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%!TEX root = index_theory.tex
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% -- commands for finetuning 
% ======================================================================================
\makeatletter
\newcommand{\raisemath}[1]{\mathpalette{\raisem@th{#1}}}
\newcommand{\raisem@th}[3]{\raisebox{#1}{$#2#3$}}
\makeatother
\makeatletter
\newcommand{\killDescendersM}[1]{\mathpalette{\killD@scendersM{#1}}}
\newcommand{\killD@scendersM}[2]{\raisebox{0pt}[\height][0pt]{$#2#1$}}
\makeatother
\DeclareRobustCommand{\minwidthbox}[2]{%
  \ifmmode%
    \expandafter\mathmakebox
  \else%
    \expandafter\makebox
  \fi
  [\ifdim#2<\width\width\else#2\fi]{#1}%
}
% ======================================================================================

% -- new relations
% ======================================================================================
\newcommand{\coloniff}{\mathrel{\vcentcolon\mkern-3mu\relax\iff}}
\newcommand{\colonLra}{\mathrel{\vcentcolon\mkern-1mu\relax\Leftrightarrow}}
% ======================================================================================


%-- Underbrace in LaTeX syntax (and without spacing problems...)
% ======================================================================================
\newcommand{\Underbrace}[2]{{\underbrace{#1}_{#2}}}
\newcommand{\Overbrace}[2]{{\overbrace{#1}^{#2}}}
\newcommand{\Underbracket}[2]{{\underbracket[0.7pt][2pt]{#1}_{#2}}}
\newcommand{\Overbracket}[2]{{\overbracket[0.7pt][2pt]{#1}^{#2}}}
% ======================================================================================


% -- K für K-Theorie
% ======================================================================================
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% \makeatletter
% \newcommand*{\IfBold}{%
%   \ifx\f@series\my@test@bx
%     \expandafter\@firstoftwo
%   \else
%     \expandafter\@secondoftwo
%   \fi
% }
% \newcommand*{\my@test@bx}{bx}
% \makeatother

%-- IfBold Command
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\makeatletter
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\DeclareRobustCommand\IfFontSeriesContextTF[1]{%
  \expand@font@defaults
  \@font@series@contextfalse
  \def\requested@test@context{#1}%
  \expandafter\edef\csname ??def@ult\endcsname{\f@family}%
  \let\@elt\test@font@series@context
      \@meta@family@list
      \@elt{??}%
  \let\@elt\relax
  \if@font@series@context
  \expandafter\@firstoftwo
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  \else
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  \expandafter\@secondoftwo
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  \fi
}
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\def\test@font@series@context#1{%
  \edef\reserved@a{\csname #1def@ult\endcsname}%
  \ifx\f@family\reserved@a
    \let\@elt\@gobble
    \expandafter\ifx
                \csname\requested@test@context series@#1\endcsname\f@series
      \@font@series@contexttrue
    \else
      \expandafter\ifx
                  \csname\requested@test@context def@ult\endcsname\f@series
        \@font@series@contexttrue
  \fi\fi\fi
}

\newif\if@font@series@context

\newcommand\IfBold{\IfFontSeriesContextTF{bf}}

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\makeatother
\newcommand{\Kay}{\IfBold{\ensuremath{\mathbold{K}}}{\ensuremath{K}}}
\newcommand{\Eee}{\IfBold{\ensuremath{\mathbold{E}}}{\ensuremath{E}}}
\newcommand{\KayAR}{\IfBold{\ensuremath{\mathbold{KR}}}{\ensuremath{KR}}}
\newcommand{\KayOh}{\IfBold{\ensuremath{\mathbold{KO}}}{\ensuremath{KO}}}
\newcommand{\KayUh}{\IfBold{\ensuremath{\mathbold{KU}}}{\ensuremath{KU}}}
\newcommand{\KayKay}{\IfBold{\ensuremath{\mathbold{KK}}}{\ensuremath{KK}}}
\newcommand{\KayKayOh}{\IfBold{\ensuremath{\mathbold{KKO}}}{\ensuremath{KKO}}}
\newcommand{\KayKayAR}{\IfBold{\ensuremath{\mathbold{KKR}}}{\ensuremath{KKR}}}
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\newcommand{\KayAxe}{\IfBold{\ensuremath{\mathbold{KX}}}{\ensuremath{KX}}}
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\newcommand{\kay}{\IfBold{\ensuremath{\mathbold{k}}}{\ensuremath{k}}}
\newcommand{\kayAR}{\IfBold{\ensuremath{\mathbold{kR}}}{\ensuremath{kR}}}
\newcommand{\kayOh}{\IfBold{\ensuremath{\mathbold{kO}}}{\ensuremath{kO}}}
\newcommand{\kayUh}{\IfBold{\ensuremath{\mathbold{kU}}}{\ensuremath{kU}}}
\robustify{\Kay}
\robustify{\Eee}
\robustify{\KayAR}
\robustify{\KayOh}
\robustify{\KayUh}
\robustify{\KayKay}
\robustify{\KayKayOh}
\robustify{\KayKayAR}
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\robustify{\KayAxe}
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\robustify{\kay}
\robustify{\kayAR}
\robustify{\kayOh}
\robustify{\kayUh}
\newcommand{\K}{\texorpdfstring{\Kay}{K}}
\newcommand{\Kconec}{\texorpdfstring{\kay}{k}}
\newcommand{\Ee}{\texorpdfstring{\Eee}{E}}
\newcommand{\KR}{\texorpdfstring{\KayAR}{KR}}
\newcommand{\KO}{\texorpdfstring{\KayOh}{KO}}
\newcommand{\KU}{\texorpdfstring{\KayUh}{KU}}
\newcommand{\KK}{\texorpdfstring{\KayKay}{KK}}
\newcommand{\KKO}{\texorpdfstring{\KayKayOh}{KKO}}
\newcommand{\KKR}{\texorpdfstring{\KayKayAR}{KKR}}
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\newcommand{\KX}{\texorpdfstring{\KayAxe}{KX}}
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\newcommand{\kR}{\texorpdfstring{\kayAR}{kR}}
\newcommand{\kO}{\texorpdfstring{\kayOh}{kO}}
\newcommand{\kU}{\texorpdfstring{\kayUh}{kU}}

\newcommand{\bbK}{\mathbb{K}}
%-- reduced only in mathmode and without "auto-bold"-feature
\newlength{\KRsize}
\settoheight{\KRsize}{$KR$}

\newcommand{\KRred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{\KR}}$}}
\newcommand{\Kred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{\K}}$}}
\newcommand{\kred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{\Kconec}}$}}
\newcommand{\KOred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{\KO}}$}}
\newcommand{\kOred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{\kO}}$}}
\newcommand{\kUred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{\kU}}$}}
\newcommand{\Hred}{\raisebox{0pt}[\KRsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{$H$}}$}}
% ======================================================================================

% -- (1,1) periodicity
% ======================================================================================
\newcommand{\onecommaone}{\IfBold{($\mathbold{1}$,$\mathbold{1}$)}{$(1,1)$}}
\robustify{\onecommaone}
\newcommand{\OneCommaOne}{\texorpdfstring{\onecommaone}{(1,1)}}
% ======================================================================================


% -- C*-algebras
% ======================================================================================
\newcommand{\Ceestar}{\IfBold{\ensuremath{\mathbold{C}^{\mathbold{*}}}}{\ensuremath{C^*}}}
\robustify{\Ceestar}
\newcommand{\Cstar}{\texorpdfstring{\Ceestar}{C*}}
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\newcommand{\RoeAlg}{\Cstar}
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\newcommand{\LocAlg}{L^*}
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\newcommand{\LocRoe}{\mathop{\Cstar_L}}
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\newcommand{\unital}{{\hspace{-1.6pt}\raisemath{0.5pt}{\sim}}}
\newcommand{\Lin}{\mathcal{L}}
\newcommand{\KH}{\mathcal{K}(\mathcal{H})}
\newcommand{\HA}{\mathcal{H}_{A}}
\newcommand{\HB}{\mathcal{H}_{B}}
\newcommand{\plus}{{\hspace{-1.6pt}+}}
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\newcommand{\Roe}{\Cstar_{\mathrm{Roe}}}
\DeclareMathOperator{\prop}{prop}
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\DeclareMathOperator{\diam}{diam}
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\DeclarePairedDelimiterX\bra[1]{\langle}{\rvert}{#1\,}
\DeclarePairedDelimiterX\ket[1]{\lvert}{\rangle}{\,#1}

