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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 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]
^ abHashimoto, 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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fi} is a monomorphism only when f is a monomorphism. Essentialmonomorphisms in a category of modules are those whose image is an essential submodule...
exists a monomorphism from X to an injective object. A monomorphism g in C {\displaystyle \mathbf {C} } is called an essentialmonomorphism if for any...
category, a morphism f : X → Y is essential if any morphism g : Y → Z is a monomorphism if and only if g ° f is a monomorphism (Porst 1981, Introduction). Taking...
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{f}{\hookrightarrow }}\;B\;{\overset {g}{\twoheadrightarrow }}\;C} where ƒ is a monomorphism and g is an epimorphism. In this case, A is a subobject of B, and the...
preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every...
(algebra) Ring homomorphism, Algebra homomorphism Ring epimorphism Ring monomorphism Ring isomorphism Skolem–Noether theorem Graded algebra Morita equivalence...
→ ∏ i ∈ I N i {\displaystyle \phi :M\to \prod _{i\in I}N_{i}\,} is a monomorphism, where each N i {\displaystyle N_{i}} is an R module, then ϕ : M → ∏...
is an associative algebra over a commutative ring R. monomorphism A morphism f is a monomorphism (also called monic) if g = h {\displaystyle g=h} whenever...
injectives, which means that every object X of the category admits a monomorphism to an injective object I. (Neither the map nor the injective object has...
suspending a weak n-category n + 2 times, further suspensions have no essential effect. The suspension functor S: nCatk→nCatk+1 is an equivalence of categories...
individuals. monadic relation See unary relation. monism See logical monism. monomorphism A morphism in category theory that is left-cancellable, meaning if two...
case that ⋀ {\displaystyle {\textstyle \bigwedge }} converts monomorphisms to monomorphisms. See Bourbaki (1989, Corollary to Proposition 12, §III.7.9)...
definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h :...
{\displaystyle [U]_{m}=[V]_{m}} , if there exists a monomorphism U → V {\displaystyle U\rightarrow V} and a monomorphism V → U {\displaystyle V\rightarrow U} . The...
makes sense to ask when a morphism f: B → C of sheaves is injective (a monomorphism) or surjective (an epimorphism). One answer is that f is injective (respectively...
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