Mathematical Structures: Categories

# Categories

Difference (from prior author revision) (major diff, minor diff)

Changed: 32c32
 \abbreviation{Abbr}
 \abbreviation{Cat}

Changed: 53c53
 dom is the domain and cod is the codomain of a morphism
 dom is the domain and cod is the codomain of a morphism.

Changed: 55,58c55,56
 This is a template. If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page. It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
 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 $\text{Hom}(a,b)=\{c\in C|\text{dom}(c)=a\text{ and }\text{cod}(c)=b\}$.

Changed: 62,63c60,62
 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 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.

Removed: 66,74d64
 \begin{definition} An ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $...$ is ...: $axiom$ $...$ is ...: $axiom$ \end{definition}

 The category of function on sets with composition.

http://mathcs.chapman.edu/structuresold/files/Categories.pdf
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\begin{document}
\textbf{\Large Categories}
% 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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\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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