\newlength{\Hsize}
\settoheight{\Hsize}{$\mathcal{H}$}
\newcommand{\Hgrad}{\raisebox{0pt}[\Hsize][\depth]{$\widehat{\raisebox{0pt}[5.9pt][\depth]{$\mathcal{H}$}}$}}
\newcommand{\Kgrad}{\raisebox{0pt}[\Hsize][\depth]{$\widehat{\raisebox{0pt}[5.9pt][\depth]{$\mathcal{K}$}}$}}
\newcommand{\KHgrad}{\mathcal{K}(\Hgrad)}


\newcommand{\q}{q}


% \newcommand{\Hgrad}{\widehat{\mathcal{H}}}
\newcommand{\HAgrad}{\Hgrad_{\mathcal{A}}}
\newcommand{\HBgrad}{\Hgrad_{\mathcal{B}}}
\newcommand{\op}{\textrm{op}}
\DeclareMathOperator{\Fred}{Fred}
\newlength{\Fredsize}
\settoheight{\Fredsize}{$\Fred$}
\DeclareMathOperator{\indeX}{ind}
\DeclareMathOperator{\IndeX}{Ind}
\newcommand{\kkprod}{\otimes}

\newcommand{\FredRed}{\raisebox{0pt}[\Fredsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{$\Fred$}}$}}
\newcommand{\ess}{\mathcal{S}}

\DeclareMathOperator{\Ext}{Ext}
\newcommand{\calkin}{\mathscr{A}}
% ======================================================================================

% -- spectra
% ======================================================================================
\newcommand{\spec}[1]{%
	\StrLeft{#1}{1}[\firstchar]%
	\IfStrEq{\firstchar}{S}{\mathbb{S}}{\mathbold{#1}}%
}
% ======================================================================================

% ======================================================================================
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\pr}{pr}
\DeclareMathOperator{\Hom}{Hom}
% \newcommand{\gradedHom}{\mathop{\mathrm{H}\widehat{\mathrm{om}}}}
\let\gradedHom\Hom
\newcommand{\grenzwX}[1]{\xrightarrow{\minwidthbox{#1}{2em}}}
\newcommand{\grenzwXY}[2]{\xrightarrow{\minwidthbox{#1}{#2}}}
\newcommand{\grenzw}[1]{%
	\mathchoice
	{\grenzwX{\raisemath{-1pt}{#1}}}%
	{\mathop{\raisemath{-.9pt}{\grenzwXY{\raisemath{-2pt}{#1}}{1.2em}}}}%
	{\grenzwX{\raisemath{-3pt}{#1}}}%
	{\grenzwX{\raisemath{-3pt}{#1}}}}%
\renewcommand{\longrightarrow}{\grenzw{}}
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\Span}{span} % Span

\newlength{\otsize}
\settoheight{\otsize}{$\otimes$}
\newcommand{\gradedtensor}{\mathchoice
	{\mathbin{\widehat{\raisebox{0pt}[4.8pt][\depth]{\ensuremath{\displaystyle\otimes}}}}}%
	{\mathbin{\widehat{\raisebox{0pt}[4.8pt][\depth]{\otimes}}}}%
	{\mathbin{\widehat{\raisebox{0pt}[3.8pt][\depth]{\ensuremath{\scriptstyle\otimes}}}}}%
	{\mathbin{\widehat{\raisebox{0pt}[2.8pt][\depth]{\ensuremath{\scriptscriptstyle\otimes}}}}}%
}
\newcommand{\gradedtensorX}{\mathbin{\raisebox{0pt}[\otsize][\depth]{$\widehat{\raisebox{0pt}[4.8pt][\depth]{\otimes}}$}}}

\newcommand{\gradedodot}{\mathbin{\raisebox{0pt}[\otsize][\depth]{$\widehat{\raisebox{0pt}[4.8pt][\depth]{\odot}}$}}}
% \newcommand{\gradedtensor}{\mathbin{\hat{\otimes}}}
% \newcommand{\gradedodot}{\mathbin{\hat{\odot}}}

