\newcommand\bm[1]{\boldsymbol{#1}} \newcommand\EA{\pmb{\bm{\mathtt{EA}}}} \newcommand\BA{\pmb{\bm{\mathtt{BA}}}} \newcommand\DL{\pmb{\bm{\mathtt{DL}}}} \newcommand\Rng{\pmb{\bm{\mathbf{Rng}}}} \newcommand\Cpu{\pmb{\bm{\mathbf{C}}^\ast}_{\hspace{-1ex}\rm PU}} \newcommand\Hilb{\pmb{\bm{\mathbf{Hilb}}}} \newcommand\HilbShort{\pmb{\bm{\mathbf{Hilb}_{\text{\tiny short}}}}} \newcommand\BanShort{\pmb{\bm{\mathbf{Ban}_{\text{\tiny short}}}}} \newcommand\Ban{\pmb{\bm{\mathbf{Ban}}}} \newcommand\Bounded{\mathop{ {\mathcal{B}}}} \newcommand\DM{\mathop{ {\mathcal{DM}}}} \newcommand\Tr{\mathop{ {\mathrm{Tr}}}} \newcommand\FProd[2][{}]{\mathord{\mathrel{\rm FProd}\parent{#2}}} \newcommand\PCM{\pmb{\bm{\mathtt{PCM}}}} \newcommand\PCMod{\pmb{\bm{\mathtt{PCMod}}}} \newcommand\EMonNorm{\pmb{\bm{\mathtt{EMonNorm}}}} \newcommand\EMod[1]{\pmb{\bm{\mathtt{EMod}}}_{#1}} \newcommand\EMon{\pmb{\bm{\mathtt{EMon}}}} \newcommand\TricoCanc{\pmb{\bm{\mathtt{TricoCanc}}}} \newcommand\D[1]{\mathop{\mathcal{D}_{#1}}} \newcommand\Endop[1]{\mathop{\mathrm{End}_{#1}}} \newcommand\G{\mathop{\mathcal{G}}} \newcommand\M[1][\mathbb{B}]{M_{#1}} \newcommand\Pred{\mathop{\mathrm {Pred}}} \newcommand\Stat{\mathop{\mathrm {Stat}}} \newcommand\Conv[1]{\mathrm{Conv}_{#1}} \newcommand\Comma[2]{#1\downarrow#2} \newcommand\lub{\bigvee} \newcommand\glb{\bigwedge} \newcommand\restrict[1]{\left.\vphantom{\int}\right\lvert_{#1}} \newcommand{\map}[1]{\require{HTML} \style{display: inline-block; border-style: none}{\mmlToken{mglyph}[src="https://younesse.net/assets/Slides/M2-Internship/Pictures/Maps/map_#1.svg" width="12mm" height="12mm" valign="-2mm" alt="Shuffle map" class="noborder" style="margin-top:15px"]{}}} \newcommand{\ovee}{\require{HTML} \style{display: inline-block; border-style: none}{ \mmlToken{mglyph}[src="https://younesse.net/assets/Slides/M2-Internship/Pictures/ovee.svg" alt="ovee" class="noborder ovee"]{}}} \newcommand\dom{\mathop{\mathrm{dom}}} \newcommand\supp{\mathop{\mathrm{supp}}} \newcommand{\closure}[2][3]{ {}\mkern#1mu\overline{\mkern-#1mu#2}} \newcommand\restr[1]{\raisebox{-.5ex}{$|$}_{#1}} \newcommand\yoneda{ {\bf y}} \newcommand\oppositeName{ {\rm op}} \newcommand\opposite[1]{ {#1}^\oppositeName} \newcommand\id[1][{}]{ {\rm id}_{#1}} \newcommand\Id[1][{}]{ {\rm Id}_{#1}} \newcommand\Cat[1]{\mathcal{#1}\/} \newcommand\Category[1]{ {\mathbf{ #1}}} \newcommand\Set{\Category{Set}} \newcommand\tensor{\otimes} \newcommand\unit{\mathbb I} \newcommand\carrier[1]{\underline{#1}} \newcommand\Comma[2]{#1\downarrow#2} \newcommand\lub{\bigvee} \newcommand\glb{\bigwedge} \newcommand\restrict[1]{\left.