Item 5e (011N) — 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 $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms

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

of $F$ at $(A,B)$ 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 $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the postcomposition functor
  
  \[ F_{*} \colon \mathsf{Fun}(\mathcal{X},\mathcal{C}) \to \mathsf{Fun}(\mathcal{X},\mathcal{D}) \]
  
  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 14: Types of Morphisms in Bicategories, Definition 14.1.1.1.1.
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
  
  \[ F^{*} \colon \mathsf{Fun}(\mathcal{D},\mathcal{X}) \to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]
  
  can fail to be faithful.
2. (b)
  Conversely, if the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}(\mathcal{D},\mathcal{X}) \to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]
  
  is faithful, then $F$ can fail to be faithful.
4.
Interaction With Precomposition II. If $F$ is essentially surjective, then the precomposition functor

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

is faithful.
5.
Interaction With Precomposition III. The following conditions are equivalent:
1. (a)
  For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}(\mathcal{D},\mathcal{X}) \to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]
  
  is faithful.
2. (b)
  For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}(\mathcal{D},\mathcal{X}) \to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]
  
  is conservative.
3. (c)
  For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}(\mathcal{D},\mathcal{X}) \to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]
  
  is monadic.
4. (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 14: Types of Morphisms in Bicategories, Definition 14.2.1.1.1.

(e)
The components

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

of the unit

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

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

(f)

The components

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

of the counit

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

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 $\mathrm{Im}(F)$:

(★)

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

Proof of Proposition 11.6.1.1.2.

Item 1: Interaction With Composition

Since the map

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

defined as the composition

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

is a composition of injective functions, it follows from Chapter 4: Constructions With Sets, Item 2a of Item 2 of Proposition 4.7.1.1.2 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: