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% font with unicode support
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\usepackage{fontspec}
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% clickable tocs
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\usepackage{hyperref}
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% use english
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\usepackage{polyglossia}
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\setmainlanguage[variant=british]{english}
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% floating figures
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\usepackage{float}
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% for appendices
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\usepackage[toc,page]{appendix}
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% for multiple cites
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\usepackage{cite}
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% fancy diagrams
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\usepackage{tikz}
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\usetikzlibrary{matrix, arrows, decorations}
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\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto}
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% simple diagrams
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\usepackage[all,cmtip]{xy}
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\usepackage{graphicx}
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\graphicspath{ {./images/} }
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\usepackage{caption}
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\usepackage{subcaption}
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% Matrices have a upper bound for its size
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\setcounter{MaxMatrixCols}{20}
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% for the fib arrow
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\usepackage{amssymb}
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% mathbb for lowercase bbs
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\usepackage{bbm}
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% Some basic objects
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\newcommand{\N}{\mathbb{N}} % natural numbers
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\newcommand{\Np}{{\mathbb{N}^{>0}}} % positive numbers
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\newcommand{\Z}{\mathbb{Z}} % integers
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\newcommand{\R}{\mathbb{R}} % reals
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\newcommand{\Q}{\mathbb{Q}} % rationals
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\renewcommand{\k}{\mathbbm{k}} % default ground ring
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% Basic category stuff
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\newcommand{\cat}[1]{\mathbf{#1}} % the category of ...
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\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category
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\newcommand{\Hom}{\mathbf{Hom}}
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\newcommand{\id}{\mathbf{id}}
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\newcommand{\Ho}{\cat{Ho}}
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% Categories
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\newcommand{\Set}{\cat{Set}} % sets
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\newcommand{\Top}{\cat{Top}} % topological spaces
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\newcommand{\Ab}{\cat{Ab}} % abelian groups
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\newcommand{\DELTA}{\boldsymbol{\Delta}}% the simplicial cat
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\newcommand{\simplicial}[1]{\cat{s{#1}}}% simplicial objects
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\newcommand{\sSet}{\simplicial{\Set}} % simplicial sets
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\newcommand{\Mod}[1]{\cat{{#1}Mod}} % modules over a ring
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\newcommand{\Alg}[1]{\cat{{#1}Alg}} % algebras over a ring
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\newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}} % graded modules over a ring
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\newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}} % graded algebras over a ring
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\newcommand{\dgMod}[1]{\cat{dg\mbox{-}{#1}Mod}} % differential graded modules over a ring
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\newcommand{\dgAlg}[1]{\cat{dg\mbox{-}{#1}Alg}} % differential graded algebras over a ring
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\newcommand{\Ch}[1]{\cat{Ch_{n\geq0}({#1})}} % chain complexes
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\newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}} % cochain complexes
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\newcommand{\DGA}{\cat{DGA}} % cochain algebras
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\newcommand{\CDGA}{\cat{CDGA}} % commutative cochain algebras
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\newcommand{\cof}{\hookrightarrow} % cofibration
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\newcommand{\fib}{\twoheadrightarrow} % fibration
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\newcommand{\we}{\tot{\simeq}} % weak equivalence
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% for use in xy diagrams
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\newcommand{\arcof}{\ar@{^{(}->}}
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\newcommand{\artcof}{\ar@{^{(}->}|\simeq}
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\newcommand{\arfib}{\ar@{->>}}
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\newcommand{\artfib}{\ar@{->>}|\simeq}
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\newcommand{\arwe}{\ar|\simeq}
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% adjunction symbol for xymatrices
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\newcommand{\xyadj}{\raisebox{0.2\height}{\scalebox{0.5}{$\perp$}}}
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% pushout and pullback for xymatrices (makes empty arrow with text)
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\newcommand{\xypo}{\ar@{}[dr]|(.75){\scalebox{1.2}{$\ulcorner$}}}
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\newcommand{\xypb}{\ar@{}[dr]|(.25){\scalebox{1.2}{$\lrcorner$}}}
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%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}}
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\newcommand{\leftadj}{\rightleftarrows}
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% Notation and operators
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\newcommand{\I}{\,\mid\,} % seperator in set notation
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\newcommand{\del}{\partial} % boundary
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\newcommand{\iso}{\cong} % isomorphic
