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\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}


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<p>Take a set of "worlds", W, and a directed acyclical graph on W,
given by a relation R &sub; W &times; W. Let's call the functions W
&rarr; {0,1} "modal truth-values", and the R-non-decreasing functions
W &rarr; {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), &sube;) 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>




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