A functor $F \colon C\to D$ is conservative if it is “isomorphism-reflecting”, i.e. if $g \colon a\to b$ is a morphism in $C$ such that $F(g)$ is an isomorphism in $D$, then $g$ is an isomorphism in $C$.
Sometimes (e.g. in the Elephant) conservative functors are assumed to be faithful as well. If $C$ has, and $F$ preserves, equalizers, then conservativity implies faithfulness.
See conservative morphism for a generalization to an arbitrary 2-category.
Every fully faithful functor, and more generally any pseudomonic functor, is a conservative functor.
(Every fully faithful functor is pseudomonic.)
But further would-be converses of Exp. fail: not every conservative functor is full or faithful:
An example of a functor that is conservative but not full is the inclusion of the groupoid core $Core(\mathcal{C}) \longrightarrow \mathcal{C}$ of a category $\mathcal{C}$ that is itself not a groupoid, into that category.
An example of a functor that is conservative but not faithful is the unique functor from any groupoid with two distinct isomorphisms $f, g : x \to y$ to the terminal groupoid.
Every monadic functor is a conservative functor (see also at monadicity theorem):
For a $T$-algebra homomorphism given by an invertible morphism $f \colon A \to B$, the inverse $f^{-1} \colon B \to A$ is easily seen to also be a $T$-algebra homomorphism.
Let $C$ be a category with pullbacks. Given any morphism $f \colon X \longrightarrow Y$ in $C$ write
for the functor of pullback along $f$ between slice categories (“base change”). If strong epimorphisms in $\mathcal{C}$ are preserved by pullback, then the following are equivalent:
$f$ is a strong epimorphism;
$f^\ast$ is conservative.
(e.g. Johnstone, lemma A.1.3.2)
When $C$ and $D$ are pretoposes, a pretopos morphism $F \colon C \to D$ is conservative if and only if for every object $c \in C$, the induced map between subobject lattices $F^{(c)} : \operatorname{Sub}(c) \to \operatorname{Sub}(F(c))$ is injective.
A conservative functor $F \colon C \to D$ reflects all limits and colimits that it preserves and which exist in the source category.
We discuss the case of limits (the argument for colimits is formally dual):
Let $K \colon J \to C$ be a diagram in $C$ whose limit $\lim K$ exists and such that $\lim (F\circ K) \,\simeq\, F (\lim K)$. Now if $const_c \to K$ is a cone in $C$ that is sent to a limiting cone $F const_c$ in $D$, then by the universal property of the limit in $D$ the morphism $F( c \to \lim K)$ is an isomorphism in $D$, hence must have been an isomorphism in $C$ (by the assumption that $F$ is conservative), hence $const_c$ must have been a limiting cone in $C$.
Geun Bin Im, Gregory Maxwell Kelly, Some remarks on conservative functors with left adjoints, J. Korean Math. Soc. 23 (1986), no. 1, 19–33, MR87i:18002b, pdf
Geun Bin Im, Gregory Maxwell Kelly, On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986), no. 1, 1–18, pdf
Geun Bin Im, Gregory Maxwell Kelly, Adjoint-triangle theorems for conservative functors, Bull. Austral. Math. Soc. 36 (1987), no. 1, 133–136, MR88k:18005, doi
Formalization in cubical Agda:
For an example of a conservative, but not faithful, functor $f: A\to Set$ having a left adjoint:
Last revised on March 6, 2024 at 12:59:45. See the history of this page for a list of all contributions to it.