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\documentclass[11pt]{article}

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\begin{document}

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\title{On a way to visualize some \\ Grothendieck Topologies}

\author{
{\large  Eduardo Ochs}\thanks{eduardoochs@gmail.com}\\
{\small UFF, Rio das Ostras, RJ, Brazil} \\
 }

\maketitle

\begin{abstract}

\def\Jcan{J_{\text{can}}}
\def\R{\mathbb{R}}
\def\Opens{\mathcal{O}}
\def\calS{\mathcal{S}}
\def\catC{\mathbf{C}}

The {\sl canonical Grothendieck topology} on $\R$, $\Jcan$, is easy to
define, but the definition takes several steps: 1) for each open set
$U \in \Opens(\R)$ a {\sl sieve on $U$} is a subset of $\Opens(U)$
that is downward-closed; 2) for each $U \in \Opens(\R)$ we write
$\Omega(U)$ for the set of all sieves on $U$; 3) we say that a sieve
$\calS \in \Omega(U)$ is {\sl covering} when $\bigcup \calS = U$; 4)
for each $U \in \Opens(\R)$ we define $\Jcan(U)$ as the set of covering
sieves on $U$.

The ``real'' definition of Grothendieck Topology generalizes this
definition of $\Jcan$ in many ways: in particular, it starts with a
category $\catC$ instead of a topological space $(\R,\Opens(\R))$, and
we can have many notions of ``covering-ness'' for the same category
--- they just have to obey the three axioms in \cite{SGL}, p.110.

In this presentation I will show how we can use some of the techniques
in \cite{O22} to understand the general definition of Grothendieck
Topology, and I will show how we can visualize all the Grothendieck
Topologies on one of the Planar Heyting Algebras of \cite{O19}. Most
of the diagrams in the presentation will be taken from \cite{O21}.

\end{abstract}



\begin{thebibliography}{00}


\bibitem{SGL} Mac Lane, S.; Moerdijk, I. {\em Sheaves in geometry and
    logic: a first introduction to topos theory}. Springer, 1992.

\bibitem{O22} Ochs, E. On the missing diagrams in Category Theory. In:
  L. Magnani (editor), {\em Handbook of Abductive Reasoning}.
  Springer, 2022. Available at \\ {\tt
    http://angg.twu.net/math-b.html\#2022-md}.

\bibitem{O21} Ochs, E. {\em Grothendieck Topologies for Children}. \\
Available at {\tt http://angg.twu.net/math-b.html\#2021-groth-tops}.

\bibitem{O19} Ochs, E. Planar Heyting Algebras for Children. {\em
    South American Journal of Logic} 5.1:125--164, 2019. Available at
  {\tt http://angg.twu.net/math-b.html\#zhas-for-children-2}.

\end{thebibliography}



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