In
mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of
homological algebra as well as possibly the
abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a common approach to such generalizations is via
abstract homotopy theory, as in
nonabelian algebraic topology In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras.
Many of the higher-dimensional algebraic structures are noncommutative and, therefore, thei ...
, and in particular the theory of
closed model categories.
This subject has received much attention in recent years due to new foundational work of
Vladimir Voevodsky
Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic va ...
,
Eric Friedlander
Eric Mark Friedlander (born January 7, 1944 in Santurce, Puerto Rico) is an American mathematician who is working in algebraic topology, algebraic geometry, algebraic K-theory and representation theory.
Friedlander graduated from Swarthmore ...
,
Andrei Suslin
Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee ...
, and others resulting in the
A1 homotopy theory for
quasiprojective varieties over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. Voevodsky has used this new algebraic homotopy theory to prove the
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' wi ...
(for which he was awarded the
Fields Medal) and later, in collaboration with
Markus Rost, the full
Bloch–Kato conjecture.
References
*
*
*
See also
*
Derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutati ...
*
Derivator
*
Cotangent complex - one of the first objects discovered using homotopical algebra
*
L∞ Algebra
*
A∞ Algebra
*
Categorical algebra
*
Nonabelian homological algebra
Non-abelian or nonabelian may refer to:
* Non-abelian group, in mathematics, a group that is not abelian (commutative)
* Non-abelian gauge theory, in physics, a gauge group that is non-abelian
See also
* Non-abelian gauge transformation, a gaug ...
External links
An abstract for a talk on the proof of the full Bloch–Kato conjecture
Algebraic topology
Topological methods of algebraic geometry
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