Image:Chain homotopy.svg

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Description

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

f^n - g^n = d_B^{n - 1} h^n + h^{n + 1} d_A^n, or simply f − g = dBh + hdA.

This can be depicted as shown in the diagram.
Source

en:Image:Chain homotopy.jpg
Date

2007-03-19, 2008-02-06
Author

en:User:Ryan Reich, User:Stannered
Permission
(Reusing this image)
Public domain This image has been (or is hereby) released into the public domain by its author, Ryan Reich at the English Wikipedia project. This applies worldwide.

In case this is not legally possible:
Ryan Reich grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
Other versions en:Image:Chain homotopy.jpg

File history

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Date/TimeDimensionsUserComment
current14:04, 6 February 2008795×208 (49 KB)Stannered ({{Information |Description=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'')
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