11.7.1 Dominant Functors

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

    Definition 11.7.1.1.1 ▶ Dominant Functors

    A functor $F\colon \mathcal{C}\to \mathcal{D}$ is dominant if every object of $\mathcal{D}$ is a retract of some object in $\mathrm{Im}\webleft (F\webright )$, i.e.:

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

        An object $A$ of $\mathcal{C}$;

      • •

        A morphism $r\colon F\webleft (A\webright )\to B$ of $\mathcal{D}$;

      • •

        A morphism $s\colon B\to F\webleft (A\webright )$ of $\mathcal{D}$;

      such that we have

    Proposition 11.7.1.1.2 ▶ Properties of Dominant Functors

    Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors and let $I\colon \mathcal{X}\to \mathcal{C}$ be a functor.

    1. 1.

      Interaction With Right Whiskering. If $I$ is full and dominant, then the map

      \[ -\mathbin {\star }\operatorname {\mathrm{id}}_{I} \colon \operatorname {\mathrm{Nat}}\webleft (F,G\webright )\to \operatorname {\mathrm{Nat}}\webleft (F\circ I,G\circ I\webright ) \]

      is a bijection.

    2. 2.

      Interaction With Adjunctions. Let $\webleft (F,G\webright )\colon \mathcal{C}\mathbin {\rightleftarrows }\mathcal{D}$ be an adjunction.

      1. (a)

        If $F$ is dominant, then $G$ is faithful.

      2. (b)

        The following conditions are equivalent:

  • (i)

    The functor $G$ is full.

  • (ii)

    The restriction

    \[ \left.G\right\vert _{\mathrm{Im}_{F}}\colon \mathrm{Im}\webleft (F\webright )\to \mathcal{C} \]

    of $G$ to $\mathrm{Im}\webleft (F\webright )$ is full.

    • Proof of Proposition 11.7.1.1.2.

    Question 11.7.1.1.3 ▶ Characterisations of Functors With Dominant Pre/Postcomposition

    Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:

    1. 1.

      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 dominant?

    2. 2.

      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 dominant?

    This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].

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