\usepackage{amsmath, amssymb, amsthm}
\usepackage[mathcal]{euscript}
\usepackage{mathtools} % contains \prescript (left sub/superscript)
\usepackage{tikz-cd}
\usepackage{mathrsfs} % math script font
\usepackage{dsfont}
\usepackage{xcolor} % coloured text
\usepackage[margin=1.25in]{geometry}
\usepackage[all]{xy}
\usepackage{adjustbox}
\usepackage{setspace} % sets line spacing
\usepackage{booktabs}
\usepackage{float}

\usepackage[style=alphabetic, backend=bibtex, doi=false, isbn=false, url=false]{biblatex}
\addbibresource{paper.bib}
% \ExecuteBibliographyOptions{url=true}
% \AtEveryBibitem{\clearfield{urldate}}

\usepackage{hyperref}
\usepackage[capitalise, nameinlink]{cleveref}

\newcommand{\alert}[1]{\textcolor{red}{[#1]}} % note to add something later

% outrageous that this is not standard tbh
\newcommand{\from}{\leftarrow}

% rm
\newcommand{\A}{\mathrm{A}}
\renewcommand{\a}{\mathrm{a}}
\newcommand{\B}{\mathrm{B}}
\newcommand{\C}{\mathrm{C}}
\renewcommand{\c}{\mathrm{c}}
\renewcommand{\d}{\mathrm{d}}
\newcommand{\D}{\mathrm{D}}
\newcommand{\e}{\mathrm{e}}
\newcommand{\E}{\mathrm{E}}
\newcommand{\f}{\mathrm{f}}
\newcommand{\F}{\mathrm{F}}
\newcommand{\G}{\mathrm{G}}
\newcommand{\g}{\mathrm{g}}
\renewcommand{\H}{\mathrm{H}}
\newcommand{\h}{\mathrm{h}}
\newcommand{\I}{\mathrm{I}}
\newcommand{\K}{\mathrm{K}}
\renewcommand{\k}{\mathrm{k}}
\renewcommand{\L}{\mathrm{L}}
\renewcommand{\l}{\mathrm{l}}
\newcommand{\M}{\mathrm{M}}
\newcommand{\m}{\mathrm{m}}
\newcommand{\N}{\mathrm{N}}
\newcommand{\n}{\mathrm{n}}
\renewcommand{\o}{\mathrm{o}}
\renewcommand{\P}{\mathrm{P}}
\newcommand{\p}{\mathrm{p}}
\newcommand{\R}{\mathrm{R}}
\renewcommand{\r}{\mathrm{r}}
\renewcommand{\S}{\mathrm{S}}
\newcommand{\s}{\mathrm{s}}
\newcommand{\T}{\mathrm{T}}
\renewcommand{\t}{\mathrm{t}}
\newcommand{\U}{\mathrm{U}}
\newcommand{\V}{\mathrm{V}}
\newcommand{\W}{\mathrm{W}}
\newcommand{\w}{\mathrm{w}}
\newcommand{\X}{\mathrm{X}}
\newcommand{\Y}{\mathrm{Y}}
\newcommand{\y}{\mathrm{y}}
\newcommand{\Z}{\mathrm{Z}}
\newcommand{\dr}{\mathrm{dR}}

% bb bold
\renewcommand{\AA}{\mathds{A}}
\newcommand{\BB}{\mathds{B}}
\newcommand{\CC}{\mathds{C}}
\newcommand{\DD}{\mathds{D}}
\newcommand{\EE}{\mathds{E}}
\newcommand{\FF}{\mathds{F}}
\newcommand{\GG}{\mathds{G}}
\newcommand{\KK}{\mathds{K}}
\newcommand{\kk}{\mathds{k}}
\newcommand{\LL}{\mathds{L}}
\newcommand{\MM}{\mathds{M}}
\newcommand{\NN}{\mathds{N}}
\newcommand{\PP}{\mathds{P}}
\newcommand{\QQ}{\mathds{Q}}
\newcommand{\RR}{\mathds{R}}
\renewcommand{\SS}{\mathds{S}}
\newcommand{\TT}{\mathds{T}}
\newcommand{\VV}{\mathds{V}}
\newcommand{\ZZ}{\mathds{Z}}


