sheaves.sty

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{sheaves}

%% Macros for if-then-else logic.
\RequirePackage{iftex}
\NewDocumentCommand\@ifmemoirloaded{m}{\@ifclassloaded{memoir}{#1}{}}
\NewDocumentCommand\@ifarticleloaded{m}{\@ifclassloaded{article}{#1}{}}

%% Use the Dutch format for dates, also in English (1 January 2000)
\RequirePackage[calc]{datetime2}
\DTMnewdatestyle{daymonthyear}{%
  \RenewDocumentCommand\DTMdisplaydate{mmmm}{\number##3~\DTMmonthname{##2} \number##1}
  \RenewDocumentCommand\DTMDisplaydate{mmmm}{\DTMdisplaydate{##1}{##2}{##3}{##4}}
}
\DTMsetdatestyle{daymonthyear}

%% Do not use wide spaces between sentences.
\frenchspacing
%% Do not indent
% \@ifclassloaded{memoir}{%
%   \setlength{\parindent}{0pt}
%   \nonzeroparskip
% }{%
%   \RequirePackage[parfill]{parskip}
% }

%% Use UTF-8 input if compiled with pdfTeX.
\ifpdftex
  \RequirePackage{textcomp}
  \RequirePackage[utf8]{inputenc}
\fi

\RequirePackage[style=british]{csquotes}

%% Reduce margins if article is loaded
% \@ifarticleloaded{\RequirePackage[scale=.7]{geometry}}
\RequirePackage[scale=0.65, vscale=0.75]{geometry}
%% Graphics
\RequirePackage{graphicx}
%% Figures
\RequirePackage{caption}
\RequirePackage{subcaption}
\@ifclassloaded{memoir}{%
  \captiondelim{\,\(\cdot\)\,}
  \captionnamefont{\sbseries}
}{%
  \DeclareCaptionLabelSeparator{dot}{\,\(\cdot\)\,}
  \DeclareCaptionFont{sb}{\sbseries}
  \captionsetup{labelfont=sb, labelsep=dot}
}
% Number subfigures with roman numerals
\RenewDocumentCommand\thesubfigure{}{\roman{subfigure}}
%% Maths
\RequirePackage{amsmath,amsthm}
\RequirePackage{mathtools}
% \mathtoolsset{centercolon}
\RequirePackage{bm} % bold maths

\RequirePackage{cases} % numcases
%% TikZ figures
\RequirePackage{tikz}
\RequirePackage{pgfplots}
\usetikzlibrary{arrows,3d}
\RequirePackage{tikz-3dplot}
%% Commutative diagrams
\RequirePackage{tikz-cd}
% Arrow style
\tikzcdset{
  arrow style=tikz,
  diagrams={>={Straight Barb[scale=0.8]}}
  % diagrams={>={Classical TikZ Rightarrow}
}
\usetikzlibrary{decorations.pathmorphing}
% Style for monos, epis, isos, weak equivalences, (acyclic) cofibrations, (acyclic) fibrations.
\tikzcdset{
  mono/.style={tail},
  epi/.style={two heads},
  iso/.style={"\cong"' sloped},
  iso'/.style={"\cong" sloped},
  pullback/.style={phantom, "\lrcorner", pos=0.025},
  pushout/.style={phantom, "\ulcorner", pos=0.025},
  weak equivalence/.style={"\sim"' sloped},
  weak equivalence'/.style={"\sim" sloped},
  cofibration/.style={hook},
  fibration/.style={two heads},
  acyclic cofibration/.style={weak equivalence, cofibration},
  acyclic cofibration'/.style={weak equivalence', cofibration},
  acyclic fibration/.style={weak equivalence, fibration},
  acyclic fibration'/.style={weak equivalence', fibration},
  inclusion/.style={hook},
}
%% Tables
\RequirePackage{booktabs}

%% Font selection
\RequirePackage{fontspec}
\RequirePackage[symbols]{kpfonts-otf}
\usepackage[only,boxslash]{stmaryrd}
\newfontfamily\KpLight{KpLight}[
  Numbers=Lowercase,
  SmallCapsFeatures={Letters=SmallCaps},
  Ligatures={TeX,Common,Required},
  Style=Swash,
  StylisticSet=1,
]
\defaultfontfeatures+[KpRoman,KpSans]{
  Numbers=Lowercase,
  SmallCapsFeatures={Letters=SmallCaps},
  Ligatures={TeX,Common,Required},
  Style=Swash,
  StylisticSet=1,
}
\defaultfontfeatures+[ETbb]{
  Numbers=Lowercase,  % oldstyle numbers
  Ligatures={TeX,Common},
  Style=Swash,
}
\newfontfamily\LaptureCaption{LaptureCaption}[
  Path = ./fonts/,
  UprightFont = {Lapture-Caption-Regular.ttf},
  BoldFont = {Lapture-Caption-Semibold.ttf},
  ItalicFont = {Lapture-Caption-Italic.ttf},
  BoldItalicFont = {Lapture-Caption-Semibold-Italic.ttf},
]
\newfontfamily\LaptureSubhead{LaptureSubhead}[
  Path = ./fonts/,
  UprightFont = {Lapture-Subhead-Regular.ttf},
  BoldFont = {Lapture-Subhead-Semibold.ttf},
  ItalicFont = {Lapture-Subhead-Italic.ttf},
  BoldItalicFont = {Lapture-Subhead-Semibold-Italic.ttf},
]
\newfontfamily\ArtigoDisplay{ArtigoDisplay}[
  Path = ./fonts/,
  UprightFont = {Artigo-Display-Regular.ttf},
  BoldFont = {Artigo-Display-Medium.ttf},
]
\defaultfontfeatures+[LaptureCaption,LaptureSubhead]{
  Numbers=Lowercase,
  Ligatures={TeX,Common,Required},
}

%% Kp
% \setmainfont{KpRoman}
% \setmonofont{KpMono}
% \newcommand*{\ltseries}{\KpLight}
% \newcommand*{\sbseries}{\KpLight\bfseries}
% \DeclareTextFontCommand{\textlt}{\ltseries}
% \DeclareTextFontCommand{\textsb}{\sbseries}

%% Fbb (Bembo-like)
\setmainfont{ETbb}
% \setmonofont[Scale=MatchLowercase]{Inconsolata}
\newcommand*{\sbseries}{\bfseries}
\DeclareTextFontCommand{\textsb}{\sbseries}

\setmathfont{KpMath-Regular}[Style=leqslant]
\setmathfont{KpMath-Regular}[range={cal,bfcal},StylisticSet=1]
%% Use mathcal from Boondox, mathbb from mathpazo.
\let\mathbfcal\undefined
\usepackage[cal=boondoxo,bb=pazo]{mathalpha}

\setsansfont[Scale=MatchLowercase]{MerriweatherSansLight}

\RequirePackage{microtype}

%% Links
\RequirePackage{hyperref}
% \definecolor{link}{RGB}{0,128,192}
\definecolor{link}{RGB}{0,0,0}
\hypersetup{
  colorlinks=true,
  urlcolor=link,
  linkcolor=link,
  citecolor=link,
  filecolor=link,
  anchorcolor=black,
}
\urlstyle{sf}
\RequirePackage[capitalise,nameinlink,noabbrev]{cleveref}
%% Refer to sections and subsections using the paragraph symbol followed by a non-breaking space.
%% Typographic reference: <https://practicaltypography.com/paragraph-and-section-marks.html>
% \crefformat{chapter}{#2\S#1#3}
% \crefmultiformat{chapter}{#2\S\S#1#3}{ and~#2#1#3}{, #2#1#3}{, and~#2#1#3}
\Crefname{chapter}{Lecture}{Lectures}
\crefname{chapter}{Lecture}{Lectures}
\crefformat{section}{#2\S~#1#3}
\Crefformat{section}{#2Section~#1#3}
\crefmultiformat{section}{#2\S\S~#1#3}{ and~#2#1#3}{, #2#1#3}{, and~#2#1#3}
\Crefmultiformat{section}{#2Sections~#1#3}{ and~#2#1#3}{, #2#1#3}{, and~#2#1#3}
\crefformat{subsection}{#2\S~#1#3}
\Crefformat{subsection}{#2Section~#1#3}
\crefmultiformat{subsection}{#2\S\S~#1#3}{ and~#2#1#3}{, #2#1#3}{, and~#2#1#3}
\Crefmultiformat{subsection}{#2Sections~#1#3}{ and~#2#1#3}{, #2#1#3}{, and~#2#1#3}
% Refer to equations by their number enclosed in brackets, not preceded by ‘Equation(s)’.
\Crefname{equation}{}{}
\crefname{equation}{}{}
% Refer to items without the word ‘Item(s)’.
\Crefname{enumi}{}{}
\crefname{enumi}{}{}
\Crefname{enumii}{}{}
\crefname{enumii}{}{}
\Crefname{enumiii}{}{}
\crefname{enumiii}{}{}
\Crefname{enumiv}{}{}
\crefname{enumiv}{}{}
\Crefname{enumv}{}{}
\crefname{enumv}{}{}

%% Itemize, enumerate, description
\RequirePackage[inline]{enumitem}  
\setlist[enumerate,1]{label=(\roman*)}
\setlist[enumerate,2]{label=(\alph*)}
%% Decrease size around itemize, enumerate, description (memoir)
\@ifmemoirloaded{\tightlists}
%%%% TODO: same thing for article?

%% To do notes
\RequirePackage{todonotes}
\newcommand{\towrite}[1]{\todo[inline,color=blue!20]{\textsf{To do: #1}}}

%% This package adds the list of references to the table of contents.
%% Not necessary for memoir.
\@ifarticleloaded{\RequirePackage[nottoc,numbib]{tocbibind}}

%% References
\usepackage[backend=biber,style=alphabetic,backref=true,backrefstyle=two,sorting=nyt]{biblatex}
\addbibresource{bibliography.bib}
\RenewDocumentCommand\RNfont{m}{\textsc{#1}}  % font of upper case roman numerals
% \DeclareBibliographyStrings{nodate={{no date}{n.d.}}}
% \DefineBibliographyStrings{english}{nodate={n.d.}}

%% These two packages are used to switch between the languages on the titlepage
\RequirePackage{ifthen} 
\RequirePackage{iflang}

%% Theorem environments
\RequirePackage{thmtools}
% \numberwithin{equation}{\@ifclassloaded{memoir}{chapter}{section}}
% \numberwithin{table}{\@ifclassloaded{memoir}{chapter}{section}}
% \numberwithin{figure}{\@ifclassloaded{memoir}{chapter}{section}}

\def\thmskip{\@ifclassloaded{memoir}{.5\baselineskip}{0}}
\declaretheoremstyle[
  spaceabove=\thmskip,
  spacebelow=\thmskip,
  headfont=\sbseries,
  bodyfont=\itshape,
  notefont=\normalfont,
  headpunct={\,\(\cdot\)},
  postheadspace=0.16667em,
  % headformat=margin,
]{myplain}
\declaretheoremstyle[
  spaceabove=\thmskip,
  spacebelow=\thmskip,
  headfont=\sbseries,
  bodyfont=\normalfont,
  notefont=\normalfont,
  headpunct={\,\(\cdot\)},
  postheadspace=0.16667em,
  % qed=\(\lozenge\),
  % headformat=margin,
]{mydefinition}
\declaretheoremstyle[
  spaceabove=\thmskip,
  spacebelow=\thmskip,
  headfont=\itshape,
  bodyfont=\normalfont,
  notefont=\normalfont,
  headpunct={\,\(\cdot\)},
  postheadspace=0.16667em,
  % qed=\(\lozenge\),
  % headformat=margin,
]{myremark}
\declaretheoremstyle[
  spaceabove=\thmskip,
  spacebelow=\thmskip,
  headfont=\itshape,
  bodyfont=\normalfont,
  notefont=\itshape,
  headpunct={.},
  % postheadspace=0.16667em,
  qed=\qedsymbol,
  numbered=no,
  % headformat=margin,
]{myproof}

\declaretheorem[
  name=Theorem,
  refname={Theorem,Theorems},
  style=myplain,
  numberwithin=\@ifclassloaded{memoir}{section}{subsection}
]{thm}
\declaretheorem[
  name=Theorem,
  refname={Theorem,Theorems},
  style=myplain,
  numbered=no
]{thm*}
\declaretheorem[
  name=Corollary,
  sharenumber=thm,
  refname={Corollary,Corollaries},
  style=myplain
]{cor}
\declaretheorem[
  name=Lemma,
  sharenumber=thm,
  refname={Lemma,Lemmas},
  style=myplain
]{lem}
\declaretheorem[
  name=Proposition,
  sharenumber=thm,
  refname={Proposition,Propositions},
  style=myplain
]{prop}
\declaretheorem[
  name=Exercise,
  sharenumber=thm,
  refname={Exercise,Exercises},
  style=mydefinition
]{exc}
\declaretheorem[
  name=Definition,
  sharenumber=thm,
  refname={Definition,Definitions},
  style=mydefinition
]{defn}
\declaretheorem[
  name=Example,
  sharenumber=thm,
  refname={Example,Examples},
  style=myremark
]{exmp}
\declaretheorem[
  name=Examples,
  sharenumber=thm,
  refname={Examples,Examples},
  style=myremark
]{exmps}
\declaretheorem[
  name=Remark,
  sharenumber=thm,
  refname={Remark,Remarks},
  style=myremark
]{rmk}
\declaretheorem[
  name=Notation,
  sharenumber=thm,
  refname={Notation,Notations},
  style=myremark
]{notn}
\declaretheorem[
  name=Convention,
  sharenumber=thm,
  refname={Convention,Conventions},
  style=myremark
]{conv}
%% Proof environment is already defined by amsthm; redefine it here.
\RenewDocumentEnvironment{proof}{}{}{}
\let\proof\@undefined
\let\endproof\@undefined
\declaretheorem[
  name=Proof,
  refname={Proof,Proofs},
  style=myproof
]{proof}

%% Use this when enumerating in a theorem environment.
\NewDocumentCommand\thmenum{}{\leavevmode}

\AddToHook{begindocument/end}{%
  %% Swap \phi and \varphi
  \let\ovarphi\varphi
  \let\ophi\phi
  \let\phi\ovarphi
  \let\varphi\ophi
  %% Swap \epsilon and \varepsilon
  \let\ovarepsilon\varepsilon
  \let\oepsilon\epsilon
  \let\epsilon\ovarepsilon
  \let\varepsilon\oepsilon
}

%% Maths macros
%% \NewMathOperator\name{symbol} defines the following commands:
%% + \nameop for the operator ‘symbol’,
%% + \name for \name,
%% + \name* for the ordinary maths version of the operator ‘symbol’ (i.e., \mathord{\name}),
%% If a star is used (\NewMathOperator*), the limits of the operator will be set above and below the symbol.
\NewDocumentCommand\NewMathOperator{smm}{%
  \IfBooleanTF{#1}{%
    \def\DeclareMathOperatorStar{\DeclareMathOperator*}
    \expandafter\DeclareMathOperatorStar\csname \string #2 op\endcsname{#3}
  }{%
    \expandafter\DeclareMathOperator\csname \string #2 op\endcsname{#3}
  }
  \NewDocumentCommand{#2}{s}{%
    \IfBooleanTF{##1}{\mathord{\csname \string #2 op\endcsname}}{\csname \string #2 op\endcsname}
  }
}
%% Change spacing of \colon:
\RenewDocumentCommand\colon{}{\nobreak \mskip 5mu\mathpunct {}\nonscript \mkern -\thinmuskip {:}\mskip 5mu\relax}
%% Use slanted ≤ and ≥
\RenewDocumentCommand\leq{}{\leqslant}
\RenewDocumentCommand\geq{}{\geqslant}
% \RenewDocumentCommand\emptyset{}{\varnothing}
%% Common paired delimiters
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\DeclarePairedDelimiter\norm{\lVert}{\rVert}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
%% Shorthand for blackboard bold
\NewDocumentCommand\bb{m}{\mathbb{#1}}
\NewDocumentCommand\vbb{m}{\varmathbb{#1}}
%% Common operators
\NewMathOperator\id{id}
% \NewDocumentCommand\id{}{1}
\NewMathOperator\dom{dom}
\NewMathOperator\cod{cod}
\NewMathOperator\ran{ran}
\NewMathOperator\im{im}
\NewMathOperator\coim{coim}
\NewMathOperator\const{const}
\NewMathOperator\incl{incl}
\NewMathOperator\eq{eq}
\NewMathOperator\coeq{coeq}
\NewMathOperator\pr{pr}
\NewMathOperator\prl{pr_1}
\NewMathOperator\prr{pr_2}
\NewMathOperator\inj{inj}
\NewMathOperator\inl{inl}
\NewMathOperator\inr{inr}
\NewMathOperator\ob{ob}
\NewMathOperator\mor{mor}
\NewMathOperator\coker{coker}
\NewMathOperator\tr{tr}
\NewMathOperator\sk{sk}
\NewMathOperator\cosk{cosk}
\NewMathOperator\ev{ev}
\NewMathOperator\Sing{Sing}
\NewMathOperator\triv{triv}
\let\lim\@undefined
\NewMathOperator*\lim{lim}
\NewMathOperator*\colim{colim}
\NewMathOperator\res{res}
\NewMathOperator\ind{ind}
\NewMathOperator\coind{coind}
\NewMathOperator\closure{cl}
\NewMathOperator\interior{int}
\NewMathOperator\cone{cone}
\NewMathOperator\cyl{cyl}
\NewMathOperator\Lan{Lan}
\NewMathOperator\Ran{Ran}
% Categories
\NewDocumentCommand\cat{m}{\mathcal{#1}}
\NewDocumentCommand\catName{m}{\mathbf{#1}}
\NewDocumentCommand\catSet{}{\catName{Set}}
\NewDocumentCommand\catPointedSet{}{\catName{Set}_*}
\NewDocumentCommand\catCategory{}{\catName{Cat}}
\NewDocumentCommand\catCATegory{}{\catName{CAT}}
\NewDocumentCommand\catPointedCategory{}{\catName{Cat}_*}
\NewDocumentCommand\catGroupoid{}{\catName{Grpd}}
\NewDocumentCommand\catMonoid{}{\catName{Mon}}
\NewDocumentCommand\catGroup{}{\catName{Grp}}
\NewDocumentCommand\catAbelianGroup{}{\catName{Ab}}
\NewDocumentCommand\catCommutativeMonoid{}{\catName{CMon}}
\NewDocumentCommand\catRing{}{\catName{Ring}}
\NewDocumentCommand\catField{}{\catName{Field}}
\NewDocumentCommand\catGraph{}{\catName{Graph}}
\NewDocumentCommand\catQuiver{}{\catName{Quiv}}
\NewDocumentCommand\catDirectedGraph{}{\catName{Graph}_{\to}}
\NewDocumentCommand\catTopologicalSpace{}{\catName{Top}}
\NewDocumentCommand\catLocalHomeomorphism{}{\catName{LocalHomeo}}
\NewDocumentCommand\catCoveringSpace{}{\catName{Cov}}
\NewDocumentCommand\catBasedTopologicalSpace{}{\catName{Top}_*}
\NewDocumentCommand\catVectorSpace{o}{\IfNoValueTF{#1}{\catName{Vect}}{\catName{Vect}_{#1}}}
\NewDocumentCommand\catModule{o}{\IfNoValueTF{#1}{\catName{Mod}}{\catName{Mod}_{#1}}}
\NewDocumentCommand\catChainComplex{o}{\IfNoValueTF{#1}{\catName{Ch}}{\catName{Ch}_{#1}}}
\NewDocumentCommand\catSimplex{}{\bm{\Delta}}
\NewDocumentCommand\catSimplicial{m}{\catName{s}#1}
\NewDocumentCommand\catSimplicialSet{}{\catSimplicial{\catSet}}
\NewDocumentCommand\catAlgebra{o}{\IfNoValueTF{#1}{\catName{Alg}}{\catName{Alg}_{#1}}}
\NewDocumentCommand\catCoalgebra{o}{\IfNoValueTF{#1}{\catName{coAlg}}{\catName{coAlg}_{#1}}}
%% TODO: kerning on other side
\NewDocumentCommand\catElements{}{\mkern-4mu\int\mkern-7mu}
\DeclareMathOperator\catHomotopy{\catName{Ho}}
\DeclareMathOperator\catK{\catName{K}}
\DeclareMathOperator\catDerived{\catName{D}}
\DeclareMathOperator\catFunctor{\catName{Fun}}
\NewDocumentCommand\catOrbit{o}{\IfNoValueTF{#1}{\catName{Orb}}{\catName{Orb}_{#1}}}
\DeclareMathOperator\catMap{\catName{Map}}
\DeclareMathOperator\catPresheaf{\catName{PSh}}
\DeclareMathOperator\catSheaf{\catName{Sh}}
\DeclareMathOperator\catLocallyConstantSheaf{\catName{Sh}^{\smash{\textup{lc}}}}
\DeclareMathOperator\catAbelianPresheaf{\catName{PAb}}
\DeclareMathOperator\catAbelianSheaf{\catAbelianGroup}
\DeclareMathOperator\catLocallyConstantAbelianSheaf{\catAbelianGroup^{\smash{\textup{lc}}}}
\NewDocumentCommand\catWalkingEdge{}{\bm{\Gamma}}
\NewDocumentCommand\catWalkingArrow{}{\vbb 2}
\NewDocumentCommand\catTerminal{}{\vbb 1}

% Labelled double arrows in all directions
\NewDocumentCommand\overunderlabels{mmO{}}{\overset{#2}{\underset{#3}{#1}}}
\NewDocumentCommand\xrightrightarrows{mO{}}{\overunderlabels{\rightrightarrows}{#1}[#2]}
\NewDocumentCommand\xrightleftarrows{mO{}}{\overunderlabels{\rightleftarrows}{#1}[#2]}
\NewDocumentCommand\xleftrightarrows{mO{}}{\overunderlabels{\leftrightarrows}{#1}[#2]}
\NewDocumentCommand\xleftleftarrows{mO{}}{\overunderlabels{\leftleftarrows}{#1}[#2]}

\NewDocumentCommand\To{}{\Rightarrow}
\NewDocumentCommand\weakequivalenceTo{}{\mathrel{\tilde{\To}}}
\NewDocumentCommand\weakequivalenceto{}{\mathrel{\tilde{\to}}}
\NewDocumentCommand\opp{}{^{\mathrm{op}}}
\NewDocumentCommand\leftadj{}{\dashv}
\NewDocumentCommand\rightadj{}{\vdash}
\NewMathOperator\Homop{Hom}
%% \HomX,Y produces \Homop(X,Y), \Hom[\cat C](X,Y) produces \Homop_{\cat C}(X,Y)
\NewDocumentCommand\Hom{o}{\IfNoValueTF{#1}{\Homop}{\Homop_{#1}}}
\RenewDocumentCommand\hom{o}{\Hom[#1]}
%% Alternative definition of Hom:
%% \Hom(X,Y) produces \Homop(X,Y), \Hom[\cat C](X,Y) produces \cat C(X,Y)
% \NewDocumentCommand\Hom{O{\Homop}}{#1}
\NewMathOperator\Autop{Aut}
%% \Aut(X,Y) produces \Autop(X,Y), \Aut[\cat C](X,Y) produces \Autop_{\cat C}(X,Y)
\NewDocumentCommand\Aut{o}{\IfNoValueTF{#1}{\Autop}{\Autop_{#1}}}
\NewMathOperator\Endop{End}
%% \End(X,Y) produces \Endop(X,Y), \End[\cat C](X,Y) produces \Endop_{\cat C}(X,Y)
\NewDocumentCommand\End{o}{\IfNoValueTF{#1}{\Endop}{\Endop_{#1}}}
\NewMathOperator\Extop{Ext}
%% \ExtX,Y produces \Extop(X,Y), \Ext[\cat C](X,Y) produces \Extop_{\cat C}(X,Y)
\NewDocumentCommand\Ext{o}{\IfNoValueTF{#1}{\Extop}{\Extop_{#1}}}
\NewMathOperator\Torop{Tor}
%% \TorX,Y produces \Torop(X,Y), \Tor[\cat C](X,Y) produces \Torop_{\cat C}(X,Y)
\NewDocumentCommand\Tor{o}{\IfNoValueTF{#1}{\Torop}{\Torop^{#1}}}
\NewDocumentCommand\inv{}{^{-1}}
\NewDocumentCommand\deloop{}{\mathbf{B}}
\NewDocumentCommand\classifying{}{E}
\NewDocumentCommand\nerve{}{N}
\NewDocumentCommand\centre{}{\mathrm{Z}}
\NewDocumentCommand\normaliser{o}{\IfNoValueTF{#1}{\mathrm{N}}{\mathrm{N}_{#1}}}
\NewDocumentCommand\initial{}{\emptyset}
\NewDocumentCommand\terminal{}{{\mathord{*}}}
\NewDocumentCommand\cotimes{}{\amalg}
%% Alternative definition for cotimes:
% \NewDocumentCommand\cotimes{}{+}
\NewDocumentCommand\tensor{}{\otimes}
\NewDocumentCommand\cotensor{}{\pitchfork}
% \NewDocumentCommand\disjointunion{}{\sqcup}
\NewDocumentCommand\disjointunion{}{\cotimes}
%% unicode-math’s \pullback and \pushout are defined in \AtBeginDocument
\AddToHook{begindocument/end}{%
  \DeclareDocumentCommand\pullback{sm}{
    \IfBooleanTF{#1}{%
      \underset{#2}{\times}
    }{%
      \mathchoice{\underset{#2}{\times}}{\times_{#2}}{\times_{#2}}{\times_{#2}}
    }
  }
  \DeclareDocumentCommand\pushout{sm}{
    \IfBooleanTF{#1}{%
      \underset{#2}{\cotimes}
    }{%
      \mathchoice{\underset{#2}{\cotimes}}{\cotimes_{#2}}{\cotimes_{#2}}{\cotimes_{#2}}
    }
  }
  \DeclareDocumentCommand\union{sm}{
    \IfBooleanTF{#1}{%
      \underset{#2}{\cup}
    }{%
      \mathchoice{\underset{#2}{\cup}}{\cup_{#2}}{\cup_{#2}}{\cup_{#2}}
    }
  }
}
\NewDocumentCommand\homotopypullback{sm}{
  \IfBooleanTF{#1}{%
    \underset{#2}{\overset{h}{\times}}
  }{%
    \mathchoice{\underset{#2}{\overset{h}{\times}}}{\times^h_{#2}}{\times^h_{#2}}{\times^h_{#2}}
  }
}
\NewDocumentCommand\chaincomplex{m}{{#1}_\bullet}
\NewDocumentCommand\cochaincomplex{m}{{#1}^\bullet}
\NewDocumentCommand\homology{o}{\IfNoValueTF{#1}{H}{H_{#1}}}
\NewDocumentCommand\cycle{o}{\IfNoValueTF{#1}{Z}{Z_{#1}}}
\NewDocumentCommand\bdary{o}{\IfNoValueTF{#1}{B}{B_{#1}}}
\NewDocumentCommand\reducedhomology{o}{\IfNoValueTF{#1}{\tilde{H}}{\tilde{H}_{#1}}}
\NewDocumentCommand\cohomology{o}{\IfNoValueTF{#1}{H}{H^{#1}}}
\NewDocumentCommand\homotopy{o}{\IfNoValueTF{#1}{\pi}{\pi_{#1}}}
\NewDocumentCommand\sphere{o}{\IfNoValueTF{#1}{S}{S^{#1}}}
\NewDocumentCommand\disk{o}{\IfNoValueTF{#1}{D}{D^{#1}}}
\NewDocumentCommand\opendisk{o}{\IfNoValueTF{#1}{\mathring{D}}{\mathring{D}^{#1}}}
\NewDocumentCommand\simplex{o}{\IfNoValueTF{#1}{\Delta}{\Delta^{#1}}}
\NewDocumentCommand\horn{oo}{\IfNoValueTF{#1}{\Lambda}{\IfNoValueTF{#2}{\Lambda^{#1}}{\Lambda^{#1}_{#2}}}}
\NewDocumentCommand\spine{o}{\IfNoValueTF{#1}{I}{I^{#1}}}
\NewDocumentCommand\simplicialsphere{o}{\catSimplicial\sphere[#1]}
\NewDocumentCommand\simplicialdisk{o}{\catSimplicial\disk[#1]}
\NewDocumentCommand\point{}{\vcenter{\hbox{\tiny\(\bullet\)}}}
\NewDocumentCommand\selfloop{}{\vcenter{\hbox{\rotatebox[origin=c]{180}{\small\(\circlearrowright\)}}}}
% \NewDocumentCommand\selfloop{}{\rotatebox[origin=c]{180}{\(\circlearrowright\)}}
\NewDocumentCommand\interval{}{I}
\NewDocumentCommand\leftderived{}{\mathbf{L}}
\NewDocumentCommand\rightderived{}{\mathbf{R}}
\DeclarePairedDelimiterX\setpred[2]{\{}{\}}{\,#1\;\delimsize\vert\;\mathopen{}#2\,}
\DeclarePairedDelimiterX\grouppres[2]{\langle}{\rangle}{\,#1\;\delimsize\vert\;\mathopen{}#2\,}
\NewDocumentCommand\opposingfunctors{mmmm}{#3 \colon #1 \rightleftarrows #2 \colon #4}
\NewDocumentCommand\leftlifting{m}{{{}^{\boxslash} #1}}
\NewDocumentCommand\rightlifting{m}{{#1}^{\boxslash}}
% for decorative numbered faces in commutative diagram
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
            \node[shape=circle,draw,inner sep=2pt] (char) {\small #1};}}

