% This file:     (find-LATEX "2020J-ops-new.tex")
% Newer version: (find-LATEX "2021planar-HAs-2.tex")
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% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020J-ops-new.tex" :end))
% (defun d () (interactive) (find-pdf-page      "~/LATEX/2020J-ops-new.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020J-ops-new.pdf"))
% (defun e () (interactive) (find-LATEX "2020J-ops-new.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020J-ops-new"))
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% (find-pdf-page   "~/LATEX/2020J-ops-new.pdf")
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% (find-LATEX "2019.mk")
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% Based on: (find-LATEX "2019J-ops.tex")
%           (find-LATEX "2019J-ops-arxiv.tex")

% «.title»	(to "title")
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% «.toc»	(to "toc")
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\documentclass[oneside]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{tocloft}                   % (find-es "tex" "tocloft")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{stmaryrd}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
%\usepackage{colorweb}                 % (find-es "tex" "colorweb")
%\usepackage{tikz}
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% (find-dednat6file "demo-write-dnt.tex")
\usepackage{ifluatex}
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% (find-dn6 "preamble6.lua" "preamble0")
\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx15}               % (find-LATEX "edrx15.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
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\addbibresource{catsem-u.bib}  % (find-LATEX "catsem-u.bib")
%
% (find-es "tex" "geometry")
\begin{document}



\input 2017planar-has-defs.tex % (find-LATEX "2017planar-has-defs.tex")
\input 2019J-ops-defs.tex      % (find-LATEX "2019J-ops-defs.tex")

\ifluatex
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  \def\pu{}
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% %L dofile "edrxtikz.lua"  -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua"  -- (find-LATEX "edrxpict.lua")
\pu

\def\ovl{\overline}



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% «title» (to ".title")
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% (jona   "title")

\title{Planar Heyting Algebras for Children 2: Local Operators, J-Operators, and Slashings}

\author{Eduardo Ochs}

\maketitle


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%                                           
% «abstract» (to ".abstract")
% (jopp 1 "abstract")
% (jop    "abstract")

\begin{abstract}

Choose a topos $\calE$. There are several different ``notions of
sheafness'' on $\calE$. How do we visualize them?

Let's refer to the classifier object of $\calE$ as $Ω$, and to its
Heyting Algebra of truth-values, $\Sub(1_\calE)$, as $H$; we will
sometimes call $H$ the ``logic'' of the topos. There is a well-known
way of representing notions of sheafness as morphisms $j:Ω→Ω$, but
these `$j$'s yield big diagrams when we draw them explicitly; here we
will see a way to represent these `$j$'s as maps $J:H→H$ in a way that
is much more manageable.

In the previous paper of this series --- called \cite{OchsPH1} from
here on --- we showed how certain toy models of Heyting Algebras,
called ``ZHAs'', can be used to develop visual intuition for how
Heyting Algebras and Intuitionistic Propositional Logic work; here we
will extend that to sheaves. The full idea is this: {\sl notions of
  sheafness} correspond to {\sl local operators} and vice-versa; {\sl
  local operators} correspond to {\sl J-operators} and vice-versa; if
our Heyting Algebra $H$ is a ZHA then {\sl J-operators} correspond to
{\sl slashings} on $H$, and vice-versa; {\sl slashings} on $H$
correspond to {\sl ``sets of question marks''} and vice-versa, and
each set of question marks induces a notion of {\sl erasing and
  reconstructing}, which induces a sheaf. Also, every ZHA $H$
corresponds to an {(acyclic) 2-column graph}, and vice-versa, and for
any two-column graph $(P,A)$ the logic of the topos $\Set^{(P,A)}$ is
exactly the ZHA $H$ associated to $(P,A)$.

The introduction of \cite{OchsPH1} discusses two different senses in
which a mathematical text can be ``for children''. The first sense
involves some precise metamathetical tools for transfering knowledge
back and forth between a general case ``for adults'' and a toy model
``for children''; the second sense is simply that the text's
presentation has few prerequisites and never becomes too abstract.
Here we will use the second sense: everything here, except for the
last section, should be accessible to students who have taken a course
on Discrete Mathematics and read \cite{OchsPH1}. This means that
categories, toposes, sheaves and the maps $j:Ω→Ω$ only appear in the
last section, and before that we deal only with the J-operators
$J:H→H$, how they correspond to slashings and sets of question marks,
and how they form an algebra.

