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diff --git a/build/preamble.tex b/build/preamble.tex
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1\documentclass[14pt,preview,border=1pt,class=extarticle]{standalone}
2
3\usepackage{amssymb}
4\usepackage{amsmath}
5\usepackage{etoolbox}
6\usepackage{xspace}
7\usepackage{mathrsfs}
8\usepackage{xifthen}
9
10\usepackage{tikz}
11\usetikzlibrary{arrows,decorations.markings,chains,calc,matrix}
12\usepackage{tikz-cd}
13\tikzset{>=cm to}
14
15\usepackage{mathtools}
16
17\undef\Re
18\undef\Im
19
20\DeclareMathOperator{\colim}{\textnormal{colim}}
21\DeclareMathOperator{\Spec}{\textnormal{Spec}}
22\DeclareMathOperator{\im}{\textnormal{im}}
23\DeclareMathOperator{\Sing}{\textnormal{Sing}}
24\DeclareMathOperator{\Map}{\textnormal{Map}}
25\DeclareMathOperator{\InnHom}{\textnormal{\textbf{Hom}}}
26\DeclareMathOperator{\Hom}{\textnormal{Hom}}
27\undef\hom
28\DeclareMathOperator{\hom}{\textnormal{hom}}
29\DeclareMathOperator{\nerve}{\textnormal{N}}
30\DeclareMathOperator{\simpnerve}{\textnormal{sN}}
31\DeclareMathOperator{\core}{\textnormal{Core}}
32\DeclareMathOperator{\id}{\textnormal{id}}
33\undef\lim
34\DeclareMathOperator{\lim}{\textnormal{lim}}
35\DeclareMathOperator{\Op}{\textnormal{op}}
36\DeclareMathOperator{\yon}{\textnormal{y}}
37\DeclareMathOperator{\HomL}{\textnormal{\textbf{Hom}}^{\textnormal{L}}}
38\DeclareMathOperator{\sk}{\textnormal{sk}}
39\DeclareMathOperator{\cosk}{\textnormal{cosk}}
40\DeclareMathOperator{\thom}{\textnormal{Th}}
41\DeclareMathOperator{\Bl}{\textnormal{Bl}}
42\DeclareMathOperator{\Stab}{\textnormal{Stab}}
43\DeclareMathOperator{\Grp}{\textnormal{\textsf{Grp}}}
44\DeclareMathOperator{\Set}{\textnormal{\textsf{Set}}}
45
46\newcommand*{\alg}[1]{\ensuremath{\overline{#1}}\xspace}
47\newcommand*{\sep}[1]{\ensuremath{#1^{\textnormal{sep}}}\xspace}
48\newcommand*{\ab}[1]{\ensuremath{#1^{\textnormal{ab}}}\xspace}
49\newcommand*{\units}[1]{\ensuremath{#1^\times}\xspace}
50\newcommand*{\lapl}{\Delta}
51\newcommand*{\grad}{\nabla}
52\newcommand*{\op}[1]{\ensuremath{#1^{\textnormal{op}}}\xspace}
53\newcommand*{\codiag}[1]{\ensuremath{\nabla\mkern-3mu_{#1}}\xspace}
54\newcommand*{\diag}[1]{\ensuremath{\Delta_{#1}}\xspace}
55
56\newcommand*{\cech}{\v{C}ech\xspace}
57\newcommand*{\mc}{\operatorname{\textnormal{\v{C}}}}
58
59\newcommand*{\normal}{\lhd}
60\newcommand*{\isom}{\cong}
61\newcommand*{\homot}{\sim}
62\newcommand*{\cequiv}{\simeq}
63
64\makeatletter
65\let\@oldsubset=\subset
66\def\@subsethelper#1#2{\mathrel{\raisebox{.5pt}{$#1\@oldsubset$}}\xspace}
67\DeclareRobustCommand*{\subset}{\mathpalette\@subsethelper\relax}
68
69\let\@oldotimes=\otimes
70\def\@otimeshelper#1#2{\mathrel{\raisebox{.5pt}{$#1\@oldotimes$}}\xspace}
71\DeclareRobustCommand*{\otimes}{\mathpalette\@otimeshelper\relax}
72\makeatother
73
74\tikzset{/tikz/commutative diagrams/arrows={thin}}
75
76\makeatletter
77\newbox\@xrat
78\renewcommand*{\xrightarrow}[2][-cm to]{%
79 \setbox\@xrat=\hbox{\ensuremath{\scriptstyle #2}}
80 \pgfmathsetlengthmacro{\@xratlen}{max(1.6em,\wd\@xrat+.6em)}
81 \pgfmathsetlengthmacro{\@xratinnersep}{.5ex-\dp\@xrat}
82 \mathrel{\tikz [#1,baseline=-.6ex]
83 \draw (0,0) -- node[auto,inner sep=\@xratinnersep] {\box\@xrat} (\@xratlen,0) ;}}
84\renewcommand*{\xleftarrow}[2][cm to-]{%
85 \setbox\@xrat=\hbox{\ensuremath{\scriptstyle #2}}
86 \pgfmathsetlengthmacro{\@xratlen}{max(1.6em,\wd\@xrat+.6em)}
87 \pgfmathsetlengthmacro{\@xratinnersep}{.5ex-\dp\@xrat}
88 \mathrel{\tikz [#1,baseline=-.6ex]
89 \draw (0,0) -- node[auto,inner sep=\@xratinnersep] {\box\@xrat} (\@xratlen,0) ;}}
90\newcommand*{\xrightarrowb}[2][-cm to]{%
91 \setbox\@xrat=\hbox{\ensuremath{\scriptstyle #2}}
92 \pgfmathsetlengthmacro{\@xratlen}{max(1.6em,\wd\@xrat+.6em)}
93 \pgfmathsetlengthmacro{\@xratinnersep}{.5ex}
94 \mathrel{\tikz [#1,baseline=-.6ex]
95 \draw (0,0) -- node[auto,inner sep=\@xratinnersep] {\box\@xrat} (\@xratlen,0) ;}}
96\newcommand*{\xleftarrowb}[2][cm to-]{%
97 \setbox\@xrat=\hbox{\ensuremath{\scriptstyle #2}}
98 \pgfmathsetlengthmacro{\@xratlen}{max(1.6em,\wd\@xrat+.6em)}
99 \pgfmathsetlengthmacro{\@xratinnersep}{.5ex}
100 \mathrel{\tikz [#1,baseline=-.6ex]
101 \draw (0,0) -- node[auto,inner sep=\@xratinnersep] {\box\@xrat} (\@xratlen,0) ;}}
102
103\pgfarrowsdeclare{my right hook}{my right hook}
104{
105\arrowsize=0.2pt
106\advance\arrowsize by .5\pgflinewidth
107\pgfarrowsleftextend{-.5\pgflinewidth}
108\pgfarrowsrightextend{3.5\arrowsize+.5\pgflinewidth}
109}
110{
111\arrowsize=0.2pt
112\advance\arrowsize by .5\pgflinewidth
113\pgfsetdash{}{0pt} % do not dash
114\pgfsetroundjoin % fix join
115\pgfsetroundcap % fix cap
116\pgfpathmoveto{\pgfpoint{0\arrowsize}{-7\arrowsize}}
117\pgfpatharc{-90}{90}{3.5\arrowsize}
118\pgfusepathqstroke
119}
120
121\tikzset{%
122 iso/.style={above,sloped,inner sep=0},
123 iso'/.style={below,sloped,inner sep=0},
124 to/.style={-cm to},
125 onto/.style={-cm double to},
126 into/.style={my right hook-cm to},
127 mapsto/.style={|-cm to},
128 clim/.style={decoration={markings,
129 mark=at position#1 with {\draw[-] (0,-3\pgflinewidth) -- (0,3\pgflinewidth);}},
130 postaction=decorate},
131 clim/.default=0.5,
132 opim/.style={decoration={markings,
133 mark=at position#1 with {\draw[-] circle(3\pgflinewidth);}},
134 postaction=decorate},
135 opim/.default=0.5
136}
137
138\newcommand*\@tikzto[2]{\mathrel{\begin{tikzpicture}[baseline]%
139 \draw[to,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (1.6em,.55ex);%
140 \end{tikzpicture}}}
141
142\newcommand*\@tikzcto[2]{\mathrel{\begin{tikzpicture}[baseline]%
143 \draw[to,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (0.8em,.55ex);%
144 \end{tikzpicture}}}
145
146\newcommand*\@tikzonto[2]{\mathrel{\begin{tikzpicture}[baseline]%
147 \draw[onto,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (1.6em,.55ex);%
148 \end{tikzpicture}}}
149
150\newcommand*\@tikzinto[2]{\mathrel{\begin{tikzpicture}[baseline]%
151 \draw[into,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (1.6em,.55ex);%
152 \end{tikzpicture}}}
153
154\newcommand*\@tikzclim[2]{\mathrel{\begin{tikzpicture}[baseline]%
155 \draw[into,clim,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (1.6em,.55ex);%
156 \end{tikzpicture}}}
157
158\newcommand*\@tikzopim[2]{\mathrel{\begin{tikzpicture}[baseline]%
159 \draw[into,opim,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (1.6em,.55ex);%
160 \end{tikzpicture}}}
161
162\newcommand*\@tikzmapsto[2]{\mathrel{\begin{tikzpicture}[baseline]%
163 \draw[mapsto,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- (1.6em,.55ex);%
164 \end{tikzpicture}}}
165
166\newcommand*\@tikziso[4]{\mathrel{\begin{tikzpicture}[baseline]%
167 \draw[to,line width={#2\pgflinewidth},scale=#1](0,.55ex) -- node[iso,pos=0.47,inner sep=#4]{$#3\sim$} (1.6em,.55ex);%
168 \end{tikzpicture}}}
169
170\newcommand*\tikzto{\mathchoice{\@tikzto{1.0}{1}}{\@tikzto{1.0}{1}}{\@tikzcto{0.8}{0.9}}{\@tikzcto{0.6}{0.75}}}
171\newcommand*\tikzcto{\mathchoice{\@tikzcto{1.0}{1}}{\@tikzcto{1.0}{1}}{\@tikzcto{0.8}{0.9}}{\@tikzcto{0.6}{0.75}}}
172\newcommand*\tikzonto{\mathchoice{\@tikzonto{1.0}{1}}{\@tikzonto{1.0}{1}}{\@tikzonto{0.8}{0.9}}{\@tikzonto{0.6}{0.75}}}
173\newcommand*\tikzinto{\mathchoice{\@tikzinto{1.0}{1}}{\@tikzinto{1.0}{1}}{\@tikzinto{0.8}{0.9}}{\@tikzinto{0.6}{0.75}}}
174\newcommand*\tikzclim{\mathchoice{\@tikzclim{1.0}{1}}{\@tikzclim{1.0}{1}}{\@tikzclim{0.8}{0.9}}{\@tikzclim{0.6}{0.75}}}
175\newcommand*\tikzopim{\mathchoice{\@tikzopim{1.0}{1}}{\@tikzopim{1.0}{1}}{\@tikzopim{0.8}{0.9}}{\@tikzopim{0.6}{0.75}}}
176\newcommand*\tikzmapsto{\mathchoice{\@tikzmapsto{1.0}{1}}{\@tikzmapsto{1.0}{1}}{\@tikzmapsto{0.8}{0.9}}{\@tikzmapsto{0.6}{0.75}}}
177\newcommand*\tikziso{\mathchoice{\@tikziso{1.0}{1}{\displaystyle}{0pt}}%
178 {\@tikziso{1.0}{1}{\textstyle}{0pt}}%
179 {\@tikziso{0.8}{0.9}{\scriptstyle}{0pt}}%
180 {\@tikziso{0.67}{0.8}{\scriptscriptstyle}{0.15ex}}}
181\makeatother
182
183\renewcommand*{\to}[1][]{\ifthenelse{\isempty{#1}}{\tikzto}{\xrightarrowb{#1}}}
184\newcommand*{\cto}{\ensuremath{\tikzcto}}
185\newcommand*{\into}[1][]{\ifthenelse{\isempty{#1}}{\tikzinto}{\xrightarrowb[into]{#1}}}
186\newcommand*{\onto}[1][]{\ifthenelse{\isempty{#1}}{\tikzonto}{\xrightarrowb[onto]{#1}}}
187\newcommand*{\clim}{\tikzclim}
188\newcommand*{\opim}{\tikzopim}
189
190\newcommand*{\iso}{\tikziso}
191
192\renewcommand*{\mapsto}{\tikzmapsto}
193
194\newcommand*{\ca}[1]{\ensuremath{\mathscr{#1}}\xspace}
