In
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 ...
, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, and 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, the ...
, 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,
Eric Friedlander,
Andrei Suslin
Andrei Suslin (, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University.
He ...
, and others resulting in the
A1 homotopy theory for
quasiprojective varieties over a
field. Voevodsky has used this new algebraic homotopy theory to prove the
Milnor conjecture (for which he was awarded the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
) and later, in collaboration with
Markus Rost, the full
Bloch–Kato conjecture.
See also
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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 commutative ...
*
Derivator In mathematics, derivators are a proposed frameworkpg 190-195 for homological algebra giving a foundation for both Abelian group, abelian and non-abelian group, non-abelian homological algebra and various generalizations of it. They were introduced ...
*
Cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic obj ...
- one of the first objects discovered using homotopical algebra
*
L∞ Algebra
*
A∞ Algebra
*
Categorical algebra
*
Nonabelian homological algebra
References
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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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