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% (find-angg "LATEX/2013sheaves-for-children.tex") % (find-dn4tex-links "2013sheaves-for-children") % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2013sheaves-for-children.tex && latex 2013sheaves-for-children.tex")) % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2013sheaves-for-children.tex && pdflatex 2013sheaves-for-children.tex")) % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && latex 2013sheaves-for-children.tex")) % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && pdflatex 2013sheaves-for-children.tex")) % (defun d () (interactive) (find-dvipage "~/LATEX/2013sheaves-for-children.dvi")) % (find-dvipage "~/LATEX/2013sheaves-for-children.dvi") % (find-xpdfpage "~/LATEX/2013sheaves-for-children.pdf") % http://angg.twu.net/LATEX/2013sheaves-for-children.pdf % (find-LATEXgrep "grep -nH -e BPM *.tex") % (find-LATEX "2011ebl-slides.tex") % (find-LATEX "2012minicats.tex") % (find-dn4ex "edrxdefs.tex") \documentclass[oneside]{article} \usepackage[latin1]{inputenc} \usepackage{edrx08} % (find-dn4ex "edrx08.sty") %L process "edrx08.sty" -- (find-dn4ex "edrx08.sty") \input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex") \begin{document} \def\BPM{\mathsf{BPM}} \def\S{\mathsf{S}} \def\catV{\mathbf{V}} \def\nec{\Box} \def\poss{\lozenge} \def\toM{\to_{\scriptscriptstyle\mathsf{M}}} \def\dagThreeH #1#2#3{[\text{ThreeH }#1#2#3]} \def\dagSquare#1#2#3#4{[\text{Square }#1#2#3#4]} \def\L{\Lambda} \input 2013sheaves-for-children.dnt Title: Sheaves for Children Author: Eduardo Ochs (eduardoochs@gmail.com) {\bf INCOMPLETE VERSION - 2013dec22 23:50} % {\bf Sheaves for children} \bigskip \begin{abstract} First-year university students -- the ``children'' of the title -- often prefer to start from an interesting particular case, and only then proceed to general statements. How can we make intuitionistic logic, toposes, and sheaves accessible to them? Let $D$ be a finite subset of $\N^2$. Draw arrows for all the ``black pawns moves'' between points of $D$, and let $\catD$ be the poset generated by that graph; $\catD$ is what we call a ``ZDAG'', and $\Set^\catD$ is a ``ZDAG-topos''. It turns out that the truth-values of a $\Set^\catD$ can be represented in a very nice way as two-dimensional ascii diagrams, and that all the operations leading to sheaves and geometric morphisms can be understood via algorithms on diagrams. In this talk we will present a computer library for performing computations interactively on the truth-values of ZDAG-toposes. The diagrams are rendered in ascii by default, but there is a module that typesets them in LaTeX. \end{abstract} \bigskip Let's start with some definitions. A non-empty subset of $\N^2$ is {\sl well-positioned} when it touches both axes; a {\sl Z-set} is a finite, non-empty, well-positioned subset of $\N^2$; a {\sl black pawn's move} is an arrow between points of $\N^2$ that moves one unit down and -1, 0 or 1 units horizontally; $\BPM(D)$ is the set of black pawns' moves between point of $D$; a {\sl ZDAG} is a graph $(D, \BPM(D))$ where $D$ is a Z-set. We will use the notation $R^*$ for the transitive-reflexive closure of the relation $R$, $\bbD$ for the graph $(D, \BPM(D)^*)$, and $\catD$ for $\bbD$ seen as a category. % (find-dn4exfile "edrx08.sty" "dagKite") A nice thing about Z-sets is that they can be ``named'' unambigously by a positional notation with bullets. Four of our favourite Z-sets --- $\dagVee***$ (``vee''), $\dagVee***$ (``lambda''), $\dagKite*****$ (``kite''), and $\dagHouse*****$ (``house'') --- also deserve letter-like names: $V$, $\L$, $K$, and $H$; by a slight abuse of language, the same diagrams will sometimes stand for $\bbV$, $\catV$, and so on. Finite models for the modal logic $\S4$ use valuations on systems of ``worlds'' (Kripke frames) whose visibility relations are partial orders. So, we have a model of $\S4$ on $\dagHouse*****$; $\dagHouse01101$ is a truth-value