%bf bold
\newcommand{\DRM}{\mathbf{dR}}
\newcommand{\Ab}{\mathbf{A}}
\newcommand{\Cb}{\mathbf{C}}
\newcommand{\Eb}{\mathbf{E}}
\newcommand{\Fb}{\mathbf{F}}
\newcommand{\Gb}{\mathbf{G}}
\newcommand{\Hb}{\mathbf{H}}
\newcommand{\Ib}{\mathbf{I}}
\newcommand{\Jb}{\mathbf{J}}
\newcommand{\Kb}{\mathbf{K}}
\newcommand{\Lb}{\mathbf{L}}
\newcommand{\Mb}{\mathbf{M}}
\newcommand{\Nb}{\mathbf{N}}
\newcommand{\Ob}{\mathbf{O}}
\newcommand{\Pb}{\mathbf{P}}
\newcommand{\Qb}{\mathbf{Q}}
\newcommand{\Rb}{\mathbf{R}}
\newcommand{\Sb}{\mathbf{S}}
\newcommand{\Tb}{\mathbf{T}}
\newcommand{\Ub}{\mathbf{U}}
\newcommand{\Vb}{\mathbf{V}}
\newcommand{\Wb}{\mathbf{W}}
\newcommand{\Xb}{\mathbf{X}}
\newcommand{\Yb}{\mathbf{Y}}
\newcommand{\Zb}{\mathbf{Z}}
\newcommand{\Omb}{\mathbf{\Omega}}

\newcommand{\Gm}{\mathbf G_m}
\newcommand{\Ga}{\mathbf G_a}

% cal
\newcommand{\gpois}{\mathcal{GP}}
\newcommand{\gcois}{\mathcal{GC}}
\newcommand{\Acal}{\mathcal A}
\newcommand{\Bcal}{\mathcal B}
\newcommand{\Ccal}{\mathcal C}
\newcommand{\Dcal}{\mathcal D}
\newcommand{\Ecal}{\mathcal E}
\newcommand{\Fcal}{\mathcal F}
\newcommand{\Gcal}{\mathcal G}
\newcommand{\Hcal}{\mathcal H}
\newcommand{\Ical}{\mathcal I}
\newcommand{\Lcal}{\mathcal L}
\newcommand{\Mcal}{\mathcal M}
\newcommand{\Ncal}{\mathcal N}
\newcommand{\Pcal}{\mathcal P}
\newcommand{\Qcal}{\mathcal Q}
\newcommand{\Rcal}{\mathcal R}
\newcommand{\Tcal}{\mathcal T}
\newcommand{\Ycal}{\mathcal Y}
\newcommand{\Xcal}{\mathcal X}
\newcommand{\Zcal}{\mathcal Z}

% frak
\newcommand{\afr}{\mathfrak a}
\newcommand{\bfr}{\mathfrak b}
\newcommand{\cfr}{\mathfrak c}
\newcommand{\ffr}{\mathfrak f}
\newcommand{\gfr}{\mathfrak g}
\newcommand{\hfr}{\mathfrak h}
\newcommand{\lfr}{\mathfrak l}
\newcommand{\pfr}{\mathfrak p}
\newcommand{\sfr}{\mathfrak s}
\newcommand{\tfr}{\mathfrak t}
\newcommand{\vfr}{\mathfrak v}
\newcommand{\wfr}{\mathfrak w}

