%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: superdiagrams as heterofunctors
%%% Primary Category Code: 00.
%%% Filename: SuperdiagramsAsHeterofunctors.tex
%%% Version: 1
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% making logically defined graphics
%\usepackage{xypic}
% define commands here
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym,color,enumerate}
\usepackage{xypic}
\xyoption{curve}
\usepackage[mathscr]{eucal}
%the next gives two direction arrows at the top of a 2 x 2 matrix
\newcommand{\directs}[2]{\def\objectstyle{\scriptstyle} \objectmargin={0pt}
\xy
(0,4)*+{}="a",(0,-2)*+{\rule{0em}{1.5ex}#2}="b",(7,4)*+{\;#1}="c"
\ar@{->} "a";"b" \ar @{->}"a";"c" \endxy }
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{conjecture}{Conjecture}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}
\theoremstyle{plain}
\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{\GL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
\newcommand{\G}{\mathcal G}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\Ce}{\mathsf{C}}
\newcommand{\Q}{\mathsf{Q}}
\newcommand{\grp}{\mathsf{G}}
\newcommand{\dgrp}{\mathsf{D}}
\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{\desp}{{\mathbb D^{\rm{es}}}}
\newcommand{\Geod}{{\rm Geod}}
\newcommand{\geod}{{\rm geod}}
\newcommand{\hgr}{{\mathbb H}}
\newcommand{\mgr}{{\mathbb M}}
\newcommand{\ob}{\operatorname{Ob}}
\newcommand{\obg}{{\rm Ob(\mathbb G)}}
\newcommand{\obgp}{{\rm Ob(\mathbb G')}}
\newcommand{\obh}{{\rm Ob(\mathbb H)}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\ghomotop}{{\rho_2^{\square}}}
\newcommand{\gcalp}{{\mathbb G(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\glob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\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{\oset}[1]{\overset {#1}{\ra}}
\newcommand{\osetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}
\newcommand{\hdgb}{\boldsymbol{\rho}^\square}
\newcommand{\hdg}{\rho^\square_2}
\renewcommand{\leq}{{\leqslant}}
\renewcommand{\geq}{{\geqslant}}
\def\<{\langle}
\def\>{\rangle}
\begin{document}
\begin{definition}
\emph{Superdiagrams} $\Sigma_S$ are defined as heterofunctors $\F_S$ that are subject to \htmladdnormallink{ETAS axioms}{http://planetphysics.us/encyclopedia/ETACAxioms.html} and link \htmladdnormallink{categorical diagrams}{http://planetphysics.us/encyclopedia/CategoricalDiagramsDefinedByFunctors.html} $\Sigma_C$ (regarded as (homo\htmladdnormallink{)functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, which are subject to the eight \htmladdnormallink{ETAC axioms}{http://planetphysics.us/encyclopedia/ETACAxioms.html}) in a manner similar to how \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/EquivalenceRelation.html} are being constructed as \emph{many-object} structures of linked \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} with all invertible \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between the linked groups. Thus, in the supercategory definition--instead of a groupoid with all invertible morphisms-- one replaces the linked groups by several $\Sigma_C$'s linked by hetero-functors $\F_S$ between such categorical diagrams or \htmladdnormallink{categorical sequences}{http://planetphysics.us/encyclopedia/HomologicalSequence2.html} with different structure. The heterofunctors corresponding to
superdiagrams also need not be invertible (as in the case of \emph{supergroupoid} structures). In this construction, one defines a supercategorical \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in terms of the \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} ``$*$'' of the heterofunctors $\F_S$ with the (homo)functors $F_C$ determined by $\Sigma_C$, so that $$\F_S * F_C := \F_S (F_C);$$
the right hand side of this equation is to be interpreted as a heterofunctor acting on the (homo)functor(s) $F_C$ determined by the categorical diagram, or the categorical sequence, $\Sigma_C$.
\end{definition}
\textbf{Remark}
In a certain sense, the superdiagrams defined here as superfunctors resemble also the \htmladdnormallink{groupoid functor}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism.html} \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html}, as well as \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} categories, if one regards the class of links between the different \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of categorical diagrams as a meta-network or \emph{\htmladdnormallink{metagraph}{http://planetphysics.us/encyclopedia/Cod.html}} (in the sense defined by Mac Lane and Moerdijk (2000).
\end{document}
