Homotopy category of chain complexes information

mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences...

manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies. Chain complexes of K-modules with chain maps form a category ChK, where...

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the...

are homotopy equivalent to each other, i.e. isomorphic in the homotopy category. Moreover, morphisms of complexes extend uniquely to a morphism of two...

singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homotopy equivalent...

relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by Daniel...

meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains...

triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short...

category if it satisfies the following axioms, motivated by the similar properties for the notions of cofibrations and weak homotopy equivalences of topological...

corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Simplicial sets are used to define quasi-categories, a basic...

generalization of chain complexes, the homology groups of the chain complex becoming the homotopy groups of the “non-commutative chain complex” or stack....

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely...

the derived category are obtained from the homotopy category of chain complexes of sheaves by taking the zeroth cohomology of the complex, i.e. [ R p...

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...

class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the...

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

Grassmannian. chain homotopy Given chain maps f , g : ( C , d C ) → ( D , d D ) {\displaystyle f,g:(C,d_{C})\to (D,d_{D})} between chain complexes of modules...

In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with...

the localization of the category of chain complexes (up to homotopy) with respect to the quasi-isomorphisms. Given an abelian category A and a Serre subcategory...

mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure...

Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topology) Betti number Euler characteristic...

an Abelian category with enough projectives. If we let C + ( A ) {\displaystyle C_{+}({\mathcal {A}})} be the category of chain complexes which are 0...

homotopy equivalence from a CW complex, this axiom reduces homology or cohomology theories on all spaces to the corresponding theory on CW complexes....