- Injective object
In
mathematics , especially in the field ofcategory theory , the concept of injective object is a generalization of the concept ofinjective module . This concept is important inhomotopy theory and in theory of model categories. The dual notion is that of aprojective object .General Definition
Let mathfrak{C} be a category and let mathcal{H} be a class of morphisms of mathfrak{C}.
An object Q of mathfrak{C} is said to be "mathcal{H}"-injective if every arrow f: A o Q and every morphisms h: A o B in mathcal{H} there exists a morphism g: B o Q extending f, i.e gh = f. In other words, Q is injective iff any mathcal{H}-morphism extends to any morphism into Q.
The morphism g in the above definition is not required to be uniquely determined by h ext{ and } f.
In a locally small category, it is equivalent to require that the
hom functor Hom_{mathfrak{C(-,Q) carries mathcal{H}-morphisms to epimorphisms (surjections).The classical choice for mathcal{H} is the class of
monomorphism s, in this case, the expression injective object is used.Abelian case
If mathfrak{C} is an
abelian category , an object "A" of mathfrak{C} is injective iff itshom functor HomC(–,"A") is exact.The abelian case was the original framework for the notion of injectivity.
Enough injectives
Let mathfrak{C} be a category, "H" a class of morphisms of mathfrak{C} ; the category mathfrak{C} is said to "have enough H-injectives" if for every object "X" of mathfrak{C}, there exist a "H"-morphism from "X" to an "H"-injective object.
Injective hull
A "H"-morphism "g" in mathfrak{C} is called "H"-essential if for any morphism "f", the composite "fg" is in "H" only if "f" is in "H".
If "f" is a "H"-essential "H"-morphism with a domain "X" and an "H"-injective codomain "G", "G" is called an "H"-injective hull of "X". This "H"-injective hull is then unique up to a canonical isomorphism.
Examples
*In the category of
Abelian group s andgroup homomorphism s, an injective object is adivisible group .
*In the category of modules andmodule homomorphism s, "R"-Mod, an injective object is aninjective module . "R"-Mod hasinjective hull s (as a consequence, R-Mod has enough injectives).
*In the category ofmetric space s andnonexpansive mapping s, an injective object is aninjective metric space .
*In the category ofT0 space s andcontinuous mapping s, an injective object is always aScott topology on acontinuous lattice therefore it is always sober andlocally compact .
*In the category ofsimplicial set s, the injective objects with respect to the class of anodyne extensions areKan complex es.
*One also talks about injective objects in more general categories, for instance in functor categories or in categories of sheaves of O"X" modules over someringed space ("X",O"X").References
*J. Rosicky, Injectivity and accessible categories
*F. Cagliari and S. Montovani, T0-reflection and injective hulls of fibre spaces
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