Injective object
Encyclopedia
In mathematics
, especially in the field of category theory
, the concept of injective object is a generalization of the concept of injective module
. This concept is important in homotopy theory and in theory of model categories
. The dual notion is that of a projective object.
be a category and let 
be a class of morphisms of 
.
An object
of 
is said to be 
-injective if for every arrow 
and every morphism 
in 
there exists a morphism 
extending (the domain of) 
, i.e 
. In other words, 
is injective iff any 
-morphism extends (via composition on the left) to any morphism into 
.
The morphism
in the above definition is not required to be uniquely determined by 
.
In a locally small category, it is equivalent to require that the hom functor
carries 
-morphisms to epimorphisms (surjections).
The classical choice for
is the class of monomorphism
s, in this case, the expression injective object is used.
is an abelian category
, an object A of
is injective iff its hom functor
HomC(–,A) is exact
.
The abelian case was the original framework for the notion of injectivity.
be a category, H a class of morphisms of 
; the category 
is said to have enough H-injectives if for every object X of 
, there exist a H-morphism from X to an H-injective object.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the field of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, the concept of injective object is a generalization of the concept of injective module
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
. This concept is important in homotopy theory and in theory of model categories
Model category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
. The dual notion is that of a projective object.
General Definition
Let be a category and let be a class of morphisms of .An object
of is said to be -injective if for every arrow and every morphism in there exists a morphism extending (the domain of) , i.e . In other words, is injective iff any -morphism extends (via composition on the left) to any morphism into .The morphism
in the above definition is not required to be uniquely determined by .In a locally small category, it is equivalent to require that the hom functor
Hom functor
In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
carries -morphisms to epimorphisms (surjections).The classical choice for
is the class of monomorphismMonomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
s, in this case, the expression injective object is used.
Abelian case
If is an abelian categoryAbelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
, an object A of
is injective iff its hom functorHom functor
In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
HomC(–,A) is exact
Exact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
.
The abelian case was the original framework for the notion of injectivity.
Enough injectives
Let be a category, H a class of morphisms of ; the category is said to have enough H-injectives if for every object X of , there exist a H-morphism from X to an H-injective object.Injective hull
A H-morphism g in 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 groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s and group homomorphismGroup homomorphismIn mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s, an injective object is a divisible groupDivisible groupIn mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n...
. - In the category of modulesModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
and module homomorphisms, R-Mod, an injective object is an injective moduleInjective moduleIn mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
. R-Mod has injective hullInjective hullIn mathematics, especially in the area of abstract algebra known as module theory, the injective hull of a module is both the smallest injective module containing it and the largest essential extension of it...
s (as a consequence, R-Mod has enough injectives). - In the category of metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and nonexpansive mappings, MetCategory of metric spacesIn category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
, an injective object is an injective metric spaceInjective metric spaceIn 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...
, and the injective hull of a metric space is its tight spanTight spanIn metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes...
. - In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice therefore it is always soberSober spaceIn mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
and locally compact. - In the category of simplicial setSimplicial setIn mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
s, the injective objects with respect to the class of anodyne extensions are Kan complexes. - In the category of partially ordered sets and monotonic functions between posets, the complete latticeComplete latticeIn mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
s form the injective objects for order-embeddingOrder-embeddingIn mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order-embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism...
s, and the Dedekind–MacNeille completionDedekind–MacNeille completionIn order-theoretic mathematics, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains the given partial order...
of a partially ordered set is its injective hull. - One also talks about injective objects in more general categories, for instance in functor categoriesFunctor categoryIn category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
or in categories of sheavesSheaf (mathematics)In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of OX modules over some ringed spaceRinged spaceIn mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
(X,OX).