail to be faithful.
4.
Interaction With Precomposition II. If $F$ is essentially surjective, then the precomposition functor

\[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

is faithful.
5.
Interaction With Precomposition III. The following conditions are equivalent:
1. (a)
  For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  is faithful.

(b)
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the precomposition functor

\[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

is conservative.

(c)

For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the precomposition functor

\[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

is monadic.

(d)

The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a corepresentably faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 13: Types of Morphisms in Bicategories, Definition 13.2.1.1.1.

(e)

The components

\[ \eta _{G}\colon G\Longrightarrow \operatorname {\mathrm{Ran}}_{F}\webleft (G\circ F\webright ) \]

of the unit

\[ \eta \colon \operatorname {\mathrm{id}}_{\mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )}\Longrightarrow \operatorname {\mathrm{Ran}}_{F}\circ F^{*} \]

of the adjunction $F^{*}\dashv \operatorname {\mathrm{Ran}}_{F}$ are all monomorphisms.

(f)

The components

\[ \epsilon _{G}\colon \operatorname {\mathrm{Lan}}_{F}\webleft (G\circ F\webright )\Longrightarrow G \]

of the counit

\[ \epsilon \colon \operatorname {\mathrm{Lan}}_{F}\circ F^{*}\Longrightarrow \operatorname {\mathrm{id}}_{\mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )} \]

of the adjunction $\operatorname {\mathrm{Lan}}_{F}\dashv F^{*}$ are all epimorphisms.

(g)

The functor $F$ is dominant (Definition 11.7.1.1.1), i.e. every object of $\mathcal{D}$ is a retract of some object in $\mathrm{Im}\webleft (F\webright )$:

(★)

•
An object $A$ of $\mathcal{C}$;
•
A morphism $s\colon B\to F\webleft (A\webright )$ of $\mathcal{D}$;
•
A morphism $r\colon F\webleft (A\webright )\to B$ of $\mathcal{D}$;

Proof of Proposition 11.6.1.1.2.

Item 1: Interaction With Composition

Since the map

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

defined as the composition

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

is a composition of injective functions, it follows from that it is also injective. Therefore $G\circ F$ is faithful.

Item 2: Interaction With Postcomposition

Omitted.

Item 3: Interaction With Precomposition I

See [Lin, Precomposition with a faithful functor] for Item 3a. Item 3b follows from Item 4 and the fact that there are essentially surjective functors that are not faithful.

Item 4: Interaction With Precomposition II

Omitted, but see for a formalised proof.

Item 5: Interaction With Precomposition III

We claim Item 5a, Item 5b, Item 5c, Item 5d, Item 5e, Item 5f, and Item 5g are equivalent: