Category 2009-04-06 14:27:19 2009-04-06 15:45:05 Definition metagraph metacategory unit law associativity axioms identity composition operations graph dom cod category theory metagraph metacategory category unit law associativity axioms identity composition operations graph dom cod % 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}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\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}} The concept of category emerged in 1943-1945 from work in Algebraic Topology and Homological Algebra by S. Eilenberg and S. Mac Lane, as a generalization of the algebraic concepts of group, semigroup, groupoid, as well as of the topological concepts and diagrams employed in homological algebra. Thus many properties of mathematical systems can be unified by a presentation with diagrams of arrows that may represent functions, transformations, distributions, operators, etc., and that-- in the case of concrete categories-- may also include objects such as class elements, sets, topological spaces, etc. ; the usefulness of such diagrams comes from the composition of the arrows and the (fundamental) axioms that define any category which allow mathematical constructions to be represented by universal properties of diagrams. To introduce the modern concept of category, according to S. MacLane \cite{MacLane2000} without using any set theory, one needs to introduce first the notions of {\em metagraph} and {\em metacategory}. \begin{definition} A concrete \emph{metagraph} $\mathcal{M}_G$ consists of objects, $A, B, C,$... and arrows $f, g, h,$... between objects, and two operations as follows: \begin{itemize} \item a {\em Domain operation}, $dom$, which assigns to each arrow $f$ an object $A~ =~dom ~f$ \item a {\em Codomain operation}, $cod$, which assigns to each arrow $f$ an object $B~ = ~cod ~f,$ represented as $f: A \to B$ or $A \stackrel{f}{\longrightarrow} B$ \end{itemize} \end{definition} \begin{definition} A \emph{metacategory} $\mathbb{C}$ is a metagraph with two additional operations: \begin{itemize} \item {\em Identity}, $id$ or {\bf 1}, which assigns to each object $A$ a unique arrow $id_A$, or $1_A$; \item {\em Composition}, $\circ$, which assigns to each pair of arrows $<g,f>$ with $dom~ g = cod~ f$ a unique arrow $g \circ f$ called their {\em composite}, such that $g \circ f : dom f \to cod g,$ \end{itemize} that are subject to two axioms: \begin{itemize} \item {\em c1. (Unit law)}: for all arrows $f: A \to B$ and $g:B \to C$ the composition with the identity arrow $1_B$ results in $ 1_B \circ f = f$ and $g \circ 1_B = g ;$ \item {\em c2. Associativity}: for given objects and arrows in the sequence: $$A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \stackrel{h}{\longrightarrow} D , $$ one always the equality $$ h \circ(g \circ f) = (h \circ g) \circ f , $$ whenever the composition $\circ$ is defined. \end{itemize} \end{definition} \begin{definition} A \emph{category} $\mathcal{C}$ is an interpretation of a metacategory within set theory. Thus, a {\em category} is a {\em graph} defined by a set $Ob \mathcal{C}:=\mathbb{O}$, a set of arrows (called also {\em morphisms}) $Mor\mathcal{C}:= \mathbb{A}$, and two functions: $$ dom: Mor \mathcal{C} \to Ob \mathcal{C}$$ and $$cod: Mor\mathcal{C} \to Ob \mathcal{C}$$ with two additional functions: $$id: Ob \mathcal{C} \to Mor \mathcal{C}$$ defined by the assignments $\mathbb{A} \times_O \mathbb{A} \longarrow \mathbb{A}$ called {\em identity}, and a composition $c := \circ $,that is $ c \to id_c$, defined by the assignments $(g,f) \longarrow g \circ f$, such that $$ dom(id_A) = A = cod(id_A), dom(g \circ g) = domf, cod (g \circ f)= codg,$$ for all objects $A \in Ob \mathcal{C}$ and all composable pairs of arrows (morphisms) $(g,f) \in \mathbb{A} \times_O \mathbb{A}, $ and also such that the unit law and associativity axioms {\em c1} and {\em c2} hold. \end{definition} For convenience one also defines a $Hom$ (or $hom$) set as: $$Hom(B,C) = [f|f \in \mathcal{C}, dom f= B, cod f = C]$$ \subsection{Alternative definitions} There are several alternative definitions of a category. Thus, as defined by W.F. Lawvere, a {\em category} is an interpretation of the ETAC axioms of the elementary theory of abstract categories. For small categories-- whose $Ob \mathcal{C}$ is a set and also $Mor\mathcal{C}$ is a set-- one has a \PMlinkexternal{\em dirtect definition}{http://planetmath.org/encyclopedia/AlternativeDefinitionOfSmallCategory.html} \subsection{Applications in Physics and Mathematical Biophysics} A `categorification' of theoretical physics (including quantum field theories) began as early as 1968 \cite{Baianu-Marinescu68,Baianu1971}, whereas categories of sets were introduced in mathematical biophysics in 1958 \cite{Rosen58a,Rosen58b}, followed by the introduction of biotheoretical models in categories with structure in 1968-1971 \cite{Baianu-Marinescu68,Baianu70, Baianu71, Baianu70}. The `categorification' process in physics continues today, especially after 1985 (\cite{Baianu87} and references cited therein).