DoubleCategory 2009-05-16 15:57:26 2009-05-16 23:46:51 Definition internal category in $Cat$ double category % this is the default PlanetPhysics preamble. as your \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} 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%\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \subsection{Background} Charles Ehresmann defined in 1963 a {\em double category} $\mathcal{D}$ as an internal category in the category of small categories $\bf{Cat}$. \begin{definition} A double category $\mathcal{D}$ consists of: \begin{itemize} \item a set of objects, \item a set of horizontal morphisms $$f: A \to B$$, \item a set of vertical morphisms $$j: A \to C$$, and \item a class of squares with source and target as shown in the following diagrams: $$\xymatrix{ {A}\ar[r]^{f}\ar[d]_{j}&{B}\ar[d]^{k}\\ {C}\ar[r]_{g}&{D} }~~~~[\alpha]$$ \end{itemize} with compositions and units of the double category that satisfy the following axioms: \begin{itemize} \item \emph{i.} Horizontal: \[ A\buildrel f_1 \over \longrightarrow B \buildrel f_2 \over \longrightarrow C = [f_1, f_2]= f_2 \circ f_1 \] \[ A\buildrel 1^h_A \over \longrightarrow A \buildrel f_1 \over \longrightarrow B = A\buildrel f_1 \over \longrightarrow B = A \buildrel f_1 \over \longrightarrow B \buildrel 1^h_B \over \longrightarrow B \] \item \emph{ii.} Vertical: \[ [A\buildrel j_1 \over \longrightarrow B \buildrel j_2 \over \longrightarrow C]_{vert} = [j_1, j_2]_{vert.}= j_2 \circ j_1 \] \[ [A\buildrel 1^v_A \over \longrightarrow A \buildrel j_1 \over \longrightarrow B = A\buildrel j_1 \over \longrightarrow B = A \buildrel j_1 \over \longrightarrow B \buildrel 1^v_B \over \longrightarrow B]_{vert.} \] \emph{Compositions for square diagrams in a double category $\mathcal{D}$:} \item \emph{iii.} Horizontal composition: $$\xymatrix{ {A}\ar[r]^{f_1}\ar[d]_{j}&{B}\ar[d]^{k}\\ {D}\ar[r]_{g_1}&{E}}~~~~[\alpha]``\circ'' \xymatrix{ {B}\ar[r]^{f_2}\ar[d]_{k}&{C}\ar[d]^{l}\\ {E}\ar[r]_{g_2}&{F}}~~~~[\beta] = \xymatrix{ {A}\ar[r]^{[f_1f_2]}\ar[d]_{j}&{C}\ar[d]^{l}\\ {D}\ar[r]_{g_1g_2}&{F}} ~~~~[\alpha \beta] $$ \item \emph{iv.} Vertical composition of squares in $\mathcal{D}$: $[\alpha \beta]_{vert.}$ = $$\xymatrix{ {A}\ar[r]^{f}\ar[d]_{[j_1 j_2]_{vert.}}&{B}\ar[d]^{[k_1 k_2]_{vert.}}\\ {E}\ar[r]_{h}&{F}}$$ \end{itemize} \end{definition}