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\newcommand{\eq}{\sim} % homotopic
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\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name
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\newcommand{\toti}[1]{\xleftarrow{\,\,{#1}\,\,}} % left arrow with name
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\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name
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\newcommand{\unit}{\eta}
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\newcommand{\counit}{\epsilon}
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\DeclareMathOperator*{\im}{im}
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\DeclareMathOperator*{\coker}{coker}
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\DeclareMathOperator*{\colim}{colim}
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\DeclareMathOperator*{\Tor}{Tor}
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\DeclareMathOperator*{\Ext}{Ext}
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\DeclareMathOperator*{\tensor}{\otimes}
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\DeclareMathOperator*{\bigtensor}{\bigotimes}
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\renewcommand{\deg}[1]{{|{#1}|}}
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\newcommand{\Char}[1]{char({#1})}
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\newcommand{\RH}{\widetilde{H}} % reduced homology
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\renewcommand{\C}{\mathcal{C}} % Serre mod C class
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\newcommand{\Apl}[0]{{A_{PL}}} % Apl simplicial set of polynomials
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% restriction of a function
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\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
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\left.\kern-\nulldelimiterspace % automatically resize the bar with \right
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#1 % the function
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\vphantom{\big|} % pretend it's a little taller at normal size
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\right|_{#2} % this is the delimiter
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}}
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% Todos in the margin
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\newcommand{\todo}[1]{
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\addcontentsline{tdo}{todo}{\protect{#1}}
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$\ast$ \marginpar{\tiny $\ast$ #1}
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}
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% Big todos in text
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\newcommand{\TODO}[1]{
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\addcontentsline{tdo}{todo}{\protect{#1}}
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{\tiny $\ast$ #1}
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}
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% TODO item, as itemize does not work
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\newcommand{\titem}[0]{\\-\qquad}
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% List of todos
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\makeatletter
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\newcommand \listoftodos{\section*{Todo list} \@starttoc{tdo}}
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\newcommand\l@todo[2]{
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\par\noindent \textit{#2}, \parbox{10cm}{#1}\par
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}
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\makeatother
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\theoremstyle{plain}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{claim}[theorem]{Claim}
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\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{notation}[theorem]{Notation}
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\newtheorem{example}[theorem]{Example}
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\newcommand{\EnvTemp}[4]{
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\begin{#1}\label{#2:#3}
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{#4}
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\end{#1}
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}
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\newcommand{\RefTemp}[3]{{#1}~\ref{#2:#3}}
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\newcommand{\Theorem}{\EnvTemp{theorem}{thm}}
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\newcommand{\Proposition}{\EnvTemp{proposition}{prop}}
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\newcommand{\Lemma}{\EnvTemp{lemma}{lem}}
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\newcommand{\Corollary}{\EnvTemp{corollary}{cor}}
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\newcommand{\Claim}{\EnvTemp{claim}{clm}}
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\newcommand{\Proof}[1]{\begin{proof}{#1}\end{proof}}
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\newcommand{\Def}{\emph}
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\newcommand{\Definition}{\EnvTemp{definition}{def}}
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\newcommand{\Notation}{\EnvTemp{notation}{not}}
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\newcommand{\Example}{\EnvTemp{example}{eg}}
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\newcommand{\TheoremRef}{\RefTemp{Theorem}{thm}}
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\newcommand{\LemmaRef}{\RefTemp{Lemma}{lem}}
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\newcommand{\CorollaryRef}{\RefTemp{Corollary}{cor}}
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\newcommand{\DefinitionRef}{\RefTemp{Definition}{def}}
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% headings for a table
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\newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}}
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% simple way to center an image
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\newcommand{\cimage}[2][]{
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\begin{center}
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\includegraphics[#1]{#2}
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\end{center}
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}
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% simple way to center a diagram
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\newcommand{\cdiagrambase}[1]{
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\begin{displaymath}
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\input{#1}
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\end{displaymath}
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}
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\newcommand{\cdiagram}[1]{
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\cdiagrambase{diagrams/#1}
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}
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