11.6.6 Essentially Surjective Functors

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

    Definition 11.6.6.1.1 ▶ Essentially Surjective Functors

    A functor $F\colon \mathcal{C}\to \mathcal{D}$ is essentially surjective1 if it satisfies the following condition:

    • (★)
      • For each $D\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{D}\webright )$, there exists some object $A$ of $\mathcal{C}$ such that $F\webleft (A\webright )\cong D$.

    1. 1Further Terminology: Also called an eso functor, meaning essentially surjective on objects.

    Question 11.6.6.1.2 ▶ Characterisations of Functors With Essentially Surjective Pre/Postcomposition

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

  • 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 essentially surjective?

  • 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 essentially surjective?

    • 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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