Conservative Functor

Idea

A conservative functor is a functor that reflects isomorphisms. In other words, it is a functor that does not “collapse” any non-isomorphic morphisms into isomorphisms in the codomain.

Definition

Let $F:\mathcal{C}\to \mathcal{D}$ be a functor. We say $F$ is conservative if for every morphism $f:A\to B$ in $\mathcal{C}$, whenever $F(f):F(A)\to F(B)$ is an isomorphism in $\mathcal{D}$, the original morphism $f$ is an isomorphism in $\mathcal{C}$.

Properties

• $F$ fully faithful $⟹$ $F$ is conservative.
• $F$ faithful $\text{centernot}⟹$ $F$ is conservative.

See also

• Beck’s Monadicity Theorem
• Reflection (Category Theory)
• Creation (Category Theory)