Conservative functor

In category theory, a branch of mathematics, a conservative functor is a functor $F:C\to D$ such that for any morphism f in C, F(f) being an isomorphism implies that f is an isomorphism.^[1]

Examples

The forgetful functors in algebra, such as from Grp to Set, are conservative. More generally, every monadic functor is conservative. In contrast, the forgetful functor from Top to Set is not conservative because not every continuous bijection is a homeomorphism.

Functor types

Additive Adjoint Conservative Derived Diagonal Enriched Essentially surjective Exact Faithful Forgetful Full Logical Monoidal Representable Smooth

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