nLab
conservative functor

Definition

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 ${\mathrm{const}}_{\theta }\to K$ is a cone in $F$ that is sent to a limiting cone $F{\mathrm{const}}_{\theta }$ in $D$, then by the universal property of the limit in $D$ the morphism $F({\mathrm{const}}_{\theta }\to \lim K)$ is an isomorphism in $D$, hence must have been an isomorphism in $C$, hence ${\mathrm{const}}_{\theta }$ must have been a limiting cone in $C$.

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 $f:A\to \mathrm{Set}$ 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