File:Chain homotopy.svg
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Size of this PNG preview of this SVG file: 403 × 106 pixels. Other resolutions: 320 × 84 pixels | 640 × 168 pixels | 1,024 × 269 pixels | 1,280 × 337 pixels | 2,560 × 673 pixels.
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LaTeX source
[edit]\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
[edit]| 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
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| Date | 19 March 2007, 2008-02-06 | ||
| Source | Image:Chain homotopy.jpg | ||
| Author | User:Ryan Reich, User:Stannered | ||
| Permission (Reusing this file) |
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| Other versions | Image:Chain homotopy.jpg | ||
| SVG development |
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 20:29, 13 January 2009 | | 403 × 106 (49 KB) | Ryan Reich (talk | contribs) | {{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 (talk | contribs) | {{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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