8.5.4 Conservative Functors
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A functor $F\colon \mathcal{C}\to \mathcal{D}$ is conservative if it satisfies the following condition:
- For each $f\in \textup{Mor}\webleft (\mathcal{C}\webright )$, if $F\webleft (f\webright )$ is an isomorphism in $\mathcal{D}$, then $f$ is an isomorphism in $\mathcal{C}$.
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
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Characterisations. The following conditions are equivalent:
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The functor $F$ is conservative.
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For each $f\in \textup{Mor}\webleft (\mathcal{C}\webright )$, the morphism $F\webleft (f\webright )$ is an isomorphism in $\mathcal{D}$ iff $f$ is an isomorphism in $\mathcal{C}$.
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Interaction With Fully Faithfulness. Every fully faithful functor is conservative.
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Interaction With Precomposition. The following conditions are equivalent:
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For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
\[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
is conservative.
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The equivalent conditions of Item 4 of Proposition 8.5.1.1.2 are satisfied.
This follows from Item 1 of Proposition 8.4.1.1.6.
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 (\text{id}_{B}\webright ) & = \text{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, $F\webleft (\text{id}_{A}\webright )=F\webleft (f^{-1}\circ f\webright )$. But since $F$ is fully faithful, we must have
\begin{align*} f\circ f^{-1} & = \text{id}_{B},\\ f^{-1}\circ f & = \text{id}_{A}, \end{align*}
showing $f$ to be an isomorphism. Thus $F$ is conservative.
Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ satisfying the following condition:
This question also appears as [MO 468125].