The Clowder Project

We have a pullback square

Interaction With Composition. If $F$ and $G$ are fully faithful, then so is $G\circ F$.

Conservativity. If $F$ is fully faithful, then $F$ is conservative.

Essential Injectivity. If $F$ is fully faithful, then $F$ is essentially injective.

Interaction With Co/Limits. If $F$ is fully faithful, then $F$ reflects co/limits.

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 $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the postcomposition functor

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

   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 14: Types of Morphisms in Bicategories, Definition 14.1.3.1.1.

Interaction With Precomposition I. If $F$ is fully faithful, then the precomposition functor

can fail to be fully faithful.

Interaction With Precomposition II. If the precomposition functor

is fully faithful, then $F$ can fail to be fully faithful (and in fact it can also fail to be either full or faithful).

Interaction With Precomposition III. If $F$ is essentially surjective and full, then the precomposition functor

is fully faithful.

Interaction With Precomposition IV. 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 fully faithful.

2. (b)

   The precomposition functor

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

   is fully faithful.

3. (c)

   The functor

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

   is fully faithful.

4. (d)

   The functor $F$ is a corepresentably fully faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 14: Types of Morphisms in Bicategories, Definition 14.2.3.1.1.

5. (e)

   The functor $F$ is absolutely dense.

6. (f)

   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 isomorphisms.

7. (g)

   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 isomorphisms.

8. (h)

   The natural transformation

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

   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

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

   is a natural isomorphism.

9. (i)

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

   • •

     An object $A_{B}$ of $\mathcal{C}$;

   • •

     A morphism $s_{B}\colon B\to F(A_{B})$ of $\mathcal{D}$;

   • •

     A morphism $r_{B}\colon F(A_{B})\to B$ of $\mathcal{D}$;

   satisfying the following conditions:

   1. (i)

      The triple $(F(A_{B}),r_{B},s_{B})$ is a retract of $B$, i.e. we have $r_{B}\circ s_{B}=\operatorname {\mathrm{id}}_{B}$.

   2. (ii)

      For each morphism $f\colon B'\to B$ of $\mathcal{D}$, we have

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

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