11.6.5 Essentially Injective Functors

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

    Definition 11.6.5.1.1 ▶ Essentially Injective Functors

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

    • (★)
      • For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, if $F\webleft (A\webright )\cong F\webleft (B\webright )$, then $A\cong B$.

    Question 11.6.5.1.2 ▶ Characterisations of Functors With Essentially Injective 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 injective, i.e. if $\phi \circ F\cong \psi \circ F$, then $\phi \cong \psi $ for all functors $\phi $ and $\psi $?

  • 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 injective, i.e. if $F\circ \phi \cong F\circ \psi $, then $\phi \cong \psi $?

    • 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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