195\renewcommand*{\cal}[1]{\ensuremath{\mathcal{#1}}\xspace}
196\newcommand*{\f}[1]{\ensuremath{\mathfrak{#1}}\xspace}
197
198\newcommand{\cl}[2][0]{{}\mkern#1mu\overline{\mkern-#1mu#2}}
199\newcommand*{\Int}[1]{\ensuremath{#1^\circ}\xspace}
200
201\newcommand*{\infcat}[1][1]{\((\infty,#1)\)–category\xspace}
202\newcommand*{\linfcat}{\(\infty\)–category\xspace}
203\newcommand*{\infcats}[1][1]{\((\infty,#1)\)–categories\xspace}
204\newcommand*{\linfcats}{\(\infty\)–categories\xspace}
205\newcommand*{\Cat}{\textnormal{\textsf{Cat}}}
206\newcommand*{\spc}{\textnormal{\textbf{Spc}}}
207\newcommand*{\ICat}{\textnormal{\textbf{Cat}}}
208\newcommand*{\infpshv}{\operatorname{\textnormal{\textbf{PShv}}}}
209\newcommand*{\infshv}{\operatorname{\textnormal{\textbf{Sh}}}}
210\newcommand*{\hypshv}{\operatorname{\widehat{\textnormal{\textbf{Sh}}}}}
211\newcommand*{\hyp}[1][]{\operatorname{\ca H}^{#1}\mkern-2mu}
212\newcommand*{\phyp}[1][]{\operatorname{\ca H}^{#1}_{\bullet}\mkern-2mu}
213\newcommand*{\sh}{\operatorname{\ca{SH}}}
214\newcommand*{\shv}{\operatorname{\textnormal{\textsf{Shv}}}}
215\newcommand*{\pshv}{\operatorname{\textnormal{\textsf{PShv}}}}
216\newcommand*{\Aff}{\operatorname{\textnormal{\textsf{Aff}}}}
217\newcommand*{\Ring}{\operatorname{\textnormal{\textsf{Ring}}}}
218\newcommand*{\Alg}{\operatorname{\textnormal{\textsf{Alg}}}}
219\newcommand*{\Disc}{\operatorname{\textnormal{\textsf{Disc}}}}
220\newcommand*{\mfd}{\operatorname{\textnormal{\textsf{Mfd}}}}
221\newcommand*{\sm}[1][k]{\mathop{\textnormal{\textsf{Sm}}/#1}}
222\newcommand*{\Simp}{\mathbf{\Delta}}
223\newcommand*{\sset}{\operatorname{\textnormal{\textsf{sSet}}}}
224\newcommand*{\ssetQ}{\operatorname{\textnormal{\textsf{sSet}}_{\textnormal{\textsf{Q}}}}}
225\newcommand*{\ssetJ}{\operatorname{\textnormal{\textsf{sSet}}_{\textnormal{\textsf{J}}}}}
226\newcommand*{\Top}{\operatorname{\textnormal{\textsf{Top}}}}
227\newcommand*{\qcat}{\operatorname{\textnormal{\textsf{qCat}}_{2}}}
228\newcommand*{\Fin}{\operatorname{\textnormal{\textsf{Fin}}_{*}}}
229\newcommand*{\tang}[1]{\textnormal{T}\mkern-2mu_{#1}}
230\newcommand*{\dual}[1]{\textnormal{D}#1}
231\newcommand*{\sch}[1][k]{\mathop{\textnormal{\textsf{Sch}}/#1}}
232
233\def\<#1>{\left\langle #1 \right\rangle}
234
235\newcommand{\dd}[1][]{%
236 \mathrm{d}%
237 \if\relax\detokenize{#1}\relax%
238 \else%
239 ^{#1}\mkern-1.5mu%
240 \fi%
241}
242
243\newcommand{\Diff}[1][]{%
244 \mathrm{D}%
245 \if\relax\detokenize{#1}\relax%
246 \else%
247 ^{#1}\m
248 kern-1.5mu%
249 \fi%
250}
251
252\newcommand{\ddd}[1][]{%
253 \mathop{}\!\dd[#1]%
254}
255
256\undef\AA
257\undef\SS
258\renewcommand*{\do}[1]{\expandafter\def\csname#1#1\endcsname{\ensuremath{\mathbb{#1}}\xspace}}
259\docsvlist{A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z}