Categories

http://mathcs.chapman.edu/structuresold/files/Categories.pdf

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\begin{document}
\textbf{\Large Categories}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Categories}{edit}
% Note: replace "Template" with Name_of_class in previous line

\abbreviation{Cat}

\begin{definition}
A \emph{category} is a structure $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ of type $\langle 2,1,1\rangle$ such that
$C$ is a class,

$\langle C,\circ\rangle$ is a (large) \href{Partial_semigroups.pdf}{partial semigroup}

$\text{dom}(x)$ is a left unit:  $\text{dom}(x)\circ x=x$

$\text{cod}(x)$ is a right unit:  $x\circ\text{cod}(x)=x$

$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$

$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$

if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$

$x\circ y$ exists iff $\text{cod}(x)=\text{dom}(y)$

Remark: The members of $C$ are called \emph{morphisms}, $\circ$ is the partial operation of \emph{composition},
dom is the \emph{domain} and cod is the \emph{codomain} of a morphism.

The set of objects of $C$ is the set $\mathbf{Obj}C=\{\text{dom}(x)|x\in C\}$. For $a,b\in C$ the set of homomorphism from $a$ to $b$ is
$\text{Hom}(a,b)=\{c\in C|\text{dom}(c)=a\text{ and }\text{cod}(c)=b\}$.
\end{definition}

\begin{morphisms}
Let $\mathbf{C}$ and $\mathbf{D}$ be categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a homomorphism: 
$h(\text{dom}(c))=\text{dom}h(c)$ and
$h(c\circ d)=h(c) \circ h(d)$ whenever $c\circ d$ is defined.
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
The category of function on sets with composition.
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
  Classtype                       & (value, see description) \cite{Ln19xx} \\\hline
  Equational theory               & \\\hline
  Quasiequational theory          & \\\hline
  First-order theory              & \\\hline
  Locally finite                  & \\\hline
  Residual size                   & \\\hline
  Congruence distributive         & \\\hline
  Congruence modular              & \\\hline
  Congruence $n$-permutable       & \\\hline
  Congruence regular              & \\\hline
  Congruence uniform              & \\\hline
  Congruence extension property   & \\\hline
  Definable principal congruences & \\\hline
  Equationally def. pr. cong.     & \\\hline
  Amalgamation property           & \\\hline
  Strong amalgamation property    & \\\hline
  Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &\\
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  f(4)= &\\
  f(5)= &\\
\end{array}$\qquad
$\begin{array}{lr}
  f(6)= &\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$

\end{finite_members}

\begin{subclasses}\ 

  \href{....pdf}{...} subvariety

  \href{....pdf}{...} expansion

\end{subclasses}

\begin{superclasses}\ 

  \href{....pdf}{...} supervariety

  \href{....pdf}{...} subreduct

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview} 

\end{thebibliography}

\end{document}
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