\vphantom{\int}\right\lvert_{#1}} \newcommand\catA{\Category A} \newcommand\catB{\Category B} \newcommand\catC{\Category C} \newcommand\catD{\Category D} \newcommand\catE{\Category E} \newcommand\Psh[1]{\widehat{#1}} \newcommand\PshStar[1]{\widehat{#1}} \newcommand{\Pfin}{\mathop{ {\mathcal P}_{\rm fin}}\nolimits} \newcommand\eqdef{≝} \newcommand\ie{\emph{i.e.~}} \newcommand\conflict{\mathrel{\sim\joinrel\sim}} \newcommand\Cocomp{\mathbf{Cocomp}} \newcommand\const[1]{\Delta_{#1}} \newcommand\Lan[2]{\mathop{\mathrm{Lan}}_{#1}(#2)} \newcommand\Ran[2]{\mathop{\mathrm{Ran}}_{#1}(#2)} \newcommand\Nerve[1]{\mathrm{N}_{#1}} \newcommand{\dinat}{\stackrel{\bullet}{\longrightarrow}} \newcommand{\End}[2][c]{\int_{#1} #2(c, c)} \newcommand{\Coend}[2][c]{\int^{#1} #2(c, c)} \newcommand{\obj}[1]{\vert #1 \vert} \newcommand{\elem}[1]{\int #1} \newcommand{\tens}[2]{#1 \cdot #2} \newcommand{\cotens}[2]{#2^{#1}} \newcommand{\Nat}{\mathop{\rm Nat}\nolimits} \newcommand{\colim}{\mathop{\rm colim}\nolimits} \newcommand{\cancolimAC}[1]{\big((i/#1)\stackrel{U}{→} \catA \stackrel{i}{→} \catC\big)} \newcommand{\canlimAC}[1]{\big((#1\backslash i)\stackrel{U}{→} \catA \stackrel{i}{→} \catC\big)} \newcommand\pair[2]{\left<{#1}, {#2}\right>} \newcommand\triple[3]{\anglebrackets{ {#1}, {#2}, {#3}}} \newcommand\anglebrackets[1]{\left<{#1}\right>} \newcommand\set[1]{\left\{#1\right\}} \newcommand\suchthat{\middle\vert} \newcommand\xto\xrightarrow \newcommand\xfrom\xleftarrow \newcommand\parent[1]{\left({#1}\right)} \newcommand\Hom[2][{}]{\mathord{\mathrel{\rm Hom}_{#1}\parent{#2}}} \newcommand{\bigovee}{ \mathop{ \mathchoice{\dobigovee\Large} {\dobigovee\large} {\dobigovee\normalsize} {\dobigovee\small} } } \newcommand{\dobigovee}[1]{\vcenter{#1\kern.2ex\hbox{$\ovee$}\kern.2ex}} \def\Bra#1{\left\langle#1\right|} \def\Ket#1{\left|#1\right\rangle} \newcommand{\EM}{\mathop{\cal EM}\nolimits} \newcommand{\Kl}{\mathop{ {\cal K}\hspace{-.1em}\ell}\nolimits} \newcommand{\KlN}{\mathop{ {\cal K}\hspace{-.1em}\ell_\mathbb{N}}\nolimits} \xymatrix @!0 @R=2pc @C=4pc { \def\adj{\ar@{}[r]|<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<{\top}} \opposite {(\EMod{\M})} \; \ar@/^1em/[rr]^{\Hom[\EMod{\M}]{-, \, \M}} \adj && \ar@/^1em/[ll]^{\Hom[\Conv {\M}]{-, \, \M}} \; \Conv {\M} ≅ \EM(\D {\M})\\ & 𝔹 \ar@/^1em/[lu]^-{\Pred ≝ \Hom[𝔹]{-, \, 1+1}\qquad} \ar@/_1em/[ru]_-{\qquad \Stat ≝ \Hom[𝔹]{1, \, -}} & }