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 $C$.

Remarks

• 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.