mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a topological half-exact functor ''F'' is a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from a fixed topological
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
(for example
CW complexes
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
or
pointed space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
s) to an
abelian 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 prototypical example of an abelian category is the category o ...
(most frequently in applications, 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 commu ...
s or category of
modules
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computer science and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
...
over a fixed ring) that has a following property: for each sequence of spaces, of the form:
: ''X'' → ''Y'' → ''C(f)''
where ''C(f)'' denotes a
mapping cone Mapping cone may refer to one of the following two different but related concepts in mathematics:
* Mapping cone (topology)
* Mapping cone (homological algebra)
{{mathdab ...
, the sequence:
: ''F(X)'' → ''F(Y)'' → ''F(C(f))''
is exact. If ''F'' is a contravariant functor, it is half-exact if for each sequence of spaces as above,
the sequence ''F(C(f))'' → ''F(Y)'' → ''F(X)'' is exact.
Homology is an example of a half-exact functor, and
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 viewed ...
(and generalized cohomology theories) are examples of contravariant half-exact functors.
If ''B'' is any fibrant topological space, the (representable) functor ''F(X)= ,B' is half-exact.