\left[h^{\text {red }}, h^{\text {blue }}\right]: \mathrm{C} \rightarrow \text { Set }
\left[h^{\text {red }}, h^{\text {blue }}\right](c)=\widehat{\mathrm{C}}\left(h^{c},\left[h^{\text {red }}, h^{\text {blue }}\right]\right)=\widehat{\mathrm{C}}\left(h^{c} \times h^{\text {red }}, h^{\text {blue }}\right) .[h^\textit{red},h^\textit{blue}](c) = \widehat{\mathsf{C}}(h^c,[h^\textit{red},h^\textit{blue}]) = \widehat{\mathsf{C}}(h^c\times h^\textit{red},h^\textit{blue}).conf 0.741
\left\{\left(h^{c} \times h^{\text {red }}\right)(d) \rightarrow h^{\text {blue }}(d)\right\}_{d \in \text { ob }(\mathrm{C})}\left\{(h^c\times h^{\textit{red}})(d)\to h^\textit{blue}(d)\right\}_{d\in \mathrm{ob}(\mathsf{C})}conf 0.769
\begin{cases}* \rightarrow * & \text { when } d \text { contains } c, \text { red }, \text { and blue } \\ \emptyset \rightarrow * & \text { when } d \text { does not contain both } c \text { and red, and } d \text { does contain blue } \\ \emptyset \rightarrow \emptyset & \text { when } d \text { does not contain both } c \text { and } \mathrm{red}, \text { and } d \text { does not contain blue. }\end{cases}\begin{cases} \ast \to \ast & \text{when $d$ contains $c$, \emph{red}, and \emph{blue}}\\ \emptyset \to \ast & \text{when $d$ does not contain both $c$ and \emph{red}, and $d$ does contain \emph{blue}}\\ \emptyset \to \emptyset & \text{when $d$ does not contain both $c$ and \emph{red}, and $d$ does not contain \emph{blue}}. \end{cases}conf 0.813
\left[h^{x}, h^{y}\right](c)= \begin{cases}* & \text { if every text that contains both } c \text { and } x \text { also contains } y \\ \emptyset & \text { otherwise }\end{cases}[h^x,h^y](c) = \begin{cases} \ast & \text{ if every text that contains both $c$ and $x$ also contains $y$}\\ \emptyset & \text{otherwise}. \end{cases}conf 0.941
\begin{gathered} 1 \leq \mathcal{C}(x, x) \\ \mathcal{C}(y, z) \otimes \mathcal{C}(x, y) \leq \mathcal{C}(x, z) \end{gathered}\begin{aligned} 1\leq \mathcal{C}(x,x) \\ \mathcal{C}(y,z)\otimes\mathcal{C}(x,y)\leq\mathcal{C}(x,z) \end{aligned}conf 0.847
x \otimes y \leq z \text { if and only if } x \leq[y, z]x\otimes y \leq z \text{ if and only if }x \leq [y,z]conf 1.000
[a, b]:= \begin{cases}b / a & \text { if } b<a \\ 1 & \text { otherwise } .\end{cases}[a,b]:= \begin{cases} b/a &\text{if } b< a,\\ 1 &\text{otherwise}. \end{cases}conf 0.989
a b \leq c \text { if and only if } a \leq[b, c]ab \leq c \text{ if and only if }a \leq [b,c]conf 1.000
\begin{aligned} & a \times b:=\min \{a, b\} \\ & a \sqcup b:=\max \{a, b\} . \end{aligned}\begin{aligned} a\times b&:= \min\{a,b\} \\ a\sqcup b&:= \max\{a,b\}. \end{aligned}conf 1.000
\mathcal{L}(x, y):=\pi(y \mid x),\mathcal{L}(x,y):=\pi(y\vert x),conf 0.873
\begin{aligned} \mathcal{L}(\text { red,red firetruck }) & =0.02 \\ \mathcal{L}(\text { red,red idea }) & =10^{-5} \\ \mathcal{L}(\text { red, blue sky }) & =0 \end{aligned}\begin{aligned} \mathcal{L}\left(\textit{red,red firetruck}\right) &=0.02\\ \mathcal{L}\left(\textit{red,red idea}\right) &=10^{-5}\\ \mathcal{L}\left(\textit{red,blue sky}\right) &=0. \end{aligned}conf 0.860
\pi(z \mid y) \pi(y \mid x)=\pi(z \mid x)
\pi(z \vert y)\pi(y\vert x)=\pi(z\vert x)
conf 0.704
\mathcal{C}(x, y) \leq \mathcal{D}(f x, f y)\mathcal{C}(x,y)\leq \mathcal{D}(fx,fy)conf 1.000
\mathcal{D}^{\mathcal{C}}(f, g):=\int_{c \in \mathcal{C}} \mathcal{D}(f c, g c)\mathcal{D}^{\mathcal{C}}(f,g):=\int_{c\in\mathcal{C}}\mathcal{D}(fc,gc)conf 1.000
\widehat{\mathcal{C}}(f, g)=\inf _{c \in \mathcal{C}}\{[f c, g c]\}\widehat{\mathcal{C}}(f,g)=\inf_{c\in\mathcal{C}}\{[fc,gc]\}conf 1.000
h^{x}:=\mathcal{C}(x,-)h^x:=\mathcal{C}(x,-)conf 1.000
\mathcal{C}(c, d) \leq\left[h^{x}(c), h^{x}(d)\right]\mathcal{C}(c,d)\leq [h^x(c),h^x(d)]conf 1.000
\widehat{\mathcal{C}}\left(h^{x}, f\right) \leq\left[h^{x}(x), f x\right]=[1, f x]=f x .\widehat{\mathcal{C}}(h^x,f)\leq [h^x(x),fx] = [1,fx] =fx.conf 1.000
c \mapsto h^{x}(c):= \begin{cases}\pi(c \mid x) & \text { if } x \leq c \\ 0 & \text { otherwise }\end{cases}c\mapsto h^x(c) := \begin{cases} \pi(c\vert x) &\text{if } x\leq c\\ 0 & \text{otherwise}. \end{cases}conf 0.939
x \xrightarrow{a} y \quad h^{y} \xrightarrow{a} h^{x}
\mathrm{C}(-, \lim F) \cong \operatorname{Set}^{\mathrm{J}}(*, \mathrm{C}(-, F))
\mathcal{E}\left(-, \lim _{W} F\right) \cong \mathcal{V}^{\mathcal{J}}(W, \mathcal{E}(-, F))\mathsf{C}(-,\lim F)\cong \mathsf{Set}^{\mathsf{J}}(\ast,\mathsf{C}(-,F))conf 0.715
\mathcal{E}\left(\operatorname{colim}_{W} F,-\right) \cong \mathcal{V}^{\mathcal{J}^{\mathrm{op}}}(W, \mathcal{E}(F,-))\mathcal{E}(-,\lim_WF)\cong\mathcal{V}^\mathcal{J}(W,\mathcal{E}(-,F))conf 0.877
\left(w_{1}, f\right) \times\left(w_{2}, g\right):=\lim _{W} F .(w_1 ,f)\times (w_2, g) := \lim_W F.
conf 1.000
\widehat{\mathcal{L}}\left(Z, \lim _{W} F\right) \cong[0,1]^{\mathcal{J}}(W, \widehat{\mathcal{L}}(Z, F)) .\mathcal{E}(\operatorname{colim}_WF,-)\cong\mathcal{V}^{\mathcal{J}^{\text{op}}}(W,\mathcal{E}(F,-))conf 0.648
\inf _{c \in \mathcal{L}}\left\{\left[Z c, \min \left\{\frac{f c}{w_{1}}, \frac{g c}{w_{2}}, 1\right\}\right]\right\}=\inf _{c \in \mathcal{L}}\left\{\frac{f c}{w_{1} Z c}, \frac{g c}{w_{2} Z c}, 1\right\} .\inf_{ c \in \mathcal{L}} \left\{\left [Zc, \min\left \{\frac{fc}{w_1},\frac{gc}{w_2},1 \right\}\right]\right\} = \inf_{c \in \mathcal{L}}\left\{\frac{fc}{w_1Zc},\frac{gc}{w_2Zc},1\right\}.conf 1.000
\widehat{\mathcal{L}}(Z, F i)=\inf _{c \in \mathcal{L}}\{[Z c, F i c]\}=\inf _{c \in \mathcal{L}}\left\{\frac{F i c}{Z c}, 1\right\} .\widehat{\mathcal{L}}(Z,Fi)=\inf_{c \in \mathcal{L}}\{[Zc,Fic]\} = \inf_{c \in \mathcal{L}}\left\{\frac{Fic}{Zc},1\right\}.conf 1.000
\begin{aligned} {[0,1]^{\mathcal{J}}(W, \widehat{\mathcal{L}}(Z, F)) } & =\min \left\{\left[w_{1}, \inf _{c \in \mathcal{L}}\left\{\frac{f c}{Z c}, 1\right\}\right],\left[w_{2}, \inf _{c \in \mathcal{L}}\left\{\frac{g c}{Z c}, 1\right\}\right]\right\} \\ & =\inf _{c \in \mathcal{L}}\left\{\frac{f c}{w_{1} Z c}, \frac{g c}{w_{2} Z c}, 1\right\} . \end{aligned}\begin{aligned} [0,1]^\mathcal{J}(W,\widehat{\mathcal{L}}(Z,F))&= \min\left\{\left[ w_1,\inf_{c \in \mathcal{L}}\left\{\frac{fc}{Zc},1\right\}\right],\left[ w_2,\inf_{c \in \mathcal{L}}\left\{\frac{gc}{Zc},1\right\}\right]\right\}\\ &= \inf_{c \in \mathcal{L}}\left\{\frac{fc}{w_1Zc},\frac{gc}{w_2Zc},1\right\}. \end{aligned}conf 0.995
\begin{aligned} \mathcal{L}(c, d) \lim _{W} F(c) & =\min \left\{\frac{\mathcal{L}(c, d) f(c)}{w_{1}}, \frac{\mathcal{L}(c, d) g(c)}{w_{2}}, \mathcal{L}(c, d)\right\} \\ & \leq \min \left\{\frac{f(d)}{w_{1}}, \frac{g(d)}{w_{2}}, 1\right\} \\ & \leq \min \left\{\frac{f(d)}{w_{1}}, \frac{g(d)}{w_{2}}, 1\right\} \\ & =\lim _{W} F(d) \end{aligned}\begin{aligned} \mathcal{L}(c,d)\lim_WF(c) &= \min\left\{ \frac{\mathcal{L}(c,d)f(c)}{w_1}, \frac{\mathcal{L}(c,d)g(c)}{w_2},\mathcal{L}(c,d) \right\} \\ &\leq \min\left\{ \frac{f(d)}{w_1} , \frac{g(d)}{w_2},1\right\}\\ &\leq \min\left\{ \frac{f(d)}{w_1},\frac{g(d)}{w_2},1\right\}\\ &=\lim_WF(d), \end{aligned}conf 0.998
\left(\left(w_{1}, h^{x}\right) \times\left(w_{2}, h^{y}\right)\right)(c)=\min \left\{\frac{h^{x}(c)}{w_{1}}, \frac{h^{y}(c)}{w_{2}}, 1\right\} .((w_1, h^x)\times (w_2, h^y))(c)=\min\left\{\frac{h^x(c)}{w_1}, \frac{h^y(c)}{w_2},1\right\}.conf 1.000