\newcommand{\even}{\textrm{ev}}
\newcommand{\odd}{\textrm{odd}}

\DeclareMathOperator{\charac}{char}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator*{\colim}{colim}
\newcommand{\bigast}{\mathop{\scalebox{1.6}{\raisebox{-0.2ex}{$\ast$}}}}
\newcommand{\pt}{\mathrm{pt}}
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\DeclareMathOperator{\Mat}{Mat}
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\DeclareMathOperator{\GL}{GL}
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\DeclareMathOperator{\SL}{SL}
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\DeclareMathOperator{\Vect}{Vect}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Spin}{Spin}
\DeclareMathOperator{\Pin}{Pin}
\DeclareMathOperator{\Ad}{Ad}
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\DeclareMathOperator{\ad}{ad}
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\newcommand{\twistedAd}{\mathchoice{\mathop{\widetilde{\raisebox{0pt}[5.8pt][\depth]{$\Ad$}}}}{\mathop{\widetilde{\raisebox{0pt}[5.8pt][\depth]{$\Ad$}}}}{\mathop{\widetilde{\raisebox{0pt}[4.5pt][\depth]{$\Ad$}}}}{\mathop{\widetilde{\raisebox{0pt}[4.0pt][\depth]{$\Ad$}}}}}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Orth}{O}
\DeclareMathOperator{\Unitary}{U}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Map}{Map}
\newcommand{\sa}{\mathrm{sa}}
\DeclareMathOperator{\spectrum}{spec}
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\DeclareMathOperator{\domain}{Domain}
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\DeclareMathOperator{\supp}{supp}
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\DeclareMathOperator{\im}{im}
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\newcommand{\Hilb}{\mathcal{H}}
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\newcommand{\Bop}{\mathcal{B}}
\newcommand{\BH}{\Bop(\Hilb)}
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\newcommand{\Inf}{I\mkern-.1mu n\mkern-.5mu f}
\newcommand{\Infhat}{I\mkern-.1mu \widehat{n}\mkern-.5mu f}
\newcommand{\RInf}{\mathcal{R}\Infhat}

\newcommand{\Repr}[1]{\Phi\mkern-2.5mu _{#1}}
\newcommand{\Reprq}{\Repr{q}}
% ======================================================================================

%-- spectra stuff
% ======================================================================================
\newcommand{\cover}[1]{\langle{#1}\rangle}
\newcommand{\reduced}[1]{\widetilde{#1}} % reduced (co)homology theory
% ======================================================================================

%-- different topologies and so on
% ======================================================================================
\newcommand{\w}{\mkern1mu\mathrm{w}}
% ======================================================================================

%-- Clifford algebras
% ======================================================================================
\usepackage{mathrsfs}
\newcommand{\Cl}{C\mkern-1.3mu\ell}
\newcommand{\Cxl}{\mathbb{C}\mkern-1.1mu\ell}
% \newcommand{\Cl}{\mathop{\mathscr{C}\mathscr{l}}}
\DeclareMathOperator{\End}{End}
% \DeclareMathOperator{\Cliff}{Cliff}
\newcommand{\Cliff}{\mathop{\mathrm{C}\mkern-1.3mu\ell\mathrm{iff}}\nolimits}
\newcommand{\M}{\mathfrak{M}}
\newcommand{\Mc}{\M^c}
\newcommand{\MR}{\M R}

\newlength{\Msize}
\settoheight{\Msize}{$M$}

\newcommand{\gradedM}{\raisebox{0pt}[\Msize][\depth]{$\widehat{\raisebox{0pt}[5.8pt][\depth]{\ensuremath{\M}}}$}}
\newcommand{\gradedMc}{\gradedM^c}
\newcommand{\gradedMR}{\raisebox{0pt}[\Msize][\depth]{$\widehat{\raisebox{0pt}[5.8pt][\depth]{\ensuremath{\MR}}}$}}
% ======================================================================================


%-- algebra stuff
% ======================================================================================
\newcommand{\intprod}{\lefthalfcup}
% ======================================================================================


