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\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}




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