conservative functor

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 $C$.

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.

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.

- 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 $f: A\to Set$ 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)