% scr
\newcommand{\Asc}{\mathscr A}
\newcommand{\Bsc}{\mathscr B}
\newcommand{\Csc}{\mathscr C}
\newcommand{\Dsc}{\mathscr D}
\newcommand{\Esc}{\mathscr E}
\newcommand{\Fsc}{\mathscr F}
\newcommand{\Gsc}{\mathscr G}
\newcommand{\Hsc}{\mathscr H}
\newcommand{\Isc}{\mathscr I}
\newcommand{\Jsc}{\mathscr J}
\newcommand{\Ksc}{\mathscr K}
\newcommand{\Lsc}{\mathscr L}
\newcommand{\Msc}{\mathscr M}
\newcommand{\Nsc}{\mathscr N}
\newcommand{\Osc}{\mathscr O}
\newcommand{\Psc}{\mathscr P}
\newcommand{\Qsc}{\mathscr Q}
\newcommand{\Rsc}{\mathscr R}
\newcommand{\Ssc}{\mathscr S}
\newcommand{\Tsc}{\mathscr T}
\newcommand{\Usc}{\mathscr U}
\newcommand{\Vsc}{\mathscr V}
\newcommand{\Wsc}{\mathscr W}
\newcommand{\Xsc}{\mathscr X}
\newcommand{\Ysc}{\mathscr Y}
\newcommand{\Zsc}{\mathscr Z}

% operads
\newcommand{\sseq}{\mathrm{SSeq}}
\newcommand{\enoperad}{\mathsf{end}}
\newcommand{\pp}{\mathsf p}
\newcommand{\ee}{\mathsf e}
\newcommand{\ass}{\mathsf{Ass}}
\newcommand{\lie}{\mathsf{Lie}}
\newcommand{\com}{\mathsf{Com}}
\newcommand{\br}{\mathrm{Br}}

% categories
\newcommand{\spaces}{\mathcal S\mathrm{pc}}
\newcommand{\cats}{\mathcal C\mathrm{at}}
\newcommand{\aff}{\mathcal A\mathrm{ff}}
\newcommand{\affsm}[1]{\mathcal A\mathrm{ff}^{\mathrm{sm}}_{/#1}}
\newcommand{\Mod}[1][k]{\mathcal M\mathrm{od}_{#1}}
\newcommand{\LMod}{\mathcal{LM}\mathrm{od}}
\newcommand{\RMod}{\mathcal{RM}\mathrm{od}}
\newcommand{\BMod}{\mathcal{BM}\mathrm{od}}
\newcommand{\coMod}[1][k]{\c\Mcal\o\d_{#1}}
\newcommand{\alg}[1][k]{\mathcal A\mathrm{lg}_{#1}}
\newcommand{\coalg}[1][k]{\mathrm{c}\mathcal A\mathrm{lg}_{#1}}
\newcommand{\calg}[1][k]{\mathcal{CA}\mathrm{lg}_{#1}}
\newcommand{\dalg}[1][k]{\mathcal{CA}\mathrm{lg}_{#1}}
\newcommand{\calgsm}[1][k]{\mathcal{CA}\mathrm{lg}^{\mathrm{art}}_{k/{#1}}}
\newcommand{\ccoalg}[1][k]{\mathrm{c}\mathcal{CA}\mathrm{lg}_{#1}} %cocommutative coalgebras
\newcommand{\balg}{\mathrm{b}\mathcal{A}\mathrm{lg}}
\newcommand{\filmod}[1][k]{\mathcal F\mathrm{il}\mathcal M\mathrm{od}_{#1}}
\newcommand{\cfilmod}[1][k]{\widehat{\mathcal F\mathrm{il}}\mathcal M\mathrm{od}_{#1}}
\newcommand{\filcalg}[1][k]{\mathcal F\mathrm{il}\mathcal{CA}\mathrm{lg}_{#1}}
\newcommand{\fildalg}[1][k]{\mathcal F\mathrm{il}\mathcal{CA}\mathrm{lg}_{#1}}
\newcommand{\cfilcalg}[1][k]{\widehat{\mathcal F\mathrm{il}}\mathcal{CA}\mathrm{lg}_{#1}}
\newcommand{\cfildalg}[1][k]{\widehat{\mathcal F\mathrm{il}}\mathcal{CA}\mathrm{lg}_{#1}}
\newcommand{\cfil}{\widehat{\Fcal\mathrm{il}}}
\newcommand{\cfillie}{\widehat{\Fcal\mathrm{il}}\Acal\mathrm{lg}_{\F\mathsf{Lie}}}
\newcommand{\grmod}[1][k]{\Gcal\r\Mcal \o\d _{#1}}
\newcommand{\grcalg}[1][k]{\Gcal\r\Ccal\mathrm{alg}_{#1}}
\newcommand{\grdalg}[1][k]{\Gcal\r\Ccal\Acal\mathrm{lg}_{#1}}
\newcommand{\gralg}[1][k]{\Gcal\r\Acal\mathrm{lg}_{#1}}
\newcommand{\sch}[1][k]{\mathrm{d}\mathcal{S}\mathrm{ch}_{#1}}
\newcommand{\asch}[1][k]{\mathrm{d}\mathcal{A}\mathrm{ff}_{#1}}
\newcommand{\dst}[1][k]{\mathrm{d}\mathcal{S}\mathrm{t}_{#1}}
\newcommand{\dpst}[1][k]{\mathrm{d}\mathcal{PS}\mathrm{t}_{#1}}
\newcommand{\fpst}[1][k]{\mathrm{f}\mathcal{PS}\mathrm{t}_{#1}}
\newcommand{\fmp}[1][\mathrm{pt}]{\mathcal{FMP}_{/#1}}
\newcommand{\fmpu}[1][\mathrm{pt}]{\mathcal{FMP}_{#1/}}
\newcommand{\sset}{\mathbf{Set}^{\Delta^\op}} % 1-category!
\newcommand{\qcoh}{\Qcal\Ccal\o\h}
\newcommand{\coh}{\Ccal\o\h}
\newcommand{\icoh}{\Ical\mathrm{nd}\Ccal\o\h}
\newcommand{\perf}{\Pcal\e\r\f}
\newcommand{\psh}{\mathcal{PS}\mathrm h}

\newcommand{\Actgp}{\Acal\mathrm{ct}_{\mathrm{Grp}}}
\newcommand{\Actmon}{\Acal\mathrm{ct}_{\mathrm{Mon}}}
\newcommand{\prlk}{\mathcal{P}\mathrm{r}^{\mathrm L}_k}
\newcommand{\liealg}{\mathcal{L}\mathrm{ie}\mathcal{A}\mathrm{bd}}

\newcommand{\dmmod}{\mathrm{Mod}_{\DD^+}}
\newcommand{\dmalg}{\mathrm{CAlg}_{\DD^+}}

% functors
\newcommand{\obl}{\mathrm{Forget}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\iend}{\underline{\mathcal E\mathrm{nd}}}
\newcommand{\imap}{\underline{\mathcal M\mathrm{ap}}}
\newcommand{\Tot}{\mathrm{Tot}}
\newcommand{\specinf}{\mathrm{Spec}^{\mathrm{inf}}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\cSym}{\mathrm{cSym}}
\newcommand{\dist}{\mathrm{Dist}}

% sub/superscripts
\newcommand{\afp}{\mathrm{afp}}
\newcommand{\fp}{\mathrm{fp}}
\newcommand{\aft}{\mathrm{aft}}
\newcommand{\laft}{\mathrm{laft}}
\newcommand{\ft}{\mathrm{ft}}
\newcommand{\red}{\mathrm{red}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\nil}{\mathrm{nil}}
\newcommand{\fil}{\mathrm{fil}}
\newcommand{\ex}{\mathrm{ex}}
\newcommand{\cl}{\mathrm{cl}}
\newcommand{\ind}{\mathrm{in}} %induced
\newcommand{\rel}{\mathrm{rel}}
\newcommand{\mix}{\mathrm{mix}}
\newcommand{\gr}{\mathrm{gr}}
\newcommand{\aug}{\mathrm{aug}} %august, I assume
\newcommand{\fol}{\mathrm{fol}}
\newcommand{\crys}{\mathrm{crys}}
\newcommand{\infml}{\mathrm{inf}}

\newcommand{\fprod}[1]{\underset{#1}{\times}}
\newcommand{\foprod}[1]{\underset{#1}{\otimes}}

\newcommand{\freela}{\mathrm{Free}_{\mathrm{LA}}}
\newcommand{\ins}{\mathrm{ins}}
\newcommand{\lsym}{\mathrm{Sym}}

\newcommand{\fhdr}{\F^*_\H\mathrm{dR}}
\newcommand{\fdr}{{\mathrm{FdR}}}
\newcommand{\poli}{\mathrm{Pol}^{\mathrm{inf}}}
\newcommand{\polc}{\mathrm{Pol}^{\mathrm{crys}}}
\newcommand{\calc}{\mathrm{Calc}}

\newcommand{\agm}{\Ab^1/\Gm}
\newcommand{\ahgm}{\widehat{\Ab}^1/\Gm}
\newcommand{\bga}{\B\Ga}
\newcommand{\grcirp}{\mathbf T}
\newcommand{\grcirn}{\mathbf T^\vee}
\newcommand{\BTp}{\B^\gr\grcirp}
\newcommand{\BTn}{\B^\gr\grcirn}

\newcommand{\tgp}{\widetilde{\mathcal{GP}}}
\newcommand{\gp}{\mathcal{GP}}
\newcommand{\isot}{\mathbf{Isot}}
\newcommand{\fmap}{\widehat{\mathcal{M}\mathrm{ap}}}
\newcommand{\faut}{\widehat{\mathrm{Aut}}}
\newcommand{\fvect}{\widehat{\VV}}
\newcommand{\fcot}{\widehat{\T}^*}
\newcommand{\ftan}{\widehat{\T}}
\newcommand{\floop}{\widehat{\mathscr L}}

\newcommand{\mapc}{\Mcal\mathrm{ap}}

\newcommand{\Gr}{\mathcal{G}\mathrm{r}}
\newcommand{\Fil}{\mathcal{F}\mathrm{il}}
\newcommand{\cFil}{\mathrm{c}\mathcal{F}\mathrm{il}}

\newcommand{\Lie}{\mathrm{Lie}}
\newcommand{\Diff}{\mathrm{Diff}}

\newcommand{\cptvv}{{\scriptscriptstyle\mathrm{CPTVV}}}

\theoremstyle{Theorem}
\newtheorem{theorem}{Theorem}
\numberwithin{theorem}{section}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{conj}[theorem]{Conjecture}
\newtheorem{problem}[theorem]{Problem}

\newtheorem{lemmanoproof}{Lemma} % lemma with no proof - borrowed from http://www.ams.org/arc/tex/howto/qed/extra-qed.pdf
\newenvironment{lemmanp}
{\renewcommand{\qedsymbol}{$\lozenge$}
\pushQED{\qed}\begin{lemmanoproof}}
{\popQED\end{lemmanoproof}}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{const}[theorem]{Construction}
\newtheorem{conv}[theorem]{Convention}

\DeclareMathOperator{\fib}{Fib}
\DeclareMathOperator{\cofib}{coFib}
\DeclareMathOperator{\hofib}{hoFib}
\DeclareMathOperator{\eq}{eq}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\pr}{pr} % projection
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Spf}{Spf}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Ann}{Ann} % annihilator
\DeclareMathOperator{\image}{im}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator{\crit}{crit}
\DeclareMathOperator{\dcrit}{dcrit}
\DeclareMathOperator{\hoch}{HH}
\DeclareMathOperator{\tor}{Tor}
\DeclareMathOperator{\ext}{Ext}
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\MC}{MC}
\DeclareMathOperator{\DR}{dR}

\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Fun}{Fun}
\DeclareMathOperator{\Map}{Map}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Mon}{Mon}
\DeclareMathOperator{\Grp}{Grp}
\DeclareMathOperator{\Arr}{Arr}
\DeclareMathOperator{\Cat}{Cat}

\DeclareMathOperator{\pt}{pt}
\DeclareMathOperator{\corr}{Corr}
\DeclareMathOperator{\Tw}{Tw}
\DeclareMathOperator{\operads}{Op}
\DeclareMathOperator{\cooperads}{coOp}
\DeclareMathOperator{\prim}{prim}

\DeclareMathOperator{\Pois}{Pois}
\DeclareMathOperator{\Cois}{Cois}
\DeclareMathOperator{\Symp}{Symp}
\DeclareMathOperator{\Pro}{Pro}

\DeclareMathOperator{\Nil}{Nil}

\DeclareMathOperator{\Ptd}{Ptd}

\DeclareMathOperator{\Res}{Res}

\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\mDer}{MDer}