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}

Summary

DescriptionChain homotopy.svg	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 ${\displaystyle h^{n}\colon A^{n}\to B^{n-1}}$ (not a map of complexes) such that ${\displaystyle f^{n}-g^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n},}$ or simply ${\displaystyle f-g=d_{B}h+hd_{A}.}$ This can be depicted as shown in the diagram.
Date	19 March 2007, 2008-02-06
Source	Image:Chain homotopy.jpg
Author	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 English Wikipedia. This applies worldwide. In some countries this may not be legally possible; if so: Ryan Reich grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse Category:PD-user#Chain%20homotopy.svg
Other versions	Image:Chain homotopy.jpg
SVG development InfoField	The SVG code is valid.   This diagram was created with an unknown SVG tool. Category:Valid SVG created with Other tools:Diagrams#Chain%20homotopy.svg