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% (find-angg "LATEX/2019ebl-abs.tex") % (defun c () (interactive) (find-LATEXsh "lualatex -record 2019ebl-abs.tex")) % (defun d () (interactive) (find-xpdfpage "~/LATEX/2019ebl-abs.pdf")) % (defun b () (interactive) (find-zsh "bibtex 2019ebl-abs; makeindex 2019ebl-abs")) % (defun e () (interactive) (find-LATEX "2019ebl-abs.tex")) % (defun u () (interactive) (find-latex-upload-links "2019ebl-abs")) % (find-xpdfpage "~/LATEX/2019ebl-abs.pdf") % (find-sh0 "cp -v ~/LATEX/2019ebl-abs.pdf /tmp/") % (find-sh0 "cp -v ~/LATEX/2019ebl-abs.pdf /tmp/pen/") % file:///home/edrx/LATEX/2019ebl-abs.pdf % file:///tmp/2019ebl-abs.pdf % file:///tmp/pen/2019ebl-abs.pdf % http://angg.twu.net/LATEX/2019ebl-abs.pdf \documentclass[11pt]{article} %use one of the following package accordingly %\usepackage[brazil]{babel} % for portuguese \usepackage[english]{babel} % for english %\usepackage[spanish]{babel} % for spanish %\usepackage[latin1]{inputenc} % for accents in portuguese %\usepackage[utf8]{inputenc} % for accents in portuguese using Unicode %% %% %% PLEASE DO NOT MAKE CHANGES TO THIS TEMPLATE %% THAT CAUSE CHANGES IN THE FORMAT OF THE TEXT %% %% \usepackage[centertags]{amsmath} \usepackage{indentfirst,amsfonts,amssymb,amsthm} \usepackage{cite} \usepackage[bottom=1.5cm,top=1.5cm,left=3cm,right=2cm]{geometry} \date{} \begin{document} %******************************************************** \title{Five applications of the ``Logic for Children'' project \\ to Category Theory} \author{ {\large Eduardo Ochs}\thanks{eduardoochs@gmail.com}\\ {\small UFF, Rio das Ostras, RJ, Brazil} \\ %{\large Author 2}\thanks{author2@email.com} \\ %{\small University, City, Country } \\ % {\large Author 3}\thanks{author@email.com}\\ % {\small University, City, Country} } \maketitle \begin{abstract} %the abstract should contain a maximum of 600 words Category Theory is usually presented in a way that is too abstract, with concrete examples of each given structure being mentioned briefly, if at all. One of the themes of the ``Logic for Children'' workshop, held in the UNILOG 2018, was a set of tools and techniques for drawing diagrams of categorical concepts in a canonical shape, and for drawing diagrams of particular cases of those concepts in essentialy the same shape as the general case; these diagrams for a general and a particular case can be draw side by side ``in parallel'' in a way that lets us transfer knowledge from the particular case to the general, and back. In this talk we will present briefly five applications of these techniques: 1) a way to visualize planar, finite Heyting Algebras --- we call them ``ZHAs'' --- and to develop a feeling for how the logic connectives in a ZHA work; 2) a way to build a topos with a given logic (when that ``logic'' is a ZHA); 3) a way to represent a closure operator on a ZHA by a ``slashing on that ZHA by diagonal cuts with no cuts stopping midway''; 4) a way to extend a slashing on a ZHA $H$ to a ``notion of sheafness'' on the associated topos; 5) a way to start from a certain very abstract factorization of geometric morphisms between toposes, described in Peter Johnstone's ``Sketches of an Elephant'' \cite{Elephant1}, and derive some intuitive meaning for what that factorization ``means'': basically, we draw the diagrams, plug in it some very simple geometric morphisms, and check which ones the factorization classifies as ``surjections'', ``inclusions'', ``closed'', and ``dense''. % Plain text version submitted to EasyChair: % % Five applications of the "Logic for Children" project to Category Theory % % Category Theory is usually presented in a way that is too abstract, % with concrete examples of each given structure being mentioned % briefly, if at all. One of the themes of the "Logic for Children" % workshop, held in the UNILOG 2018, was a set of tools and techniques % for drawing diagrams of categorical concepts in a canonical shape, % and for drawing diagrams of particular cases of those concepts in % essentialy the same shape as the general case; these diagrams for a % general and a particular case can be draw side by side "in parallel" % in a way that lets us transfer knowledge from the particular case to % the general, and back. % % In this talk we will present briefly five applications of these % techniques: 1) a way to visualize planar, finite Heyting Algebras % --- we call them "ZHAs" --- and to develop a feeling for how the % logic connectives in a ZHA work; 2) a way to build a topos with a % given logic (when that "logic" is a ZHA); 3) a way to represent a % closure operator on a ZHA by a "slashing on that ZHA by diagonal % cuts with no cuts stopping midway"; 4) a way to extend a slashing on % a ZHA $H$ to a "notion of sheafness" on the associated topos; 5) a % way to start from a certain very abstract factorization of geometric % morphisms between toposes, described in Peter Johnstone's "Sketches % of an Elephant", and derive some intuitive meaning for what that % factorization "means": basically, we draw the diagrams, plug in it % some very simple geometric morphisms, and check which ones the % factorization classifies as "surjections", "inclusions", "closed", % and "dense". \end{abstract} % Sample for references: % % Book \cite{Itala} % % Chapter \cite{Anjolina} % % Dissertation \cite{Elaine} % % Proceeding \cite{Daniele} % % Published paper \cite{Valeria} % % Thesis \cite{Renata} % % Unpublished paper \cite{Mariangela} \begin{thebibliography}{00} % \bibitem{Daniele} % Ayala-Rincón, M.; Fernández, M.; Nantes-Sobrinho, D.; Nominal % Narrowing. In: Conference on Formal Structures for Computation and % Deduction (FSCD 2016), Porto, Portugal, 2016. {\em LIPIcs-Leibniz % International Proceedings in Informatics}. Vol. 52. pp. 11--17, % Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. % % \bibitem{Valeria} % Bierman, G. M.; de Paiva V. C. V. On an intuitionistic modal logic. % {\em Studia Logica} 65:383--416, 2000. \bibitem{Elephant1} Johnstone, P. T.; {\em Sketches of an Elephant: A Topos Theory Compendium (Volume 1)}. Oxford University Press, 2002. % \bibitem{Itala} % Cignoli, R. L. O.; D'ottaviano I. M. L.; Mundici D. {\em Algebraic % foundations of many-valued reasoning}. Kluwer Academic Publishers, % 2000. % % \bibitem{Renata} % de Freitas, R. {\em Lógica Modal da Bifurcação}. Tese de Doutorado, % COPPE-UFRJ, 2002. % % \bibitem{Anjolina} % de Oliveira. A. G.; de Queiroz, R. J. G. B. Geometry of deduction via % graphs of proofs. In: R.J.G.B de Queiroz (Editor), {\em Logic for % Concurrency and Synchronisation}. Springer, Trends in Logic, Volume % 18, chapter 1, pages 3--88, 2003. % % \bibitem{Elaine} % Pimentel, E. G. {\em Alguns Resultados sobre Estabilidade de % Hipersuperfícies de Curvatura Média Constante}. Dissertação de % Mestrado, UFMG, 1994. % % \bibitem{Mariangela} % Weiss. M. A. {\em A Polynomial Algorithm for 3-sat: draft version}. \\ % Disponível em {\tt https://www.ime.usp.br/$\sim$weiss/P=NP-12-06.pdf}, % acessado em $01\slash 09 \slash 2018$. \end{thebibliography} \end{document} % Local Variables: % coding: utf-8-unix % End: