11.6.4 Conservative Functors

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

    Definition 11.6.4.1.1 ▶ Conservative Functors

    A functor $F\colon \mathcal{C}\to \mathcal{D}$ is conservative if it satisfies the following condition:1

    • (★)
      • For each $f\in \operatorname {\mathrm{Mor}}\webleft (\mathcal{C}\webright )$, if $F\webleft (f\webright )$ is an isomorphism in $\mathcal{D}$, then $f$ is an isomorphism in $\mathcal{C}$.

    1. 1Slogan: A functor $F$ is conservative if it reflects isomorphisms.

    Proposition 11.6.4.1.2 ▶ Properties of Conservative Functors

    Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

    1. 1.

      Characterisations. The following conditions are equivalent:

      1. (a)

        The functor $F$ is conservative.

      2. (b)

        For each $f\in \operatorname {\mathrm{Mor}}\webleft (\mathcal{C}\webright )$, the morphism $F\webleft (f\webright )$ is an isomorphism in $\mathcal{D}$ iff $f$ is an isomorphism in $\mathcal{C}$.

    2. 2.

      Interaction With Fully Faithfulness. Every fully faithful functor is conservative.

  • 3.

    Interaction With Precomposition. 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 conservative.

    2. (b)

      The equivalent conditions of Item 5 of Proposition 11.6.1.1.2 are satisfied.

    • Proof of Proposition 11.6.4.1.2.

    Item 1: Characterisations
    This follows from Item 1 of Proposition 11.5.1.1.6.

    Item 2: Interaction With Fully Faithfulness
    Let $F\colon \mathcal{C}\to \mathcal{D}$ be a fully faithful functor, let $f\colon A\to B$ be a morphism of $\mathcal{C}$, and suppose that $F_{f}$ is an isomorphism. We have

    \begin{align*} F\webleft (\operatorname {\mathrm{id}}_{B}\webright ) & = \operatorname {\mathrm{id}}_{F\webleft (B\webright )}\\ & = F\webleft (f\webright )\circ F\webleft (f\webright )^{-1}\\ & = F\webleft (f\circ f^{-1}\webright ). \end{align*}

    Similarly, $F\webleft (\operatorname {\mathrm{id}}_{A}\webright )=F\webleft (f^{-1}\circ f\webright )$. But since $F$ is fully faithful, we must have

    \begin{align*} f\circ f^{-1} & = \operatorname {\mathrm{id}}_{B},\\ f^{-1}\circ f & = \operatorname {\mathrm{id}}_{A}, \end{align*}

    showing $f$ to be an isomorphism. Thus $F$ is conservative.

    Question 11.6.4.1.3 ▶ Characterisations of Functors With Conservative Pre/Postcomposition

    Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ satisfying the following condition:

    • (★)
      • 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 conservative?

    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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