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% (find-angg "LATEX/2008sheaves-abs1.tex") % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && latex 2008sheaves-abs1.tex")) % (defun c () (interactive) (find-zsh "cd ~/LATEX/ && pdflatex 2008sheaves-abs1.tex")) % (eev "cd ~/LATEX/ && Scp 2008sheaves-abs1.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/") % (find-dvipage "~/LATEX/2008sheaves-abs1.dvi") % (find-pspage "~/LATEX/2008sheaves-abs1.pdf") % (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2008sheaves-abs1.ps 2008sheaves-abs1.dvi") % (find-pspage "~/LATEX/2008sheaves-abs1.ps") % (ee-cp "~/LATEX/2008sheaves-abs1.pdf" (ee-twupfile "LATEX/2008sheaves-abs1.pdf") 'over) % (ee-cp "~/LATEX/2008sheaves-abs1.pdf" (ee-twusfile "LATEX/2008sheaves-abs1.pdf") 'over) % (find-angg "LATEX/2008filterp-abs.tex") \documentclass{book} \usepackage{amsfonts} \usepackage{hyperref} \usepackage{url} \begin{document} \def\O{\mathcal{O}} Sheaves for non-categorists - part 2 Eduardo Ochs - UFF \url{http://angg.twu.net/math-b.html} \bigskip Take a set of ``worlds'', $W$, and a directed acyclical graph on $W$, given by a relation $R \subset W \times W$. Let's call the functions $W \to \{0,1\}$ ``modal truth-values'', and the $R$-non-decreasing functions $W \to \{0,1\}$ ``intuitionistic truth-values''. If we see $W$ as a topological space with the order topology induced by $R$, the intuitionistic truth-values correspond to open sets. The pair $(\O(W), \subseteq)$ is a Heyting algebra --- meaning that we can interpret intuitionistic propositional logic on it --- and it is a (bigger) DAG, and so we can repeat the above process with it, to generate a (bigger) topological space $(\O(W), \O(\O(W)))$, which is the natural setting for talking about ``covers'', ``saturated covers'', and ``unions of covers''. This presentation will be focused on understanding all these ideas (and more!), mainly in the case where $W$ has three worlds forming a ``V'', and $R$ has two arrows pointing downwards. The operation of ``taking the union of a cover'' turns out to be a particular case of a ``Lawvere-Tierney modality''; the double negation is another LT-modality. \end{document} #*[# (eeblogme-now-bounded) # (find-equailfile "sgml-input.el" ";; SUBSET OF") # (find-equailfile "sgml-input.el" ";; MULTIPLICATION SIGN") # (find-equailfile "sgml-input.el" ";; RIGHTWARDS ARROW\n") # (find-fline "/tmp/ee.html") #] <p>Take a set of "worlds", W, and a directed acyclical graph on W, given by a relation R ⊂ W × W. Let's call the functions W → {0,1} "modal truth-values", and the R-non-decreasing functions W → {0,1} "intuitionistic truth-values". If we see W as a topological space with the order topology induced by R, the intuitionistic truth-values correspond to open sets.</p> <p>The pair (O(W), ⊆) is a Heyting algebra --- meaning that we can interpret intuitionistic propositional logic on it --- and it is a (bigger) DAG, and so we can repeat the above process with it, to generate a (bigger) topological space (O(W), O(O(W))), which is the natural setting for talking about "covers", "saturated covers", and "unions of covers".</p> <p>This presentation will be focused on understanding all these ideas (and more!), mainly in the case where W has three worlds forming a "V", and R has two arrows pointing downwards. The operation of "taking the union of a cover" turns out to be a particular case of a "Lawvere-Tierney modality"; the double negation is another LT-modality.</p> #*[# # Local Variables: # coding: raw-text-unix # modes: (latex-mode fundamental-mode blogme-mode) # End: #]