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In homological algebra in 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. It lies intermediate between the category of chain complexes Kom(A) of A and the derived category D(A) of A when A is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) makes isomorphisms of 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).
## Definitions

Let A be an additive category. 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 $h^n\; colon\; A^n\; to\; B^\{n\; -\; 1\}$ (not a map of complexes) such that
## Remarks

Two chain maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general). This shows that there is a canonical functor $K(A)\; rightarrow\; D(A)$ to the derived category (if A is abelian).
## The triangulated structure

The shift A[1] of a complex A is the following complex
^{+}(A), K^{-}(A) and K^{b}(A). Although triangles make sense in Kom(A) as well, that category is not triangulated with respect to these distinguished triangles; for example,
## Generalization

More generally, the homotopy category Ho C of a differential graded category C is defined to have the same objects as C, but morphisms are defined by
$Hom\_\{Ho\; C\}(X,\; Y)\; =\; H^0\; Hom\_C\; (X,\; Y)$. (This boils down to the homotopy of chain complexes if C is the category of complexes whose morphisms do not have to respect the differentials). If C has cones and shifts in a suitable sense, then Ho C is a triangulated category, too.
## References

- $f^n\; -\; g^n\; =\; d\_B^\{n\; -\; 1\}\; h^n\; +\; h^\{n\; +\; 1\}\; d\_A^n,$ or simply $f\; -\; g\; =\; d\_B\; h\; +\; h\; d\_A.$

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

- $f\; sim\; g$ if f is homotopic to g

- $operatorname\{Hom\}\_\{K(A)\}(A,\; B)\; =\; operatorname\{Hom\}\_\{Kom(A)\}(A,B)/sim$

The following variants of the definition are also widely used: if one takes only bounded-below (A^{n}=0 for n<<0), bounded-above (A^{n}=0 for n>>0), or bounded (A^{n}=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 K^{b}(A), respectively.

A morphism $f\; :\; A\; rightarrow\; B$ which is an isomorphism in K(A) is called a homotopy equivalence. In detail, this means there is another map $g\; :\; B\; rightarrow\; A$, such that the two compositions are homotopic to the identities: $f\; circ\; g\; sim\; Id\_B$ and $g\; circ\; f\; sim\; Id\_A$.

The name "homotopy" comes from the fact that homotopic maps of topological spaces induce homotopic (in the above sense) maps of singular chains.

- $A[1]:\; ...\; to\; A^\{n+1\}\; xrightarrow\{d\_\{A[1]\}^n\}\; A^\{n+2\}\; to\; ...$ (note that $(A[1])^n\; =\; A^\{n\; +\; 1\}$),

For the cone of a morphism f we take the mapping cone. There are natural maps

- $A\; xrightarrow\{f\}\; B\; to\; C(f)\; to\; A[1]$

- $X\; xrightarrow\{id\}\; X\; to\; 0\; to$

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Last updated on Wednesday February 06, 2008 at 07:04:56 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday February 06, 2008 at 07:04:56 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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