Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Injective object
Summary
In mathematics, especially in the field of , the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of . The dual notion is that of a projective object.
An in a is said to be injective if for every monomorphism and every morphism there exists a morphism extending to , i.e. such that .
That is, every morphism factors through every monomorphism .
The morphism in the above definition is not required to be uniquely determined by and .
In a category, it is equivalent to require that the hom functor carries monomorphisms in to surjective set maps.
The notion of injectivity was first formulated for , and this is still one of its primary areas of application. When is an abelian category, an object Q of is injective if and only if its hom functor HomC(–,Q) is exact.
If is an exact sequence in such that Q is injective, then the sequence splits.
The category is said to have enough injectives if for every object X of , there exists a monomorphism from X to an injective object.
A monomorphism g in is called an essential monomorphism if for any morphism f, the composite fg is a monomorphism only if f is a monomorphism.
If g is an essential monomorphism with domain X and an injective codomain G, then G is called an injective hull of X. The injective hull is then uniquely determined by X up to a non-canonical isomorphism.
In the category of abelian groups and group homomorphisms, Ab, an injective object is necessarily a divisible group. Assuming the axiom of choice, the notions are equivalent.
In the category of (left) modules and module homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives).
In the , Met, an injective object is an injective metric space, and the injective hull of a metric space is its tight span.
In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice, and therefore it is always sober and locally compact.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood