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In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, 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, 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, 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 Markus Rost, the full Bloch–Kato conjecture.
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In mathematics, homotopicalalgebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects...
is always a homotopy equivalence—is null-homotopic. Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant...
Homological algebra. Abstract nonsense, a term for homological algebra and category theory Derivator Homotopicalalgebra List of homological algebra topics...
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical...
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts...
homology of the original space. A much more general concept from homotopicalalgebra, including as special cases both the localization of spaces and of...
G. (1967). Homotopicalalgebra. Berlin: Springer-Verlag. ISBN 978-3-540-03914-3. OCLC 294862881. Peter May, "A Concise Course in Algebraic Topology" :...
invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as...
Categories of abstract algebraic structures including representation theory and universal algebra; Homological algebra; Homotopicalalgebra; Topology using categories...
the study of closed model categories is sometimes thought of as homotopicalalgebra. The definition given initially by Quillen was that of a closed model...
coefficients"... "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopicalalgebra Tame topology Yoga of...
homotopy theory of simplicial sets can be developed using standard homotopicalalgebra methods. Furthermore, the geometric realization and singular functors...
Archived from the original on 2015-04-20. Quillen, Daniel G. (1967). HomotopicalAlgebra. Lecture Notes in Mathematics. Vol. 43. Berlin, New York: Springer-Verlag...
mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept...
constructions are derivatorspg 193 which are a new framework for homotopicalalgebra. The concept of homotopy colimitpg 4-8 is a generalization of homotopy...
information in the algebra. The study of A ∞ {\displaystyle A_{\infty }} -algebras is a subset of homotopicalalgebra, where there is a homotopical notion of associative...
category of sheaves on X {\displaystyle X} using the methods of homotopicalalgebra. Restricted versions of cotangent complexes were first defined in...
mathematical physics, algebraic geometry, Lie groups and Lie algebras, conformal field theory, homological and homotopicalalgebra. In 1969, Feigin graduated...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework...
In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely...
In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information...
the cone construction) and provide at the same time a language for homotopicalalgebra. Derivators were first introduced by Alexander Grothendieck in his...
categories and derived algebraic geometry. Derived algebraic geometry is a way of infusing homotopical methods into algebraic geometry, with two purposes:...
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