In mathematics, an effaceable functor is an

additive functor In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every ho ...

''F'' between

abelian categories In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...

''C'' and ''D'' for which, for each object ''A'' in ''C'', there exists a

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

u: A \to M

, for some ''M'', such that

F(u) = 0

. Similarly, a coeffaceable functor is one for which, for each ''A'', there is an epimorphism into ''A'' that is killed by ''F''. The notions were introduced in Grothendieck's Tohoku paper. A theorem of Grothendieck says that every effaceable δ-functor (i.e., effaceable in each degree) is universal.

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Meaning of “efface” in “effaceable functor” and “injective effacement”
Functors {{categorytheory-stub