In category theory, a branch of mathematics, a **conservative functor** is a functor 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]}

## References

**^**Riehl, Emily (2016).*Category Theory in Context*. Courier Dover Publications. ISBN 048680903X. Retrieved 18 February 2017.**^**Grandis, Marco (2013).*Homological Algebra: In Strongly Non-Abelian Settings*. World Scientific. ISBN 9814425931. Retrieved 14 January 2017.