Conservative functor

This article defines a functor property: a property that can be evaluated to true/false given a functor between two categories.
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Definition

Symbol-free definition

A functor is termed conservative if it reflects isomorphisms: in other words, any morphism that gets mapped to an isomorphism must be an isomorphism to begin with.

Definition with symbols

Suppose ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ is a functor between two categories. We say that ${\mathcal {F}}$ is conservative if, whenever $A,B\in \operatorname {Ob} {\mathcal {C}}$ and $f\in {\mathcal {C}}(A,B)$ such that $Ff$ is an isomorphism from $FA$ to $FB$, $f$ is an isomorphism.

Relation with other properties

Stronger properties

• Strongly conservative functor