%From bbold.sty, except I don't want it to pollute the nicer \mathbb macro
 %from amsfonts and bbold.sty overrides \mathbb.
%\newcommand{\mybbfamily}{\fontencoding{U}\fontfamily{bbold}\selectfont}
%\newcommand{\mytextbb}[1]{{\mybbfamily#1}}
%\DeclareMathAlphabet{\mymathbb}{U}{bbold}{m}{n}

\newcommand\mymathbb[1]{{#1}}
\newcommand\bbeta{\eta}
\newcommand\bbmu{\mu}
\newcommand\bbalpha{\alpha}
\newcommand\bbtheta{\theta}

\newcommand{\catc}{{\pmb{C}}} 
\newcommand{\catd}{{\mathbb{C}}}
\newcommand{\comma}[2]{{(#1 \downarrow #2)}}
\newcommand{\funbaraa}[2]{{\left| #2 \right|_{#1}}}
\newcommand{\funbara}[1]{{\funbaraa{}{#1}}}
\newcommand{\funbar}{{\funbara{\_}}}
\newcommand{\forfun}{{\forfunalg{}}}
\newcommand{\tc}{{\pmb{T}}}
\newcommand{\td}{{\mathbb{T}}}
\newcommand{\tS}{{\lift{T}}}
\newcommand{\unitc}[1]{{\pmb{\eta}_{#1}}}
\newcommand{\unitd}[1]{{\bbeta_{#1}}}
\newcommand{\unitS}[1]{{\lift{\eta}_{#1}}}
\newcommand{\multc}[1]{{\pmb{\mu}_{#1}}}
\newcommand{\multd}[1]{{\bbmu_{#1}}}
\newcommand{\multS}[1]{{\lift{\mu}_{#1}}}
\newcommand{\dist}[1]{{\disttwoarg{#1}{}}}
\newcommand{\disttwoarg}[2]{{\sigma^{#2}_{#1}}}
%\newcommand{\catset}{\mathrm{Set}}
\newcommand{\setcat}{\pmb{Set}}
\newcommand{\catset}{\setcat}
\newcommand{\natto}{{\Rightarrow}}
\newcommand\scone[3]{({#2} \downarrow {#3})}
%\newcommand{\scone}[2]{{{\mathrm{Scone}}_{#1}^{#2}}}
%\newcommand\subsc{\curlywedgedownarrow}
\newcommand\subsc{\rlap{$\shortdownarrow$}{\raise0.2ex\hbox{$\mkern2.7mu^{\scriptscriptstyle\cap}$}}}
%\newcommand\subsc{\rlap{$\shortdownarrow$}{\raise0.2ex\hbox{$\mkern-2.25mu^{\scriptscriptstyle\cap}$}}}
%\newcommand\subsc{\rlap{$\mkern0.5mu\shortdownarrow$}{\raise0.6ex\hbox{$\Ydown$}}}
\newcommand\subscone[3]{({#2} \;\subsc\; {#3})}
%\newcommand{\subscone}[2]{{{\mathrm{Subscone}}_{#1}^{#2}}}
\newcommand\laxsubsc{\rlap{$\downharpoonright$}{\raise0.2ex\hbox{$\mkern2.7mu^{\scriptscriptstyle\cap}$}}}
\newcommand\laxsubscone[3]{({#2} \;\laxsubsc\; {#3})}
\newcommand{\homset}[3]{{#1(#2,#3)}}
\newcommand{\termobj}{{1}}
\newcommand{\termobjd}{\mymathbb{1}}
\newcommand{\termobjc}{\pmb{1}}
\newcommand{\catalg}[2]{{{#1}_{#2}}}
\newcommand{\forfunalg}[1]{{U_{#1}}}
\newcommand{\lift}[1]{{\widetilde{#1}}}
\newcommand{\proj}{{\pi}}
\newcommand{\projrel}{{\pi^{S}}}
\newcommand{\st}{{\mathrm{St}}}
\newcommand{\pfin}[1]{{\pow_{\mathrm{fin}} (#1)}}
\newcommand{\pfs}[1]{\pfsna (#1)}
\newcommand{\pfsna}{\pow_{\mathrm{fs}}}
\newcommand{\alphac}[3]{{\pmb{\alpha}_{{#1},{#2},{#3}}}}
\newcommand{\alphacna}{{\pmb{\alpha}}}
\newcommand{\alphad}[3]{{\alphadna_{{#1},{#2},{#3}}}}
\newcommand{\alphadna}{{\bbalpha}}
\newcommand{\alphaS}[3]{{\lift{\alpha}_{{#1},{#2},{#3}}}}
\newcommand{\alphaSna}{{\lift{\alpha}}}
\newcommand{\rc}[1]{{\pmb{r}_{#1}}}
\newcommand{\rd}[1]{\rdna_{#1}}
\newcommand\rdna{{\mymathbb{r}}}
\newcommand{\rS}[1]{{\lift{r}_{#1}}}
\newcommand{\strc}[2]{{\pmb{t}}_{{#1},{#2}}}
\newcommand{\strd}[2]{\strdna_{{#1},{#2}}}
\newcommand{\strS}[2]{{\lift{t}_{{#1},{#2}}}}
\newcommand{\strcna}{{\pmb{t}}}
\newcommand{\strdna}{{\mymathbb{t}}}
\newcommand{\strSna}{{\lift{t}}}
\newcommand{\morid}[1]{{{\mathrm id}_{#1}}}
%\newcommand{\}{{}}

\newcommand{\bistrc}[2]{{\pmb{d}_{{#1},{#2}}}}
\newcommand{\bistrd}[2]{\bistrdna_{{#1},{#2}}}
\newcommand{\bistrS}[2]{{\lift{d}_{{#1},{#2}}}}
\newcommand{\bistrcna}{{\pmb{d}}}
\newcommand{\bistrdna}{{\mymathbb{d}}}
\newcommand{\bistrSna}{{\lift{d}}}

\newcommand{\comment}[1]{}


\newtheorem{fact}{Fact}

\newcommand{\identity}[1]{\mathrm{id}_{#1}}
\newcommand{\ordinalcat}{\mathcal I}
\newcommand{\relcat}{\pmb{Rel}}
\newcommand{\monad}{\tc}
\newcommand{\monadrel}{\td}
\newcommand{\unit}{\pmb{\eta}}
\newcommand{\mult}{\pmb{\mu}}
\newcommand{\strength}{\pmb{t}}
%\newcommand{\inl}[2]{i_{{#1}, {#2}}}
%\newcommand{\inr}[2]{j_{{#1}, {#2}}}
\newcommand{\distr}{\sigma}

\newcommand{\comp}{\pmb{Comp}}

\newcommand{\pow}{\mathbb{P}}
\newcommand{\Mes}{\pmb{Mes}}
%\newenvironment{hackequation}{\hfill$\refstepcounter{equation}}{$\fill\eqno \hbox{\@eqnnum}$\@ignoretrue}
\newcommand\Eval[1]{\left\llbracket{#1}\right\rrbracket}
\newcommand\colim{\mathop{\mathrm{colim}}}
\newcommand{\ordcat}{\pmb{Ord}}
\newcommand{\Top}{\pmb{Top}}
\newcommand{\CCTop}{\pmb{CCTop}}
\newcommand{\discr}{\pmb{D}}
\newcommand\lad{{\mathrm{lad}}}
\newcommand\rad{{\mathrm{rad}}}
\newcommand\unita{{\dot \eta}}
\newcommand\counita{{\dot \epsilon}}

\newcommand\preshf[1]{\widehat{#1}}
\newcommand\hDelta{\preshf\Delta}
\newcommand\nerve{N}
\newcommand\nat{\mathbb{N}}
\newcommand\rat{\mathbb{Q}}
\newcommand\contig{\rightarrowtriangle}
\newcommand\openset[1]{{#1}^\circ}
\newcommand\suppset[1]{{#1}^\star}
\newcommand\invord{f_0^\prec}
\newcommand\gamord{f_0^\succ}
%\newcommand\Im{\mathrm{Im}}
\newcommand\trace[1]{\mathrm{Tr} (#1)}

\newcommand{\tstr}[2]{\tstrna_{#1, #2}}
\newcommand{\tstrna}{\bbtheta}
\newcommand{\tstri}{\mymathbb{i}}
\newcommand{\tstrc}[2]{\tstrcna_{#1, #2}}
\newcommand{\tstrcna}{\pmb{\theta}}
\newcommand{\tstrci}{\pmb{i}}
\newcommand\Ic{\pmb{I}}
\newcommand\Id{\mathbb{I}}
\newcommand\IS{\lift{I}}
\newcommand\tensorc{\pmb{\otimes}}
\newcommand\tensord{\otimes}
\newcommand\tensorS{\lift{\otimes}}
\newcommand{\lc}[1]{{\lcna_{#1}}}
\newcommand\lcna{{\pmb{\ell}}}
\newcommand{\ld}[1]{\ldna_{#1}}
\newcommand\ldna{{\mymathbb{l}}}
\newcommand{\lS}[1]{{\lift{\ell}_{#1}}}
\newcommand{\cc}[2]{{\pmb{c}_{#1, #2}}}
\newcommand{\ccna}{{\pmb{c}}}
\newcommand{\cd}[2]{{\cdna_{#1, #2}}}
\newcommand{\cdna}{\mymathbb{c}}
\newcommand{\cS}[2]{{\lift{\ell}_{#1, #2}}}
\newcommand{\cSna}{{\lift {\ell}}}

\newcommand\Talg[1]{{#1}\mathbf{\mbox{-}Alg}}
\newcommand\Kleisli[1]{\mathbf{Kleisli} (#1)}

\newcommand\AlgF[1]{F^{#1}}
\newcommand\AlgU[1]{U^{#1}}
\newcommand\KlF[1]{F_{#1}}
\newcommand\KlU[1]{U_{#1}}

\newcommand\pair[2]{[{#1}, {#2}]}
\newcommand\ihom[2]{{#2}^{#1}} %{\langle {#1} \to {#2} \rangle}
\newcommand\App{\mathsf{App}}
\newcommand\ihoml[2]{{}^{#1}{#2}}
\newcommand\Appl{\mathsf{qqA}}

\newcommand\BB[2]{{#1}^* {#2}} %{\mathfrak{B}_{#1} ({#2})}
\newcommand\BBna[1]{{#1}^*} %{\mathfrak{B}_{#1}}
\newcommand\BBU[1]{\mathcal{U} ({#1})}
\newcommand\BBUna{\mathcal{U}}
\newcommand\K[1]{{#1}_*}
\newcommand\decalaged{\mathop{\uparrow}} %{\mathop{\twoheaduparrow}} %{\K \td}
\newcommand{\unitBBd}[1]{{\bbeta_{*\;#1}}}
\newcommand{\multBBd}[1]{{\bbmu_{*\;#1}}}
\newcommand\upc{\mathop{\uparrow}\nolimits}
\newcommand\dc{\mathop{\downarrow}\nolimits}
\newcommand\uuarrow{\rlap{$\uparrow$}\raise.5ex\hbox{$\uparrow$}}%%
\newcommand\ddarrow{\rlap{$\downarrow$}\raise.5ex\hbox{$\downarrow$}}%%
\newcommand\Nearrow{\rlap{$\nearrow$}\hskip.5ex\raise.5ex\hbox{$\nearrow$}}
\newcommand\Searrow{\rlap{$\searrow$}\hskip.5ex\raise-.5ex\hbox{$\searrow$}}
\newcommand\twoheaduparrow{\uuarrow}
\newcommand\decalagec{\mathop{\pmb{\uparrow}}} %{\K \tc}
\newcommand{\unitBBc}[1]{{\pmb{\eta}_{*\;#1}}}
\newcommand{\multBBc}[1]{{\pmb{\mu}_{*\;#1}}}
\newcommand{\funbarBBaa}[2]{{\left| #2 \right|_{*\;#1}}}
\newcommand{\funbarBBa}[1]{{\funbarBBaa{}{#1}}}
\newcommand{\funbarBB}{{\funbarBBa{\_}}}
\newcommand{\distBB}[1]{{\distBBtwoarg{#1}{}}}
\newcommand{\distBBtwoarg}[2]{{\sigma^{#2}_{*\;#1}}}
\newcommand{\tstrBB}[2]{{\tstrBBthreearg{#1}{#2}{}}}
\newcommand{\tstrBBthreearg}[3]{{\bbtheta^{#3}_{*\; #1, #2}}}
\newcommand{\tstrBBna}{\bbtheta_*}
\newcommand{\tstrBBi}{\mymathbb{i}_*}
\newcommand{\tstrBBc}[2]{{\tstrBBcthreearg{#1}{#2}{}}}
\newcommand{\tstrBBcthreearg}[3]{{\pmb{\theta}^{#3}_{*\; #1, #2}}}
\newcommand{\tstrBBcna}{\pmb{\theta}_*}
\newcommand{\tstrBBci}{\pmb{i}_*}
\newcommand{\bistrBBd}[2]{\mymathbb{d}_{*\; {#1},{#2}}}
\newcommand{\bistrBBdna}{{\mymathbb{d}_*}}
\newcommand{\bistrBBc}[2]{\pmb{d}_{*\; {#1},{#2}}}
\newcommand{\bistrBBcna}{{\pmb{d}_*}}
\newcommand\IBBd{\mathbb{I}_*}
\newcommand\IBBc{\pmb{I}_*}
\newcommand{\ccBB}[2]{{\pmb{c}_{*\;#1, #2}}}
\newcommand{\ccBBna}{{\pmb{c}_*}}
\newcommand{\cdBB}[2]{{\mymathbb{c}_{*\;#1, #2}}}
\newcommand{\cdBBna}{\mymathbb{c}_*}

\newcommand{\real}{\mathbb{R}}
\newcommand{\creal}{\overline{\real^+}}
\newcommand{\Dcpo}{\pmb{Cpo}}
\newcommand{\Pcpo}{\pmb{PCpo}}
\newcommand{\CONT}{\pmb{Cont}}
\newcommand{\DOM}{\pmb{Dom}}
\newcommand{\cL}{\pmb{cL}}
\newcommand{\RB}{\pmb{RB}}
\newcommand{\FS}{\pmb{FS}}
\newcommand{\Met}{\pmb{Met}}
\newcommand{\Pol}{\pmb{Pol}}
\newcommand{\CSMet}{\pmb{CSMet}}
\newcommand{\CMet}{\pmb{CMet}}
\newcommand{\PERDOM}{\pmb{PERDom}}
\newcommand{\PERCLat}{\pmb{PERCLat}}
\newcommand{\PEQU}{\pmb{PEqu}}
\newcommand{\EQU}{\pmb{Equ}}
\newcommand\OBall[2]{{\cal B} (#1, #2)}
\newcommand\cbf[1]{{\cal C}_b ({#1})}
\def\llimp{\mathbin{-\mkern-3mu\circ}}%

\newcommand\kw[1]{\mbox{\sf {#1}}}
\def\letkw{\mathop{\kw{let}}}
\def\letin#1#2{\letkw{#1}\mathbin{\kw{in}}{#2}}
\def\valu{\mathop{\kw{val}}}

\newcommand\ForAuthors[1]%          %  temporary remark for the
{\par\smallskip                     %  authors:
  \begin{center}%                    %
    \fbox%                            %    --------
    {\parbox{0.9\linewidth}%          %    |  #1  |
      {\raggedright\sc--- #1}%         %    --------
      }%                                %
  \end{center}%                      %
  \par\smallskip                     %
  }

\newcommand\limproj{{\underleftarrow{\lim}}}

\newcommand\lamc{\lambda_{\mathsf c}}

% From Pakin's comprehensive latex symbol list:
\def\Xint#1{\mathchoice
  {\XXint\displaystyle\textstyle{#1}}%
  {\XXint\textstyle\scriptstyle{#1}}%
  {\XXint\scriptstyle\scriptscriptstyle{#1}}%
  {\XXint\scriptscriptstyle\scriptscriptstyle{#1}}%
  \!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$}
    \vcenter{\hbox{$#2#3$}}\kern-.5\wd0}}
\def\ddashint{\Xint=}
\def\dashint{\Xint-}