HOME

TheInfoList



OR:

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of
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. Specifically, if ''Q'' is a submodule ...
. This concept is important in
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
, in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
and in the theory of
model categories In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' sati ...
. The dual notion is that of a
projective object In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object ...
.


Definition

An object Q in a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
\mathbf is said to be injective if for every monomorphism f: X \to Y and every
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
g: X \to Q there exists a morphism h: Y \to Q extending g to Y, i.e. such that h \circ f = g. That is, every morphism X \to Q factors through every monomorphism X \hookrightarrow Y. The morphism h in the above definition is not required to be uniquely determined by f and g. In a locally small category, it is equivalent to require that the
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 ...
\operatorname_(-,Q) carries monomorphisms in \mathbf to
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
set maps.


In Abelian categories

The notion of injectivity was first formulated for abelian categories, and this is still one of its primary areas of application. When \mathbf is an abelian category, an object ''Q'' of \mathbf is injective
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
its
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 ...
HomC(–,''Q'') is exact. If 0 \to Q \to U \to V \to 0 is an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...
in \mathbf such that ''Q'' is injective, then the sequence splits.


Enough injectives and injective hulls

The category \mathbf is said to ''have enough injectives'' if for every object ''X'' of \mathbf, there exists a monomorphism from ''X'' to an injective object. A monomorphism ''g'' in \mathbf is called an
essential monomorphism In mathematics, specifically category theory, an essential monomorphism is a monomorphism ''f'' in a category ''C'' such that for a morphism ''g'' in ''C'', the morphism g \circ f is a monomorphism only when ''g'' is a monomorphism. Essential mon ...
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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
a non-canonical isomorphism.


Examples

*In the category of
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is com ...
s 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 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. Specifically, if ''Q'' is a submodule ...
. ''R''-Mod has injective hulls (as a consequence, ''R''-Mod has enough injectives). *In the category of metric spaces, 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 mapping In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
s, an injective object is always a Scott topology on a continuous lattice, and therefore it is always sober and
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
.


Uses

If an abelian category has enough injectives, we can form injective resolutions, i.e. for a given object ''X'' we can form a long exact sequence :0\to X \to Q^0 \to Q^1 \to Q^2 \to \cdots and one can then define the
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in var ...
s of a given functor ''F'' by applying ''F'' to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
theories in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
and algebraic geometry. The categories being used are typically functor categories or categories of sheaves of ''O''''X'' modules over some
ringed space In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
(''X'', ''O''''X'') or, more generally, any Grothendieck category.


Generalization

Let \mathbf be a category and let \mathcal be a class of morphisms of \mathbf. An object Q of \mathbf is said to be ''\mathcal''-injective if for every morphism f: A \to Q and every morphism h: A \to B in \mathcal there exists a morphism g: B \to Q with g \circ h = f. If \mathcal is the class of monomorphisms, we are back to the injective objects that were treated above. The category \mathbf is said to ''have enough \mathcal-injectives'' if for every object ''X'' of \mathbf, there exists an ''\mathcal''-morphism from ''X'' to an ''\mathcal''-injective object. A ''\mathcal''-morphism ''g'' in \mathbf is called ''\mathcal''-essential if for any morphism ''f'', the composite ''fg'' is in ''\mathcal'' only if ''f'' is in ''\mathcal''. If ''g'' is a ''\mathcal''-essential morphism with domain ''X'' and an ''\mathcal''-injective codomain ''G'', then ''G'' is called an \mathcal-injective hull of ''X''.


Examples of -injective objects

*In the category of
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s, the injective objects with respect to the class ''\mathcal'' of anodyne extensions are
Kan complex In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
es. *In the category of
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
s and
monotone map In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...
s, the
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...
s form the injective objects for the class ''\mathcal'' of order-embeddings, and the
Dedekind–MacNeille completion In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and construc ...
of a partially ordered set is its ''\mathcal''-injective hull.


See also

*
Projective object In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object ...


Notes

{{reflist


References

*Jiri Adamek, Horst Herrlich, George Strecker. Abstract and concrete categories: The joy of cats, Chapter 9, Injective Objects and Essential Embeddings
Republished in Reprints and Applications of Categories, No. 17 (2006) pp. 1-507
Wiley (1990). *J. Rosicky, Injectivity and accessible categories *F. Cagliari and S. Montovani, T0-reflection and injective hulls of fibre spaces Category theory de:Injektiver Modul#Injektive Moduln