Essential monomorphism: Difference between revisions

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In mathematics, specifically category theory, an essential monomorphism is a monomorphism i in an abelian category C such that for a morphism f in C, the composition $fi$ is a monomorphism only when f is a monomorphism.^[1] Essential monomorphisms in a category of modules are those whose image is an essential submodule of the codomain. An injective hull of an object A is an essential monomorphism from A to an injective object.^[1]

References

^ ^a ^b Hashimoto, Mitsuyasu (November 2, 2000). Auslander-Buchweitz Approximations of Equivariant Modules. Cambridge University Press. p. 19. ISBN 9780521796965. Retrieved February 3, 2024 – via Google Books.

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[auto-1] Hashimoto, Mitsuyasu (November 2, 2000). Auslander-Buchweitz Approximations of Equivariant Modules. Cambridge University Press. p. 19. ISBN 9780521796965. Retrieved February 3, 2024 – via Google Books.

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-In [[mathematics]], specifically [[category theory]], an '''essential monomorphism''' is a [[monomorphism]] ''f'' in a [[category (mathematics)|category]] ''C'' such that for a morphism ''g'' in ''C'', the morphism <math>g \circ f</math> is a monomorphism only when ''g'' is a monomorphism. Essential monomorphisms in a [[category of modules]] are those whose image is an [[essential submodule]] of the [[codomain]]. An injective hull of an object ''X'' is an essential monomorphism from ''X'' to an [[injective object]].
+In [[mathematics]], specifically [[category theory]], an '''essential monomorphism''' is a [[monomorphism]] ''i'' in an [[abelian category]] ''C'' such that for a morphism ''f'' in ''C'', the composition <math>fi</math> is a monomorphism only when ''f'' is a monomorphism.<ref name="auto">{{cite book|last=Hashimoto|first=Mitsuyasu|title=Auslander-Buchweitz Approximations of Equivariant Modules|url=https://www.google.com/books/edition/Auslander_Buchweitz_Approximations_of_Eq/OOBkgAmsWnsC?gbpv=1|page=19|isbn=9780521796965|publisher=[[Cambridge University Press]]|via=Google Books|publication-date=November 2, 2000|access-date=February 3, 2024}}</ref> Essential monomorphisms in a [[category of modules]] are those whose image is an [[essential submodule]] of the [[codomain]]. An injective hull of an object ''A'' is an essential monomorphism from ''A'' to an [[injective object]].<ref name="auto" />
 ==References==
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 [[Category:Category theory]]