In category theory, an object Q is said to be injective if every arrow to Q can be pushed forward across monomorphisms. That is, Q is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...

In the category of modules and module homomorphisms, an injective object is an injective module. In the category of metric spaces and nonexpansive mappings, an injective object is an injective metric space. 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 some ringed space (X,O_X). In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a short map, or nonexpansive map, is a special kind of continuous function between metric spaces. ... In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a ringed space is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...