Incategory theory, a branch ofmathematics, aconservative functor is afunctor${\displaystyle F:C\to D}$ such that for anymorphismf inC,F(f) being anisomorphism implies thatf is an isomorphism.

Theforgetful functors inalgebra, such as fromGrp toSet, are conservative. More generally, everymonadic functor is conservative.^[1] In contrast, the forgetful functor fromTop toSet is not conservative because not everycontinuous bijection is ahomeomorphism.

Everyfaithful functor from abalanced category is conservative.^[2]

• Conservative functor at thenLab

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