on it, and $\nec\dagHouse01101 = \dagHouse00101$, and $\poss\dagHouse01101 = \dagHouse01111$. Note how a lot of information can be recovered from the shapes of our diagrams. {\sl Modal truth-values} are ZDAG bullet diagrams with `0's and `1's in place of `*'s. There is a well-known translation of intuitionistic logic into $\S4$ ([1]), in which the ``intuitionistic truth-values'' correspond to $\nec$-stable modal truth-values. When we put this in terms of Z-sets, we get the following: the modal truth-values on a Z-set $D$ are exactly (the characteristic functions of elements of) $\Pts(D)$, and the intuitionistic truth-values on $D$ are a subset of that --- if we consider the order topology on $D$ induced by $\BPM(D)$ we get a topological space $(D, \Opens(D))$, and the intuitionistic truth-values on $D$ are exacty the (characteristic functions of) elements of $\Opens(D)$. So, for each Z-set $D$ we have a finite Heyting algebra associated to it, whose elements have a very nice diagrammatic representation --- and, using `$\to$' for the intuitionistic implication and `$\toM$' for the modal implication, we can say things like this: % $$ (\dagHouse00111 \to \dagHouse01011) = \nec(\dagHouse00111 \toM \dagHouse00111) = \nec \dagHouse01011 = \dagHouse01011 $$ Anyone with a lot of practice in Topos Theory may find all this trivial --- we are only establishing a nice notation for working with the truth-values of a topos $\Set^\bbD$, which would be standard if the \LaTeX{} macros for Z-sets and valuations on Z-sets were in wide use. The only things here that may be new theorems are these: call a ZDAG ``thin'' if it doesn't have $\dagThreeH***$ as a subgraph; then $(\Opens(D), \subseteq)$ is isomorphic to a ZDAG iff $D$ is thin, and that ZDAG is thin iff $D$ doesn't contain $\dagSquare****$ as a subgraph... {\sl ...but mathematics also has a social side} --- and when we take it into account we become able to admit that {\sl theorems are of little use when too few people are able to understand them.} ZDAGs are tools for {\it social} mathematics --- they can be used to analyze particular cases of many theorems, and to let people understand the general theorems from these particular cases. In the paper [2] I have laid down some a method for doing that precisely, in which the particular and the general cases are written down in parallel, and I have had this conjecture for several years: % \begin{quote} Categories of the form $\Set^\catD$ are the archetypal models for a sizeable fragment of basic sheaf theory, \end{quote} % but {\sl what fragment}? I have since then realized that I won't be able to answer that by working alone --- \end{document} % ---------------------------------------------------------------------------- \documentclass[oneside]{book} \usepackage{amssymb} \begin{document} \def\N{\mathbb{N}} \def\Set{\mathbf{Set}} \def\catD{\mathbf{D}} {\bf Sheaves for children} \medskip First-year university students -- the ``children'' of the title -- often prefer to start from an interesting particular case, and only then proceed to general statements. How can we make intuitionistic logic, toposes, and sheaves accessible to them? Let $D$ be a finite subset of $\N^2$. Draw arrows for all the ``black pawns moves'' between points of $D$, and let $\catD$ be the poset generated by that graph; $\catD$ is what we call a ``ZDAG'', and $\Set^\catD$ is a ``ZDAG-topos''. It turns out that the truth-values of a $\Set^\catD$ can be represented in a very nice way as two-dimensional ascii diagrams, and that all the operations leading to sheaves and geometric morphisms can be understood via algorithms on diagrams. In this talk we will present a computer library for performing computations interactively on the truth-values of ZDAG-toposes. The diagrams are rendered in ascii by default, but there is a module that typesets them in LaTeX. \end{document} % Local Variables: % coding: raw-text-unix % ee-anchor-format: "«%s»" % End: