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% (find-angg "LATEX/2017visualizing-gms.tex") % (defun c () (interactive) (find-LATEXsh "lualatex -record 2017visualizing-gms.tex")) % (defun d () (interactive) (find-xpdfpage "~/LATEX/2017visualizing-gms.pdf")) % (defun e () (interactive) (find-LATEX "2017visualizing-gms.tex")) % (defun u () (interactive) (find-latex-upload-links "2017visualizing-gms")) % (find-xpdfpage "~/LATEX/2017visualizing-gms.pdf") % (find-sh0 "cp -v ~/LATEX/2017visualizing-gms.tex /tmp/") % (find-sh0 "cp -v ~/LATEX/2017visualizing-gms.pdf /tmp/") % (find-sh0 "cp -v ~/LATEX/2017visualizing-gms.pdf /tmp/pen/") % file:///home/edrx/LATEX/2017visualizing-gms.pdf % file:///tmp/2017visualizing-gms.pdf % file:///tmp/pen/2017visualizing-gms.pdf % http://angg.twu.net/LATEX/2017visualizing-gms.pdf \documentclass[oneside]{book} \usepackage[colorlinks]{hyperref} \usepackage{amsfonts} \begin{document} \def\Set{\mathbf{Set}} \def\Z{\mathbb{Z}} \def\A{\mathbf{A}} {\bf Visualizing Geometric Morphisms} Eduardo Ochs Departamento de CiÃªncias da Natureza Universidade Federal Fluminense (UFF), Brazil eduardoochs@gmail.com \medskip Different people have different ways of remembering theorems. A person with a very visual mind may remember a theorem in Category Theory mainly by the shape of a diagram and the order in which its objects are constructed. For such a person most books on Category Theory feel as if they have lots of missing diagrams, that she has to reconstruct if she wants to understand the subject. The shape of a categorical diagram remains the same if we specialize it to a particular case --- and this means that we can sometimes remember a general diagram, and the theorems associated to it, from the diagram of a particular case. In this talk we will present the general technique above and one application: reconstructing the statements, and some of the proofs, of two factorizations of geometric morphisms between toposes described in section A4 of [1], from particular cases that are easy to draw explicitly --- in which our toposes are of the form $\Set^\A$, where $\A$ is a finite category whose objects are certain points of $\Z^2$. The tricks for visualizing sheaves on these `$\Set^\A$'s are described in [2]. \bigskip References: [1]: ``Sketches of an Elephant: A Topos Theory Compendium''. P.T.\ Johnstone, Oxford, 2002. [2]: ``Planar Heyting Algebras for Children, 3: Geometric Morphisms''. E.\ Ochs, 20017. Preprint available at: \noindent \url{http://angg.twu.net/math-b.html#zhas-for-children-2}. \end{document} % Local Variables: % coding: utf-8-unix % End: