In
mathematics, especially in the field of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the concept of injective object is a generalization of the concept of
injective module. 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 and in the theory of
model categories. The dual notion is that of a
projective object.
Definition
An
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
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)
*C ...
is said to be injective if for every
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
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, morphisms ...
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
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 ...
carries monomorphisms in
to
surjective 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
is an abelian category, an object ''Q'' of
is injective
if and only if 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 ...
Hom
C(–,''Q'') is
exact.
If
is an
exact sequence in
such that ''Q'' is injective, then the
sequence splits.
Enough injectives and injective hulls
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 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 comm ...
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. ''R''-Mod has
injective hulls (as a consequence, ''R''-Mod has enough injectives).
*In the
category of metric spaces In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of tw ...
, 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 valu ...
s, an injective object is always a
Scott topology
Scott may refer to:
Places Canada
* Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec
* Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380
* Rural Municipality of Scott No. 98, Sask ...
on a
continuous lattice
In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it.
Definitions
Let a,b\in P be two elements of a preordered set (P,\lesssim). Then we say that a approximat ...
, 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 ...
.
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
:
and one can then define the
derived functors 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
Ext, ext or EXT may refer to:
* Ext functor, used in the mathematical field of homological algebra
* Ext (JavaScript library), a programming library used to build interactive web applications
* Exeter Airport (IATA airport code), in Devon, England
...
, 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 groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
,
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 classify ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
. 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 In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves ...
.
Generalization
Let
be a category and let
be a
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
of morphisms of
.
An object
of
is said to be ''
''-injective if for every morphism
and every morphism
in
there exists a morphism
with
.
If
is the class of
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
s, we are back to the injective objects that were treated above.
The category
is said to ''have enough
-injectives'' if for every object ''X'' of
, there exists an ''
''-morphism from ''X'' to an ''
''-injective object.
A ''
''-morphism ''g'' in
is called ''
''-essential if for any morphism ''f'', the composite ''fg'' is in ''
'' only if ''f'' is in ''
''.
If ''g'' is a ''
''-essential morphism with domain ''X'' and an ''
''-injective codomain ''G'', then ''G'' is called an
-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 ''
'' of anodyne extensions are
Kan complexes.
*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 binary ...
s and
monotone maps, the
complete lattices form the injective objects for the class ''
'' of
order-embedding In order theory, a branch of mathematics, 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 stric ...
s, and the
Dedekind–MacNeille completion of a partially ordered set is its ''
''-injective hull.
See also
*
Projective 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, T
0-reflection and injective hulls of fibre spaces
Category theory
de:Injektiver Modul#Injektive Moduln