In category theory, a branch of mathematics, a conservative functor is a functor F : C → D {\displaystyle 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
External links
- Conservative functor at the nLab
References
Riehl, Emily (2016). Category Theory in Context. Courier Dover Publications. ISBN 048680903X. Retrieved 18 February 2017. 048680903X ↩
Grandis, Marco (2013). Homological Algebra: In Strongly Non-Abelian Settings. World Scientific. ISBN 9814425931. Retrieved 14 January 2017. 9814425931 ↩