Mathematical Structures: Categories

# Categories

Difference (from prior major revision) (author diff)

Changed: 55c55
 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$is  The set of objects of$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$is http://mathcs.chapman.edu/structuresold/files/Categories.pdf %%run pdflatex %  \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \hyperbaseurl{http://math.chapman.edu/structures/files/} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Categories} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Categories}{edit} % Note: replace "Template" with Name_of_class in previous line \abbreviation{Cat} \begin{definition} A \emph{category} is a structure$\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)= &\\
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}
%