\left(\left(w_{1}, h^{x}\right) \times\left(w_{2}, h^{y}\right)\right)(c)= \begin{cases}\min \left\{\frac{\pi(c \mid x)}{w_{1}}, \frac{\pi(c \mid y)}{w_{2}}, 1\right\} & \text { if } x \leq c \text { and } y \leq c \\ 0 & \text { otherwise }\end{cases}\begin{aligned} ((w_1,h^x)\times (w_2,h^y))(c) &= \begin{cases} \min\left\{\frac{\pi(c\vert x)}{w_1},\frac{\pi(c\vert y)}{w_2},1\right\} & \text{ if } x\leq c \text{ and } y\leq c\\[5pt] 0 & \text{otherwise}. \end{cases} \end{aligned}conf 0.925
\widehat{\mathcal{L}}(h, f \times g)=\widehat{\mathcal{L}}(h, f) \times \widehat{\mathcal{L}}(h, g) .\widehat{\mathcal{L}}(h,f\times g)=\widehat{\mathcal{L}}(h,f)\times\widehat{\mathcal{L}}(h,g).conf 1.000
\begin{aligned} \widehat{\mathcal{L}}(h, f \times g) & =\inf _{c \in \mathcal{L}}\{[h c,(f \times g) c]\} \\ & =\inf _{c \in \mathcal{L}}\{[h c, \min \{f c, g c\}]\} \\ & =\inf _{c \in \mathcal{L}}\left\{\frac{f c}{h c}, \frac{g c}{h c}, 1\right\} \\ & \left.=\min _{\left\{\inf _{c \in \mathcal{L}}\right.}\left\{\frac{f c}{h c}, 1\right\}, \inf _{c \in \mathcal{L}}\left\{\frac{g c}{h c}, 1\right\}, 1\right\} \\ & =\widehat{\mathcal{L}}(h, f) \times \widehat{\mathcal{L}}(h, g) \end{aligned}\begin{aligned} \widehat{\mathcal{L}}(h,f\times g) &= \inf_{c\in \mathcal{L}}\{[hc,(f\times g)c]\} \\ &= \inf_{c\in \mathcal{L}}\{[hc, \min\{fc,gc\}] \}\\ &= \inf_{c\in \mathcal{L}}\left\{\frac{fc}{hc},\frac{gc}{hc},1\right\} \\ &= \min\left\{\inf_{c\in \mathcal{L}}\left\{ \frac{fc}{hc},1\right \} ,\inf_{c\in \mathcal{L}}\left\{\frac{gc}{hc},1\right\},1\right\} \\ &=\widehat{\mathcal{L}}(h,f)\times \widehat{\mathcal{L}}(h,g) \end{aligned}conf 0.903
\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right):=\operatorname{colim}_{W} F .(w_1, f)\sqcup (w_2, g) := \operatorname{colim}_W F.conf 1.000
\widehat{\mathcal{L}}\left(\operatorname{colim}_{W} F, Z\right)=[0,1]^{\mathcal{J}}(W, \widehat{\mathcal{L}}(F, Z)) .\widehat{\mathcal{L}}(Z,\text{lim}_WF)\cong[0,1]^\mathcal{J}(W,\widehat{\mathcal{L}}(Z,F)).conf 0.889
\begin{aligned} \widehat{\mathcal{L}}\left(\operatorname{colim}_{W} F i, Z\right) & =\inf _{c \in \mathcal{L}}\left\{\left[\max \left\{w_{1} f c, w_{2} g c\right\}, Z c\right]\right\} \\ & =\inf _{c \in \mathcal{L}}\left\{\frac{Z c}{\max \left\{w_{1} f c, w_{2} g c\right\}}, 1\right\} \\ & =\inf _{c \in \mathcal{L}}\left\{\frac{Z c}{w_{1} f c}, \frac{Z c}{w_{2} g c}\right\} . \end{aligned}\begin{aligned} \widehat{\mathcal{L}}(\operatorname{colim}_WFi,Z) &= \inf_{c \in \mathcal{L}}\{[\max\left \{w_1fc,w_2gc\right\},Zc]\}\\ &= \inf_{c \in \mathcal{L}}\left\{\frac{Zc}{\max\left \{w_1fc,w_2gc\right\}},1 \right\}\\ &= \inf_{c \in \mathcal{L}}\left\{\frac{Zc}{w_1fc},\frac{Zc}{w_2gc}\right\}. \end{aligned}conf 1.000
\widehat{\mathcal{L}}(F i, Z)=\inf _{c \in \mathcal{L}}\{[F i c, Z c]\}=\inf _{c \in \mathcal{L}}\left\{\frac{Z c}{F i c}, 1\right\} .\widehat{\mathcal{L}}(Fi,Z)=\inf_{c\in \mathcal{L}}\{[Fic,Zc]\} =\inf_{c\in \mathcal{L}} \left\{ \frac{Zc}{Fic}, 1 \right\}.conf 1.000
