nLab conservative functor (Rev #7, changes)

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Context

Category theory

Definition

A functor $F:C\to D$ 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.

Properties

Proposition

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

Proof

Let $K : J \to C$ be a diagram in $X$ whose limit $\lim K$ exists and such that $\lim F\circ K \simeq F \lim K$ . Then if $const_\theta \to K$ is a cone in $F$ that is sent to a limiting cone $F const_\theta$ in $D$ , then by the universal property of the limit in $D$ the morphism $F( const_\theta \to \lim K)$ is an isomorphism in $D$ , hence must have been an isomorphism in $C$ , hence $const_\theta$ must have been a limiting cone in $C$ .

The arguments for colimits is analogous.

Literature

Geun Bin Im,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
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 $f: A\to Set$ 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

Revision on April 12, 2011 at 16:45:14 by Zoran Škoda See the history of this page for a list of all contributions to it.

nLab conservative functor (Rev #7, changes)

Context

Category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Contents

Definition

Definition

Remark

Properties

Proposition

Proof

Literature