ToposAxioms 2009-03-02 08:59:10 2009-03-02 09:19:17 Definition topos axioms of toposes or topoi % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % 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\red{\textcolor{red}} \def\magenta{\textcolor{magenta}} \def\blue{\textcolor{blue}} \def\<{\langle} \def\>{\rangle} \begin{definition} The two axioms that define elementary topoi, or standard toposes, $\tau$ as categories are: \begin{itemize} \item {\bf i.} $\tau$ has finite limits \item {\bf ii.} $\tau$ has power objects $\Omega(A)$ for objects $A$ in $\tau$. \end{itemize} \end{definition} To complete the axiomatic definition of topoi, one needs to add the ETAC axioms which allow one to define a category as an interpretation of ETAC. The above axioms imply that any topos has finite colimits, a subobject classifier (such as a Heyting logic algebra), as well as several other properties. Alternative definitions of topoi have also been proposed, such as: \begin{definition} A \emph{topos} is a category $\tau$ subject to the following axioms: \begin{itemize} \item {\bf $\mathbb{T}_1$}. $\tau$ is cartesian closed \item {\bf $\mathbb{T}_2$}. $\tau$ has a subobject classifier. \end{itemize} \end{definition} One can show that axioms {\em i.} and {\em ii.} also imply axioms $\mathbb{T}_1$ and $\mathbb{T}_2$; one notes that property $\mathbb{T}_2$ can also be expressed as the existence of a representable subobject functor. \begin{thebibliography}{99} \bibitem{RJW2k4} R.J. Wood. 2004. Ordered Sets via Adjunctions, in Categorical Foundations., \bibitem{MCP-WT} M. C. Pedicchio and W. Tholen, Eds. 2000. Cambridge, UK: Cambridge University Press. \bibitem{WFL63} W.F. Lawvere. 1963. Functorial Semantics of Algebraic Theories. Proc. Natl. Acad. Sci. USA, 50: 869-872 \bibitem{WFL66} W. F. Lawvere. 1966. The Category of Categories as a Foundation for Mathematics. , In Proc. Conf. Categorical Algebra-La Jolla, 1965, Eilenberg, S et al., eds. Springer-Verlag: Berlin, Heidelberg and New York, pp. 1-20. \bibitem{JL-PJS2k} J. Lambek and P. J. Scott. Introduction to higher order categorical logic. Cambridge University Press. \bibitem{SML97} S. Mac Lane. 1997. Categories for the Working Mathematician, 2nd ed. Springer-Verlag. \bibitem{SML-IM92} S. Mac Lane and I. Moerdijk. 1992. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag: Berlin. \end{thebibliography}