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 category...

measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal...

the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;...

homological algebra. The dual notion of a projective object is that of an injective object. An object P {\displaystyle P} in a category C {\displaystyle...

{\displaystyle s\in S} to itself) is injective. In particular, the identity function X → X {\displaystyle X\to X} is always injective (and in fact bijective). If...

enough injectives. Embedding X into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution...

Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary...

mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest...

algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact,...

flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a module M over a ring...

I^{0}\to I^{1}\to I^{2}\to \cdots } where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence...

Instead, the receiving "client" (object or function) is provided with its dependencies by external code (an "injector"), which it is not aware of. Dependency...

concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation...

quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous...

When some object X {\displaystyle X} is said to be embedded in another object Y {\displaystyle Y} , the embedding is given by some injective and structure-preserving...

monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions...

there are no zero objects in Met. The injective objects in Met are called injective metric spaces. Injective metric spaces were introduced and studied...

arrows, replacing injective objects with projective ones, and so on. Suppose that A is an abelian category with enough injectives and F a left exact...

lemma: Lemma — If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n, H n...

submodule of the codomain. An injective hull of an object A is an essential monomorphism from A to an injective object. Hashimoto, Mitsuyasu (November...

then f {\displaystyle f} is bijective. In fact, f {\displaystyle f} is injective, as f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x = g ( f ( x...

short map) on a bounded injective space has a fixed point. A metric space is injective if and only if it is an injective object in the category of metric...

Ring are the injective epimorphisms. The inclusion Z → Q is an example of a bimorphism which is not an isomorphism. The only injective object in Ring up...

enough injectives, F {\displaystyle F} a left-exact functor, and G {\displaystyle G} sending injective objects to F {\displaystyle F} -acyclic objects, then...

Hopfian objects and cohopfian objects have an elementary interaction with projective objects and injective objects. The two results are: An injective hopfian...

p are injective and l is surjective. Let c in C be such that n(c) = 0. t(n(c)) is then 0. By commutativity, p(h(c)) = 0. Since p is injective, h(c) =...

category with enough injectives is the least non-negativer integer n such that every object in the category admits an injective resolution of length at...

Object-oriented analysis and design (OOAD) is a technical approach for analyzing and designing an application, system, or business by applying object-oriented...

The distinction between subject and object is a basic idea of philosophy. A subject is a being that exercises agency, undergoes conscious experiences,...

faithful functor that is injective on objects. Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F is...