%% Classes of maps in model categories
\NewDocumentCommand\weakequivalences{}{\mathcal{W}}
\NewDocumentCommand\cofibrations{}{\mathcal{C}}
\NewDocumentCommand\fibrations{}{\mathcal{F}}

%% Do not fill to the bottom of the page
% \@ifmemoirloaded{\raggedbottom}

%% Fonts for chapter and section heads
\@ifmemoirloaded{%
  \RenewDocumentCommand\printchaptername{}{\LaptureCaption\LARGE\textls[120]{\textsc{lecture}}}
  \RenewDocumentCommand\chapternamenum{}{\enskip\space}
  \RenewDocumentCommand\printchapternum{}{\LaptureCaption\LARGE\textsc{\thechapter}}
  \RenewDocumentCommand\midchapskip{}{12pt}
  \RenewDocumentCommand\chaptitlefont{}{\LaptureCaption\Huge\itshape}
  \setsecheadstyle{\sbseries\Large}
  \setsubsecheadstyle{\sbseries\large}
  % \newcommand{\afterchapternum}{\par\nobreak\vskip \midchapskip}
  % \RenewDocumentCommand\printchaptername{}{\textls[120]{\chapnamefont\@chapapp}}
  % \chapterstyle{hangnum}
  % \hangsecnum{}
  % \RenewDocumentCommand\chaptitlefont{}{\sffamily\Huge\bfseries}
}
\@ifarticleloaded{%
  \RequirePackage{sectsty}
  \allsectionsfont{\LaptureSubhead\bfseries}
}

%% Set page numbers at left resp. right on odd resp. even pages.
% \@ifmemoirloaded{%
%   \makeevenfoot{plain}{}{}{\thepage}
%   \makeoddfoot{plain}{\thepage}{}{}
% }

% \@ifmemoirloaded{%
%   \let\oldchapter\chapter
%   \let\oldsection\section
%   \let\oldsubsection\subsection
%   \let\section\oldchapter
%   \let\subsection\oldsection
% }

\AddToHook{begindocument/end}{%
  \let\subset\subseteq
}

\tikzset{
  dashtwocolors/.code args={#1,#2}{
    \pgfkeysalso{dash pattern=on 4pt off 4pt, #1, postaction={draw=#2, dash phase=4pt}}
  },
  dashthreecolors/.code args={#1,#2,#3}{
    \pgfkeysalso{dash pattern=on 4pt off 8pt, #1, postaction={draw=#2, dash phase=4pt}, postaction={draw=#3, dash phase=8pt}}
  }
}

% Sheaf related commands
% poset of opens
\NewMathOperator\open{Open}
\NewMathOperator\Map{Map}
% continuous maps
% \NewMathOperator\cont{Cont}
\NewDocumentCommand\cont{o}{\IfNoValueTF{#1}{\catTopologicalSpace}{\catTopologicalSpace_{/{#1}}}}
% constant sheaf
\NewDocumentCommand\constSheaf{m}{\underline{#1}}
\NewMathOperator\etalespace{sp}

\RequirePackage{epigraph}
\setlength\epigraphrule{0pt}
\setlength\beforeepigraphskip{0pt}
\setlength\afterepigraphskip{0pt}
\NewDocumentCommand\remyquote{m}{\epigraph{\makebox[0pt][r]{`}{#1}'}{}}

\DeclareMathOperator\cofilteredlim{\varprojlim}
\newcommand{\colim@}[2]{%
  \vtop{\m@th\ialign{##\cr
    \hfil$#1\operator@font colim$\hfil\cr
    \noalign{\nointerlineskip\kern1.5\ex@}#2\cr
    \noalign{\nointerlineskip\kern-\ex@}\cr}}%
}
\newcommand{\filteredcolim}{%
  \mathop{\mathpalette\colim@{\rightarrowfill@\scriptscriptstyle}}\nmlimits@
}
\renewcommand{\varprojlim}{%
  \mathop{\mathpalette\varlim@{\leftarrowfill@\scriptscriptstyle}}\nmlimits@
}
\renewcommand{\varinjlim}{%
  \mathop{\mathpalette\varlim@{\rightarrowfill@\scriptscriptstyle}}\nmlimits@
} % Source: https://tex.stackexchange.com/questions/284059/new-command-for-filtered-colimits-and-limits

% for subfile compilation
\usepackage{subfiles} % best loaded last in preamble according to internet although do not know if that means here (?)