In the mathematical disciplines of
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
homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given
diagram for a particular concept and reversing the direction of all arrows, much as in
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 ...
with the idea of the
opposite category. A significantly deeper form argues that the fact that the dual notion of a
limit
Limit or Limits may refer to:
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* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
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is a
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions suc ...
allows us to change the
Eilenberg–Steenrod axioms for
homology to give axioms for
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 ...
. It is named after
Beno Eckmann
Beno Eckmann (31 March 1917 – 25 November 2008) was a Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry.
Life
Born in Bern, Eckmann received his master's degree from ...
and
Peter Hilton
Peter John Hilton (7 April 1923Peter Hilton, "On all Sorts of Automorphisms", ''The American Mathematical Monthly'', 92(9), November 1985, p. 6506 November 2010) was a British mathematician, noted for his contributions to homotopy theory and ...
.
Discussion
An example is given by
currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f th ...
, which tells us that for any object
, a map
is the same as a map
, where
is the
exponential object, given by all maps from
to
. In the case of
topological spaces, if we take
to be the unit interval, this leads to a duality between
and
, which then gives a duality between the
reduced suspension , which is a quotient of
, and the
loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topol ...
, which is a subspace of
. This then leads to the
adjoint relation , which allows the study of
spectra, which give rise to
cohomology theories.
We can also directly relate
fibrations and
cofibration In mathematics, in particular homotopy theory, a continuous mapping
:i: A \to X,
where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
s: a fibration
is defined by having the
homotopy lifting property, represented by the following diagram
and a cofibration
is defined by having the dual
homotopy extension property In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual ...
, represented by dualising the previous diagram:
The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration
we get the sequence
:
and given a cofibration
we get the sequence
:
and more generally, the duality between the exact and coexact
Puppe sequences.
This also allows us to relate
homotopy and cohomology: we know that
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s are
homotopy classes of maps from the
''n''-sphere to our space, written
, and we know that the sphere has a single nonzero (reduced)
cohomology group
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 ...
. On the other hand, cohomology groups are homotopy classes of maps to spaces with a single nonzero homotopy group. This is given by the
Eilenberg–MacLane spaces
and the relation
:
A formalization of the above informal relationships is given by
Fuks duality.
See also
*
Model category
References
*
*
{{DEFAULTSORT:Eckmann-Hilton Duality
Duality theories
Algebraic topology