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Concept# Homotopy category of chain complexes

Summary

In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, K(A) is more understandable than D(A).
Let A be an . The homotopy category K(A) is based on the following definition: if we have complexes A, B and maps f, g from A to B, a chain homotopy from f to g is a collection of maps (not a map of complexes) such that
or simply
This can be depicted as:
We also say that f and g are chain homotopic, or that is null-homotopic or homotopic to 0. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition.
The homotopy category of chain complexes K(A) is then defined as follows: its objects are the same as the objects of Kom(A), namely chain complexes. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation
if f is homotopic to g
and define
to be the quotient by this relation. It is clear that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps.
The following variants of the definition are also widely used: if one takes only bounded-below (An=0 for n0), or bounded (An=0 for |n|>>0) complexes instead of unbounded ones, one speaks of the bounded-below homotopy category etc. They are denoted by K+(A), K−(A) and Kb(A), respectively.
A morphism which is an isomorphism in K(A) is called a homotopy equivalence.

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Homotopy category of chain complexes

In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence.

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In homological algebra, the mapping cone is a 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 and cokernel: if the chain complexes take their terms in an , so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi-isomorphism; if we pass to the of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism.

Triangulated category

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