The Clowder Project

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.

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.

Interaction With Precomposition II. If $F$ is essentially surjective, then the precomposition functor

is faithful.

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.

2. (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.

3. (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.

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 13: Types of Morphisms in Bicategories, Definition 13.2.1.1.1.

5. (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.

6. (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.

7. (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 $\mathrm{Im}\webleft (F\webright )$:

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

     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 $\mathcal{D}$;

   such that $r\circ s=\operatorname {\mathrm{id}}_{B}$.