Homotopical algebra
Encyclopedia
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 and in particular the theory of closed model categories.
This subject has received much attention in recent years due to new foundational work of Voevodsky
, Friedlander, Suslin
, 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
) and later, in collaboration with M. Rost
, the full Bloch-Kato conjecture.
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...
, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
as well as possibly the abelian
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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
aspects as special cases. The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory and in particular the theory of closed model categories.
This subject has received much attention in recent years due to new foundational work of Voevodsky
Vladimir Voevodsky
Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...
, Friedlander, Suslin
Andrei Suslin
Andrei Suslin is a Russian mathematician who has made major contributions to the field of algebra, especially algebraic K-theory and its connections with algebraic geometry. He is currently a Trustee Chair and Professor of mathematics at Northwestern University.He was born on December 27, 1950,...
, and others resulting in the A1 homotopy theory for quasiprojective varieties
Quasiprojective variety
In mathematics, a quasiprojective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. Voevodsky has used this new algebraic homotopy theory to prove the Milnor conjecture
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by .-Statement of the theorem:Let F be a field of...
(for which he was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
) and later, in collaboration with M. Rost
Markus Rost
Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China...
, the full Bloch-Kato conjecture.