nLab
conservative functor

Contents

Definition

Definition

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

Remark

Sometimes conservative functors are assumed to be faithful as well. If CC has, and FF preserves, equalizers, then conservativity implies faithfulness.

See conservative morphism for a generalization to an arbitrary 2-category.

Examples

Proposition

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

f *:𝒞 /Y𝒞 /X f^\ast \colon \mathcal{C}_/Y \longrightarrow \mathcal{C}_{/X}

for the functor of pullback along ff between slice categories (base change). If strong epimorphisms in 𝒞\mathcal{C} are preserved by pullback, then the following are equivalent:

  1. ff is a strong epimorphism;

  2. f *f^\ast is conservative.

(e.g. Johnstone, lemma 1.3.2)

Properties

Proposition

A conservative functor reflects all limits and colimits that it preserves and which exist in the source category.

Proof

Let K:JCK : J \to C be a diagram in CC whose limit limK\lim K exists and such that limFKFlimK\lim F\circ K \simeq F \lim K. Then if const θKconst_\theta \to K is a cone in FF that is sent to a limiting cone Fconst θF const_\theta in DD, then by the universal property of the limit in DD the morphism F(const θlimK)F( const_\theta \to \lim K) is an isomorphism in DD, hence must have been an isomorphism in CC, hence const θconst_\theta must have been a limiting cone in CC.

The arguments for colimits is analogous.

Literature

For an example of a conservative, but not faithful, functor f:ASetf: 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

Revised on March 25, 2016 12:35:31 by Ingo Blechschmidt (31.15.67.4)