Item 10c (012T) — The Clowder Project

11.6.3 Fully Faithful Functors

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

Definition 11.6.3.1.1 ▶ Fully Faithful Functors

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is fully faithful if $F$ is full and faithful, i.e. 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 bijective.

Proposition 11.6.3.1.2 ▶ Properties of Fully Faithful Functors

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

1.
Characterisations. The following conditions are equivalent:
1. (a)
  The functor $F$ is fully faithful.
2. (b)
  We have a pullback square
  in $\mathsf{Cats}$.
2.
Interaction With Composition. If $F$ and $G$ are fully faithful, then so is $G\circ F$.
3.
Conservativity. If $F$ is fully faithful, then $F$ is conservative.
4.
Essential Injectivity. If $F$ is fully faithful, then $F$ is essentially injective.
5.
Interaction With Co/Limits. If $F$ is fully faithful, then $F$ reflects co/limits.
6.
Interaction With Postcomposition. The following conditions are equivalent:
1. (a)
  The functor $F\colon \mathcal{C}\to \mathcal{D}$ is fully 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 fully faithful.
3. (c)
  The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably fully faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 13: Types of Morphisms in Bicategories, Definition 13.1.3.1.1.
7.
Interaction With Precomposition I. If $F$ is fully 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 fully faithful.
8.
Interaction With Precomposition II. If the precomposition functor

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

is fully faithful, then $F$ can fail to be fully faithful (and in fact it can also fail to be either full or faithful).
9.
Interaction With Precomposition III. If $F$ is essentially surjective and full, then the precomposition functor

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

is fully faithful.
10.
Interaction With Precomposition IV. 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 fully faithful.
2. (b)
  The precomposition functor
  
  \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathsf{Sets}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathsf{Sets}\webright ) \]
  
  is fully faithful.

(c)
The functor

\[ \operatorname {\mathrm{Lan}}_{F}\colon \mathsf{Fun}\webleft (\mathcal{C},\mathsf{Sets}\webright ) \to \mathsf{Fun}\webleft (\mathcal{D},\mathsf{Sets}\webright ) \]

is fully faithful.

(d)

The functor $F$ is a corepresentably fully faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 13: Types of Morphisms in Bicategories, Definition 13.2.3.1.1.

(e)

The functor $F$ is absolutely dense.

(f)

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 isomorphisms.

(g)

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 isomorphisms.

(h)

The natural transformation

\[ \alpha \colon \operatorname {\mathrm{Lan}}_{h_{F}}\webleft (h^{F}\webright )\Longrightarrow h \]

with components

\[ \alpha _{B',B}\colon \int ^{A\in \mathcal{C}}h^{B'}_{F_{A}}\times h^{F_{A}}_{B}\to h^{B'}_{B} \]

given by

\[ \alpha _{B',B}\webleft (\webleft [\webleft (\phi ,\psi \webright )\webright ]\webright )=\psi \circ \phi \]

is a natural isomorphism.

(i)

For each $B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{D}\webright )$, there exist:

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

satisfying the following conditions:

(i)
The triple $\webleft (F\webleft (A_{B}\webright ),r_{B},s_{B}\webright )$ is a retract of $B$, i.e. we have $r_{B}\circ s_{B}=\operatorname {\mathrm{id}}_{B}$.
(ii)
For each morphism $f\colon B'\to B$ of $\mathcal{D}$, we have

\[ \webleft [\webleft (A_{B},s_{B'},f\circ r_{B'}\webright )\webright ]=\webleft [\webleft (A_{B},s_{B}\circ f,r_{B}\webright )\webright ] \]

in $\int ^{A\in \mathcal{C}}h^{B'}_{F_{A}}\times h^{F_{A}}_{B}$.

Proof of Proposition 11.6.3.1.2.

Item 1: Characterisations

Omitted.

Item 2: 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 bijective functions, it follows from that it is also bijective. Therefore $G\circ F$ is fully faithful.

Item 3: Conservativity

This is a repetition of Item 2 of Proposition 11.6.4.1.2, and is proved there.

Item 4: Essential Injectivity

Omitted.

Item 5: Interaction With Co/Limits

Omitted.

Item 6: Interaction With Postcomposition

This follows from Item 2 of Proposition 11.6.1.1.2 and