The Clowder Project

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

1. 1.

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

2. 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. 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. 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. 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.

   5. (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.

   6. (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.

   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 $\mathrm{Im}(F)$:

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

        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 $\mathcal{D}$;

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