\begin{aligned} {[0,1]^{\mathcal{J}}(W, \widehat{\mathcal{L}}(F, Z)) } & =\min \left\{\left[w_{1}, \inf _{c \in \mathcal{L}}\left\{\frac{Z c}{f c}, 1\right\}\right],\left[w_{2}, \inf _{c \in \mathcal{L}}\left\{\frac{Z c}{g c}, 1\right\}\right]\right\} \\ & =\inf _{c \in \mathcal{L}}\left\{\frac{Z c}{w_{1} f c}, \frac{Z c}{w_{2} g c}, 1\right\} . \end{aligned}\begin{aligned} [0,1]^\mathcal{J}(W,\widehat{\mathcal{L}}(F,Z))&=\min\left\{\left[w_1,\inf_{c\in \mathcal{L}} \left\{ \frac{Zc}{fc}, 1 \right\}\right],\left[w_2,\inf_{c\in \mathcal{L}} \left\{ \frac{Zc}{gc}, 1 \right\}\right]\right\}\\ &=\inf_{c\in \mathcal{L}} \left\{ \frac{Zc}{w_1fc}, \frac{Zc}{w_2gc}, 1 \right\}. \end{aligned}conf 0.995
\begin{aligned} \mathcal{L}(c, d) \operatorname{colim}_{W} F(c) & =\max \left\{\mathcal{L}(c, d) w_{1} f(c), \mathcal{L}(c, d) w_{2} g(c)\right\} \\ & \leq \max \left\{\frac{f(d)}{w_{1}}, \frac{g(d)}{w_{2}}\right\} \\ & =\operatorname{colim}_{W} F(d) \end{aligned}\begin{aligned} \mathcal{L}(c,d)\operatorname{colim}_WF(c) &= \max\left\{\mathcal{L}(c,d)w_1f(c), \mathcal{L}(c,d)w_2g(c)\right\} \\ &\leq \max\left\{ \frac{f(d)}{w_1} , \frac{g(d)}{w_2}\right\}\\ &=\operatorname{colim}_WF(d), \end{aligned}conf 0.996
[f, g](c):=\widehat{\mathcal{L}}\left(h^{c} \times f, g\right) .[f,g](c):=\widehat{\mathcal{L}}(h^c\times f,g).conf 1.000
\widehat{\mathcal{L}}(h \times f, g)=\widehat{\mathcal{L}}(h,[f, g]) .\widehat{\mathcal{L}}(\operatorname{colim}_WF,Z)=[0,1]^\mathcal{J}(W,\widehat{\mathcal{L}}(F,Z)).conf 0.677
\left[h^{x}, h^{y}\right](c)=\inf _{d \in \mathcal{L}}\left\{\frac{\pi(d \mid y)}{\min \{\pi(d \mid c), \pi(d \mid x)\}}, 1\right\} .[h^x,h^y](c) = \inf_{d \in \mathcal{L}}\left\{\frac{\pi(d|y)}{\min\{\pi(d|c) , \pi(d|x)\}},1\right\}.conf 0.912
\begin{aligned} {\left[h^{x}, h^{y}\right](c) } & =\widehat{\mathcal{L}}\left(h^{c} \times h^{x}, h^{y}\right) \\ & =\inf _{d \in \mathcal{L}}\left\{\left[\left(h^{c} \times h^{x}\right)(d), h^{y}(d)\right]\right\} \\ & =\inf _{d \in \mathcal{L}}\left\{\frac{\pi(d \mid y)}{\pi(d \mid c) \times \pi(d \mid x)}, 1\right\} \\ & =\inf _{d \in \mathcal{L}}\left\{\frac{\pi(d \mid y)}{\min \{\pi(d \mid c), \pi(d \mid x)\}}, 1\right\} . \end{aligned}\begin{aligned} [h^x,h^y](c) &= \widehat{\mathcal{L}}(h^c\times h^x,h^y)\\ &= \inf_{d \in \mathcal{L}} \left\{\left[(h^c \times h^x)(d),h^y(d)\right]\right\}\\ &= \inf_{d \in \mathcal{L}}\left\{\frac{\pi(d|y)}{\pi(d|c) \times \pi(d|x)},1\right\}.\\ &= \inf_{d \in \mathcal{L}}\left\{\frac{\pi(d|y)}{\min\{\pi(d|c), \pi(d|x)\}},1\right\}. \end{aligned}conf 0.627
