% (find-LATEX "2020hosgood.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020hosgood.tex" :end))
% (defun d () (interactive) (find-pdf-page      "~/LATEX/2020hosgood.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020hosgood.pdf"))
% (defun e () (interactive) (find-LATEX "2020hosgood.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020hosgood"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page   "~/LATEX/2020hosgood.pdf")
% (find-sh0 "cp -v  ~/LATEX/2020hosgood.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2020hosgood.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2020hosgood.pdf
%               file:///tmp/2020hosgood.pdf
%           file:///tmp/pen/2020hosgood.pdf
% http://angg.twu.net/LATEX/2020hosgood.pdf
% (find-LATEX "2019.mk")

\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
%\usepackage{colorweb}                 % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (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")
%
% (find-es "tex" "geometry")
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")





% https://thosgood.com/
% https://thosgood.com/papers/
% https://arxiv.org/a/hosgood_t_1.html
% https://arxiv.org/pdf/1609.00920.pdf




%L forths["->>"] = function () pusharrow("->>") end
\pu

\def\ann#1{\;\text{[#1]}}
\def\t{\text}

For all diagrams like this
%
%D diagram ??
%D 2Dx     100   +50
%D 2D  100 A0 -> A1   
%D 2D      |      |   
%D 2D      v      v   
%D 2D  +25 A2 -> A3   
%D 2D
%D ren A0 A1 ==> \{F_1×_{F_0}X_i\}_{i∈I}\ann{Scheme}_2 \{X_i\}_{i∈I}
%D ren A2 A3 ==>  F_1\ann{Sheaf}   F_0\ann{Scheme}
%D
%D (( A1 A3 ->> .plabel= r \psm{\t{Zariski}\\\t{open}\\\t{cover}}
%D    A2 A3 ->
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

we have:
%
%D diagram ??
%D 2Dx     100   +60
%D 2D  100 A0 -> A1   
%D 2D      |      |   
%D 2D      v      v   
%D 2D  +25 A2 -> A3   
%D 2D
%D 2D  +20 B0 -> B1
%D 2D      |      |
%D 2D      v      v
%D 2D  +25 B2 -> B3
%D 2D
%D ren A0 A1 ==> \{F_1×_{F_0}X_i\}_{i∈I}\ann{Scheme}_2 \{X_i\}_{i∈I}
%D ren A2 A3 ==>  F_1\ann{Sheaf}   F_0\ann{Scheme}
%D ren B0 B1 ==>  F_1×_{F_0}X_i\ann{Scheme}_1        X_i
%D ren B2 B3 ==>  F_1\ann{Sheaf}   F_0\ann{Scheme}
%D
%D (( A0 A1 ->
%D    A0 A2 ->
%D    A1 A3 ->> .plabel= r \psm{\t{Zariski}\\\t{open}\\\t{cover}}
%D    A2 A3 ->
%D    A0 relplace 8 8 \pbsymbol{7}
%D    
%D    B0 B1 ->
%D    B0 B2 ->
%D    B1 B3 -> .plabel= r \psm{\t{Zariski}\\\t{open}}
%D    B2 B3 ->
%D    B0 relplace 8 8 \pbsymbol{7}
%D
%D    B2 B3 midpoint xy+= 0 15 .TeX= (∀i∈I.\ann{Scheme}_1)→\ann{Scheme}_2 place
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

i.e.,

If $∀i∈I$ the pullback $F_1×_{F_0}X_i$ ``has schemeness''

then the pullback $\{F_1×_{F_0}X_i\}_{i∈I}$ ``has schemeness'' too.


\end{document}

%  __  __       _        
% |  \/  | __ _| | _____ 
% | |\/| |/ _` | |/ / _ \
% | |  | | (_| |   <  __/
% |_|  |_|\__,_|_|\_\___|
%                        
% <make>

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020hosgood veryclean
make -f 2019.mk STEM=2020hosgood pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "NONE"
% End:
