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.

Properties

Proposition

A conservative functor reflects all limits and colimits that it preserves.

Proof

Let K:JCK : J \to C be a diagram in XX 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

  • Geun Bin Im, George 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: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 January 14, 2014 03:56:28 by Urs Schreiber (89.204.138.148)