Editing Homotopical algebra (section)

{{Short description|Branch of mathematics}}
{{inline |date=May 2024}}
In [[mathematics]], '''homotopical algebra''' is a collection of concepts comprising the ''nonabelian'' aspects of [[homological algebra]], and possibly the [[abelian category|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 [[Higher-dimensional algebra#Nonabelian algebraic topology|nonabelian algebraic topology]], and in particular the theory of [[closed model category|closed model categories]].

This subject has received much attention in recent years due to new foundational work of [[Vladimir Voevodsky]], [[Eric Friedlander]], [[Andrei Suslin]], and others resulting in the [[A1 homotopy theory|'''A'''<sup>1</sup> homotopy theory]] for [[quasiprojective variety|quasiprojective varieties]] over a [[field (mathematics)|field]].  Voevodsky has used this new algebraic homotopy theory to prove the [[Milnor conjecture (K-theory)|Milnor conjecture]] (for which he was awarded the [[Fields Medal]]) and later, in collaboration with [[Markus Rost]], the full [[Bloch–Kato conjecture]].