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'category example (arrow category)'
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| Title of object: |
category example (arrow category) |
| Canonical Name: |
ExampleOfCategory |
| Type: |
Example |
| Created on: |
2002-06-01 08:56:32-04 |
| Modified on: |
2002-06-02 07:00:08-04 |
| Classification: |
msc:18A05 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
| Let $\mathcal{C}$ be a category, and let $\mathcal{D}$ be the category whose objects are the arrows of $\mathcal{C}$. A morphism between two morphisms $f: A \rightarrow B$ and $g: A' \rightarrow B'$ is defined to be a couple of morphisms $(h,k)$, where $h \in \hom(A,A')$ and $k \in \hom(B,B')$ such that $g \circ h = k \circ f$. The resulting category $\mathcal{D}$ is called the \emph{arrow category} of $\mathcal{C}$. |
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