Contents

category theory

# Contents

## Definition

###### Definition

A functor $F \colon C\to D$ is conservative if it is “isomorphism-reflecting”, i.e. if $g:a\to b$ is a morphism in $C$ such that $F(g)$ is an isomorphism in $D$, then $g$ is an isomorphism in $C$.

###### Remark

Sometimes 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.

## Examples

###### Proposition

Let $\mathcal{C}$ be a category with pullbacks. Given any morphism $f \colon X \longrightarrow Y$ in $\mathcal{C}$ write

$f^\ast \colon \mathcal{C}_{/Y} \longrightarrow \mathcal{C}_{/X}$

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:

1. $f$ is a strong epimorphism;

2. $f^\ast$ is conservative.

(e.g. Johnstone, lemma A.1.3.2)

###### Example

Every fully faithful functor is a conservative functor. An example of a functor that is conservative but not fully faithful is the inclusion of the groupoid core of a category into the category.

###### Example

When $C$ and $D$ are pretoposes, a pretopos morphism $F : C \to D$ 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.

###### Example

Every monadic functor is a conservative functor: given a $T$-algebra homomorphism $f : A \to B$, the inverse $f^{-1} : B \to A$ is easily seen to also be a $T$-algebra homomorphism.

## Properties

###### Proposition

A conservative functor $F : C \to D$ reflects all limits and colimits that it preserves and which exist in the source category.

###### Proof

Let $K : J \to C$ be a diagram in $C$ whose limit $\lim K$ exists and such that $\lim F\circ K \simeq F \lim K$. Then 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 $C$, hence $const_c$ must have been a limiting cone in $C$.

The arguments for colimits is analogous.

## Literature

• 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; On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986), no. 1, 1–18, Adjoint-triangle theorems for conservative functors, Bull. Austral. Math. Soc. 36 (1987), no. 1, 133–136, MR88k:18005, doi

For an example of a conservative, but not faithful, functor $f: A\to Set$ having a left adjoint see Example 2.4 in

• Reinhard Börger, Walter Tholen, Strong regular and dense generators, Cahiers de Topologie et Géométrie Différentielle Catégoriques 32, no. 3 (1991), p. 257-276, MR1158111, numdam

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