The Clowder Project

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

1. 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. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 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 $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, 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.

7. 7.

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

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

   can fail to be fully faithful.

8. 8.

   Interaction With Precomposition II. If the precomposition functor

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

   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. 9.

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

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

   is fully faithful.

10. 10.

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