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...

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...

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...

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...

areas of mathematics such as: Algebraic geometry (e.g., A1 homotopy theory) Category theory (specifically the study of higher categories) In homotopy theory...

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

respectively obtained from singular chain complexes, Morse complexes, Khovanov complexes, and Hochschild complexes. In other cases, such as for group homology...

construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel...

simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory...