File:Chain homotopy.svg

[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 (not a map of complexes) such that or simply 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 file)	Public domainPublic domainfalsefalse This work has been 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. العربية | Català | Česky | Dansk | Deutsch | Ελληνικά | English | Español | Eesti | فارسی | Suomi | Français | עברית | Magyar | Հայերեն | Italiano | 日本語 | 한국어 | Lietuvių | Македонски | മലയാളം | Plattdüütsch | Nederlands | Polski | Português | Português do Brasil | Română | Русский | Српски / Srpski | Svenska | ไทย | Vèneto | Tiếng Việt | 中文 | ‪中文(简体)‬ | ‪中文(繁體)‬ | +/−
Other versions	en:Image:Chain homotopy.jpg

Click on a date/time to view the file as it appeared at that time.

	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''

• Homotopy category of chain complexes