# Conservative functor

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.

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