Conservative functor explained

F: C \to D such that for any morphism f in C, F(f) being an isomorphism implies that f is an isomorphism.

Examples

The forgetful functors in algebra, such as from Grp to Set, are conservative. More generally, every monadic functor is conservative.^[1] In contrast, the forgetful functor from Top to Set is not conservative because not every continuous bijection is a homeomorphism.

Every faithful functor from a balanced category is conservative.^[2]

Notes and References

1. Book: Riehl, Emily. Emily Riehl. 2016. Category Theory in Context. Courier Dover Publications. 048680903X. 18 February 2017.
2. Book: Grandis, Marco. 2013. Homological Algebra: In Strongly Non-Abelian Settings. World Scientific. 9814425931. 14 January 2017.