WellPointedTopos 2007-01-23 17:30:40 2007-01-23 17:30:40 Definition topos complemented supports split % this is the default PlanetMath 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} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{satsifies} The concept of well-pointed topoi was introduced by Freyd in~\cite{Fr}. A topos is \emph{well-pointed} if it satisfies either the following equivalent conditions: \begin{enumerate} \item The terminal object $1$ distinguishes morphisms in the sense that if the diagram \[\xymatrix{ 1\ar[r]^x & A\ar@<1ex>[r]^f\ar@<-1ex>[r]_g & B }\] commutes for every morphism $x\colon 1\to A$, then in fact \[\xymatrix{ A\ar@<1ex>[r]^f\ar@<-1ex>[r]_g & B }\] commutes, that is, $f=g$. Moreover, $1$ is not isomorphic to the initial object. \item The topos $\mathcal{T}$ is complemented and supports split, and the truth object $\Omega$ of $\mathcal{T}$ has exactly two elements, $\top\colon 1\to\Omega$, and $\bot\colon 1\to\Omega$. To say that $\mathcal{T}$ is \emph{complemented} means that if $m\colon X\to Y$ is a monomorphism, then there exists a monomorphism $m'\colon X'\to Y$ such that $m\sqcup m'\colon X\sqcup X'\to Y$ is an isomorphism. To say that $\mathcal{T}$ \emph{supports split} means that every subobject of $1$ is projective. \end{enumerate} Every well-pointed topos is a Boolean topos. \begin{thebibliography}{99} \bibitem{Fr} P.~Freyd. Aspects of topoi. {\it Bull. Austral. Math. Soc.} {\bf 7} (1972), 1--76. \bibitem{Jo} P.~T.~Johnstone. {\it Topos theory}. Academic Press, 1977. \bibitem{MaMo} S.~Mac~Lane and I.~Moerdijk. {\it Sheaves and Geometry in Logic: A First Introduction to Topos Theory}, Springer-Verlag, 1992. \end{thebibliography}