\end{abstract}


% (find-books "__cats/__cats.el" "bell")
% (find-belltpage (+ 14 163) "modality")
% (find-belltpage (+ 14 164) "universal closure operation")
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% (find-books "__cats/__cats.el" "johnstone-elephant")
% (find-elephantpage (+ 17 195) "A4.4 Local Operators")
%
% (find-books "__cats/__cats.el" "fourman-scott")
% (find-slnm0753page (+ 16 324) "J-operators")

% TODO: add this (and an abstract):
% (ph2p 4 "piccs-and-slashings")
% (ph2    "piccs-and-slashings")


% «abstract-arxiv»  (to ".abstract-arxiv")
%
% Choose a topos $E$. There are several different "notions of
% sheafness" on $E$. How do we visualize them?
% 
% Let's refer to the classifier object of $E$ as $\Omega$, and to its
% Heyting Algebra of truth-values, $Sub(1_E)$, as $H$; we will
% sometimes call $H$ the "logic" of the topos. There is a well-known
% way of representing notions of sheafness as morphisms $j:\Omega\to
% \Omega$, but these `$j$'s yield big diagrams when we draw them
% explicitly; here we will see a way to represent these `$j$'s as maps
% $J:H\to H$ in a way that is much more manageable.
% 
% In the previous paper of this series we showed how certain toy
% models of Heyting Algebras, called "ZHAs", can be used to develop
% visual intuition for how Heyting Algebras and Intuitionistic
% Propositional Logic work; here we will extend that to sheaves. The
% full idea is this: notions of sheafness correspond to local
% operators and vice-versa; local operators correspond to J-operators
% and vice-versa; if our Heyting Algebra $H$ is a ZHA then J-operators
% correspond to slashings on $H$, and vice-versa; slashings on $H$
% correspond to "sets of question marks" and vice-versa, and each set
% of question marks induces a notion of erasing and reconstructing,
% which induces a sheaf. Also, every ZHA $H$ corresponds to an
% (acyclic) 2-column graph, and vice-versa, and for any two-column
% graph $(P,A)$ the logic of the topos $\mathbf{Set}^{(P,A)}$ is
% exactly the ZHA $H$ associated to $(P,A)$.


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%                   
% «toc»  (to ".toc")
% (jonp 2 "toc")
% (jona   "toc")
% (favp 2 "toc")
% (fav    "toc")

% (find-es "tex" "tocloft")
\renewcommand{\cfttoctitlefont}{\bfseries}
\setlength{\cftbeforesecskip}{2.5pt}

\tableofcontents



% \end{document}


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% «parts»  (to ".parts")
% (jonp 2 "parts")
% (jon    "parts")

\newpage
\input 2019J-ops-slashings.tex    % (find-LATEX "2019J-ops-slashings.tex")
\newpage
\input 2019J-ops-logic.tex        % (find-LATEX "2019J-ops-logic.tex")
\newpage
\input 2019J-ops-midway.tex       % (find-LATEX "2019J-ops-midway.tex")
\newpage
\input 2019J-ops-cubes.tex        % (find-LATEX "2019J-ops-cubes.tex")
\newpage
\input 2019J-ops-valuations.tex   % (find-LATEX "2019J-ops-valuations.tex")
\newpage
\input 2019J-ops-algebra.tex      % (find-LATEX "2019J-ops-algebra.tex")
\newpage
\input 2019J-ops-categories.tex   % (find-LATEX "2019J-ops-categories.tex")
\newpage
\input 2019J-ops-classifier.tex   % (find-LATEX "2019J-ops-classifier.tex")
\newpage
\input 2019J-ops-kan.tex          % (find-LATEX "2019J-ops-kan.tex")

%L write_dnt_file()
\pu


% (find-LATEX "2019ilha-grande-poster-a4.tex" "references")


\newpage

\printbibliography




\end{document}

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% <make>

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# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020J-ops-new veryclean
make -f 2019.mk STEM=2020J-ops-new pdf

% Local Variables:
% coding: utf-8-unix
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% End:
