File:Chain homotopy.svg

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Chain_homotopy.svg‎ (SVG file, nominally 403 × 106 pixels, file size: 49 KB)

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[edit] LaTeX source

\documentclass{amsart}
\usepackage{amsmath,amssymb,nopageno}
\usepackage[all]{xy}
\begin{document}
\begin{equation*}
\xymatrix@+3em{
{\dots} \ar[r]^{d_A^{n - 2}}
	& A^{n - 1}
		\ar[r]^{d_A^{n - 1}}
		\ar@<0.5ex>[d]^{g^{n - 1}}
		\ar@<-0.5ex>[d]_{f^{n - 1}}
		\ar[dl]|*+<1ex,1ex>{\scriptstyle h^{n - 1}}
	& A^n
		\ar[r]^{d_A^n}
		\ar@<0.5ex>[d]^{g^n}
		\ar@<-0.5ex>[d]_{f^n}
		\ar[dl]|*+<1ex,1ex>{\scriptstyle h^n}
	& A^{n + 1}
		\ar[r]^{d_A^{n + 1}}
		\ar@<0.5ex>[d]^{g^{n + 1}}
		\ar@<-0.5ex>[d]_{f^{n + 1}}
		\ar[dl]|*+<1ex,1ex>{\scriptstyle h^{n + 1}}
	& {\dots}
		\ar[dl]|*+<1ex,1ex>{\scriptstyle h^{n + 2}}\\
{\dots} \ar[r]^{d_B^{n - 2}}
	& B^{n - 1} \ar[r]^{d_B^{n - 1}}
	& B^n \ar[r]^{d_B^n}
	& B^{n + 1} \ar[r]^{d_B^{n + 1}}
	& {\dots}
}
\end{equation*}
\end{document}

[edit] Summary

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 = d B h + h d A .

This can be depicted as shown in the diagram.

Date

2007-03-19, 2008-02-06

Source

en:Image:Chain homotopy.jpg

Author

en:User:Ryan Reich, User:Stannered

Permission
(Reusing this image)

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/Time	Thumbnail	Dimensions	User	Comment
current	20:29, 13 January 2009		403×106 (49 KB)	Ryan Reich	({{Information \|Description=A depiction of a homotopy of two maps of chain complexes \|Source=Created it myself \|Date=01-13-2009 \|Author=~~~ \|Permission=See below \|other_versions= }})
	14:04, 6 February 2008		795×208 (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'')