homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure...

a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure. In geometry...

mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept...

up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems...

operads in algebra and algebraic topology, an A∞-operad is a parameter space for a multiplication map that is homotopy coherently associative. (An operad...

mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology...

an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are...

introduced another approach to constructing algebraic K-theory under the name of Γ-objects. Segal's approach is a homotopy analog of Grothendieck's construction...

mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained...

topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored...

In algebraic geometry and algebraic topology, branches of mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of...

which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring". The most familiar example of a ring is...

\pi _{n}(B)\to \cdots } Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle C (...

{\displaystyle [x,y]} . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same...

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....

goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

in defining the counting of holomorphic disks. Homotopy associative algebra Kenji Fukaya, Morse homotopy, A ∞ {\displaystyle A_{\infty }} category and...

The algebras over the associative operad are precisely the semigroups: sets together with a single binary associative operation. The k-linear algebras over...

(that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from simplices and...

Quillen as an axiomatization of homotopy theory that applies to topological spaces, but also to many other categories in algebra and geometry. The example that...

its development. Briefly, Noether subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic...

"product" Production (disambiguation) This disambiguation page lists articles associated with the title Product. If an internal link led you here, you may wish...

Suspension functor Stable homotopy theory Spectrum (homotopy theory) Morava K-theory Hodge conjecture Weil conjectures Directed algebraic topology Example: DE-9IM...

These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called stable groups...

{\displaystyle U(1)} -action on the dual Lie algebra is trivial. The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically...

Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory)...

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental...

the homotopy groups stabilize, and πk(O(n + 1)) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups...

in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or...