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.


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]


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