Topological half-exact functor
Encyclopedia
In mathematics
, a topological half-exact functor F is a functor
from a fixed topological category
(for example CW complexes or pointed space
s) to an abelian category (most frequently in applications, category of abelian groups or category of modules
over a fixed ring) that has a following property: for each sequence of the form:
where C(f) denotes a mapping cone
, a sequence:
is exact.
Examples include functors of homology
.
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...
, a topological half-exact functor F is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from a fixed topological category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
(for example CW complexes or pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
s) to an abelian category (most frequently in applications, category of abelian groups or category of modules
Module (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...
over a fixed ring) that has a following property: for each sequence of the form:
- X → Y → C(f)
where C(f) denotes a mapping cone
Mapping cone
In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf.-Definition:...
, a sequence:
- F(X) → F(Y) → F(C(f))
is exact.
Examples include functors of homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
.