In category theory, a branch of mathematics, a '''conservative functor''' is a functor <math>F: C \to D</math> 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.<ref>{{cite book|last=Riehl|first=Emily|author-link=Emily Riehl|year=2016|title=Category Theory in Context|publisher=Courier Dover Publications|url=https://books.google.com/books?id=Sr09DQAAQBAJ|isbn=048680903X|access-date=18 February 2017}}</ref> 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.<ref>{{cite book|last=Grandis|first=Marco|year=2013|title=Homological Algebra: In Strongly Non-Abelian Settings|publisher=World Scientific|url=https://books.google.com/books?id=kvW6CgAAQBAJ|isbn=9814425931|access-date=14 January 2017}}</ref>
==References== {{reflist}}
==External links== * {{nlab|id=conservative+functor|title=Conservative functor}}
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Category:Category theory
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