%-- classifying spaces universal bundles an so on
% ======================================================================================
\newcommand{\PB}{\mathop{\mathcal{PB}}}
\newcommand{\BU}{B\mathrm{U}}
\newcommand{\BO}{B\mathrm{O}}
\newcommand{\BSp}{B{\Sp}}
\newcommand{\trivB}{\theta}
% ======================================================================================


%-- differential, integrals and so on
% ======================================================================================
\newcommand{\mathd}{\mathrm{d}\mkern-0.7mu} % for differentials
\newcommand{\intmathd}{\mkern4mu\mathd} % for differentials
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\newcommand{\Int}[1]{\int_{\mathrlap{\raisemath{-2pt}{#1}}}\,\,\,} % integral without spacing in index
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\newcommand{\diff}[2]{\ensuremath{\frac{{\partial #1}}{{\partial #2}} }}
\newcommand{\diffd}[2]{\ensuremath{\frac{\mathd #1}{\mathd #2} }}
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\newcommand{\Tang}{T}
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\newcommand{\D}{D}
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\DeclareMathOperator{\DiffOp}{Diff}
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\DeclareMathOperator{\Sym}{Sym}
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\DeclareMathOperator{\Smbl}{Smbl}
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\newcommand{\Schwartz}{\mathbold{S}}
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% ======================================================================================


%-- bracing commands
% ======================================================================================
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\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
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% \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
% \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
\DeclarePairedDelimiter\norm{\Vert}{\Vert}
\DeclarePairedDelimiter\enbrace{(}{)}
\DeclarePairedDelimiter\benbrace{[}{]}
\DeclarePairedDelimiter\bbenbrace{\llbracket}{\rrbracket}
% \DeclarePairedDelimiter\lenbrace{<}{>}
\DeclarePairedDelimiter\angbrace{\langle}{\rangle}
\newcommand{\ssbrace}[1]{{\scriptscriptstyle\enbrace{#1}}}
% \newcommand{\ssbbrace}[1]{{\scriptscriptstyle\benbrace{#1}}}
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\DeclarePairedDelimiterX\skal[2]{\langle}{\rangle}{#1\mkern2mu,#2}
\DeclarePairedDelimiterX\skalRound[2]{(}{)}{#1\mkern2mu,#2}
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% ======================================================================================


%-- modified Stackrel
% ======================================================================================
\newcommand{\StackText}[2]{\stackrel{\mbox{\scriptsize #1}}{#2}}
\newcommand{\StackTextClap}[2]{\stackrel{\mathclap{\mbox{\scriptsize #1}}}{#2}}
% ======================================================================================


%-- set command
% ======================================================================================
\newcommand\SetSymbol[1][]{\nonscript\:#1\vert\allowbreak\nonscript\:\mathopen{}}
\providecommand\given{} % to make it exist
\DeclarePairedDelimiterX\set[1]\{\}{\renewcommand\given{\SetSymbol[\delimsize]}#1}
% ======================================================================================

%-- definition of mappings
% ======================================================================================
\newcommand{\mapdef}[5]{%
	\[
		\begin{array}{rcl}
			\textstyle #1 &\xrightarrow{\minwidthbox{#5}{2em}} & \textstyle #2 \\[0.5ex]
			\textstyle #3 &\xmapsto{\minwidthbox{\mbox{ }}{2em}} & \textstyle #4
		\end{array}
	\]
}
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\newcommand{\To}[1]{\xrightarrow{\minwidthbox{#1}{2em}}}
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% ======================================================================================