\widehat{\mathcal{M}}(f, g)=\sup _{x \in \mathcal{M}} d_{[0, \infty]}(f x, g x)=\sup _{x \in \mathcal{M}} \max \{g x-f x, 0\}\widehat{\mathcal{M}}(f,g)=\sup_{x\in\mathcal{M}}d_{[0,\infty]}(fx,gx)=\sup_{x\in\mathcal{M}}\max\{gx-fx,0\}conf 1.000
\begin{gathered} \left(\left(w_{1}, f\right) \times\left(w_{2}, g\right)\right)(c)=\max \left\{f c-w_{1}, g c-w_{2}, 0\right\} \\ \left(\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right)\right)(c)=\min \left\{f c+w_{1}, g c+w_{2}\right\} \end{gathered}\begin{aligned} ((w_1, f) \times (w_2, g))(c) = \max \{fc-w_1, gc-w_2,0\} \\ ((w_1, f) \sqcup (w_2, g))(c) = \min \{fc+w_1, gc+w_2\} \end{aligned}conf 0.881
\left(\left(w_{1}^{\prime}, f^{\prime}\right) \times\left(w_{2}^{\prime}, g^{\prime}\right)\right)(c)=\min \left\{\frac{f^{\prime} c}{w_{1}^{\prime}}, \frac{g^{\prime} c}{w_{2}^{\prime}}, 1\right\}((w_1', f') \times (w_2', g') )(c)= \min\left\{\frac{f'c}{w'_1},\frac{g'c}{w'_2},1\right\}conf 0.670
\begin{aligned} \left(\left(w_{1}, f\right) \times\left(w_{2}, g\right)\right)(c) & =-\ln \left(\left(\left(w_{1}^{\prime}, f^{\prime}\right) \times\left(w_{2}^{\prime}, g^{\prime}\right)\right)(c)\right) \\ & =-\ln \left(\min \left\{\frac{f^{\prime} c}{w_{1}^{\prime}}, \frac{g^{\prime} c}{w_{2}^{\prime}}, 1\right\}\right) \\ & =\max \left\{-\ln \left(\frac{f^{\prime} c}{w_{1}^{\prime}}\right),-\ln \left(\frac{g^{\prime} c}{w_{2}^{\prime}}\right), 0\right\} \\ & =\max \left\{-\ln \left(f^{\prime} c\right)+\ln \left(w_{1}^{\prime}\right),-\ln \left(g^{\prime} c\right)+\ln \left(w_{2}^{\prime}\right), 0\right\} \\ & =\max \left\{f c-w_{1}, g c-w_{2}, 0\right\} \end{aligned}\begin{aligned} \left(\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right)\right)(c) & =-\ln \left(\left(\left(w_{1}^{\prime}, f^{\prime}\right) \sqcup\left(w_{2}^{\prime}, g^{\prime}\right)\right)(c)\right) \\ & =-\ln \left(\max \left\{w_{1}^{\prime} f^{\prime} c, w^{\prime} 2 g^{\prime} c\right\}\right) \\ & =\min \left\{-\ln \left(w_{1}^{\prime} f^{\prime} c\right),-\ln \left(w_{2}^{\prime} g^{\prime} c\right)\right\} \\ & =\min \left\{-\ln \left(f^{\prime} c\right)-\ln \left(w_{1}^{\prime}\right),-\ln \left(g^{\prime} c\right)-\ln \left(w_{2}^{\prime}\right)\right\} \\ & =\min \left\{f c+w_{1}, g c+w_{2}\right\} . \end{aligned}
s_{1} \oplus s_{2}=\min \left\{s_{1}, s_{2}\right\} \text { and } s_{1} \odot s_{2}=s_{1}+s_{2}s_1\oplus s_2=\min\{s_1,s_2\} \text{ and }s_1 \odot s_2=s_1+s_2conf 1.000
\left(\left(w_{1}, f\right) \sqcup\left(w_{2}, g\right)\right)(c)=\min \left\{f c+w_{1}, g c+w_{2}\right\}=w_{1} \odot f c \oplus w_{2} \odot g c .((w_1, f) \sqcup ( w_2, g))(c) = \min\left\{fc+w_1,gc+w_2\right\} = w_1 \odot fc \oplus w_2 \odot gc.conf 1.000