% -- direct and projective limit
% ======================================================================================
\newcommand{\MyTo}[1]{\mathbin{\tikz[baseline] \draw[-to,line width=.4pt] (0ex,0.94ex) -- (#1,0.94ex);}}
\newcommand{\MyToX}[1]{\mathbin{\tikz[baseline] \draw[to-,line width=.4pt] (0ex,0.94ex) -- (#1,0.94ex);}}
\newcommand{\dlim}{%
    \mathchoice
      {\lim\limits_{\MyTo{4.2ex}}\,}% \displaystyle
      {\lim\limits_{\MyTo{2.8ex}}}% \textstyle
      {\lim\limits_{\MyTo{2.3ex}}}% \scriptstyle
      {\lim\limits_{\MyTo{2.3ex}}}% \scriptscriptstyle
}
\newcommand{\plim}{%
    \mathchoice%
      {\lim\limits_{\MyToX{4.2ex}}\,}% \displaystyle
      {\lim\limits_{\MyToX{2.8ex}}}% \textstyle
      {\lim\limits_{\MyToX{2.3ex}}}% \scriptstyle
      {\lim\limits_{\MyToX{2.3ex}}}% \scriptscriptstyle
}
\newcommand{\Dlim}{\killDescendersM{\dlim}}
\newcommand{\Plim}{\killDescendersM{\plim}}
% ======================================================================================

%-- categories
% ======================================================================================
\newcommand{\Cat}[1]{\mathscr{#1}}
\newcommand{\TOP}{\Cat{T\mkern-4mu o\mkern-2mu p}}
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\newcommand{\SET}{\Cat{S\mkern-2mu e\mkern-2mu t}}
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\newcommand{\HoTOP}{\Ho\TOP}
\newcommand{\CW}{\Cat{CW}}
\newcommand{\hCW}{h\Cat{CW}}
\newcommand{\hCWpointed}{\hCW\mkern-3mu_*}
\newcommand{\Ab}{\Cat{Ab}}
\newcommand{\SPECTRA}{\Cat{S}}
\newcommand{\Ho}{\Cat{H}}
\newcommand{\HoSPECTRA}{\Ho\SPECTRA}
\newcommand{\RealBundles}{\Cat{F}}
\let\F\RealBundles
\newcommand{\E}{\Cat{E}}

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\newcommand{\fdSPECTRA}{\SPECTRA_{\mathrm{fd}}}
\newcommand{\fSPECTRA}{\SPECTRA_{\mathrm{f}}}
\newcommand{\sSPECTRA}{\SPECTRA_{\mathrm{s}}}
\newcommand{\sfdSPECTRA}{\SPECTRA_{\mathrm{sfd}}}
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\newcommand{\LC}{\mathcal{L}\mathcal{C}}
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\newcommand{\Coa}{\Cat{C\mkern-3.2mu o\mkern-2.5mu a}}
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% ======================================================================================


%-- font shenanigans 
% ======================================================================================
\DeclareFontFamily{U} {MnSymbolC}{}
\DeclareFontShape{U}{MnSymbolC}{m}{n}{
  <-6> MnSymbolC5
  <6-7> MnSymbolC6
  <7-8> MnSymbolC7
  <8-9> MnSymbolC8
  <9-10> MnSymbolC9
  <10-12> MnSymbolC10
  <12-> MnSymbolC12}{}
\DeclareFontShape{U}{MnSymbolC}{b}{n}{
  <-6> MnSymbolC-Bold5
  <6-7> MnSymbolC-Bold6
  <7-8> MnSymbolC-Bold7
  <8-9> MnSymbolC-Bold8
  <9-10> MnSymbolC-Bold9
  <10-12> MnSymbolC-Bold10
  <12-> MnSymbolC-Bold12}{}
  
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{MnLargeSymbols}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{MnLargeSymbols}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12
}{}
\DeclareFontShape{OMX}{MnSymbolE}{b}{n}{
    <-6>  MnSymbolE-Bold5
   <6-7>  MnSymbolE-Bold6
   <7-8>  MnSymbolE-Bold7
   <8-9>  MnSymbolE-Bold8
   <9-10> MnSymbolE-Bold9
  <10-12> MnSymbolE-Bold10
  <12->   MnSymbolE-Bold12
}{}

\DeclareSymbolFont{MnSyC} {U} {MnSymbolC}{m}{n}
\DeclareMathSymbol{\lefthalfcup}{\mathbin}{MnSyC}{183}

\DeclareMathDelimiter{\llbracket}{\mathopen}%
                     {MnLargeSymbols}{'102}{MnLargeSymbols}{'102}
\DeclareMathDelimiter{\rrbracket}{\mathclose}%
                     {MnLargeSymbols}{'107}{MnLargeSymbols}{'107}
% ======================================================================================