14.45377pt\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}.$

Note that if $\alpha>-1/2$ then all the terms on the left–hand side of (A.43)
are positive. We now control the term $\iint h^{\alpha}h_{x}^{2}$ on the
right-hand side of (A.43).

By integration by parts and the periodic boundary conditions

$\iint\limits_{Q_{T}}h^{\alpha+m-n}h_{x}^{2}\;dxdt=-\tfrac{1}{\alpha+m-n+1}\iint\limits_{Q_{T}}h^{\alpha+m-n+1}h_{xx}\;dxdt$
(A.44)

Applying the Cauchy-Schwartz inequality and the Cauchy inequality to (A.44)
yields

$a_{1}\iint\limits_{Q_{T}}h^{\alpha+m-n}h_{x}^{2}\;dxdt\leq\tfrac{a_{0}(1+2\alpha)}{2(1-\alpha)}\iint\limits_{Q_{T}}h^{\alpha}h_{xx}^{2}\
\;dxdt+\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\iint\limits_{Q_{T}}h^{\alpha+2(m-n+1)}\;dxdt$
(A.45)

Using inequality (A.45) in (A.43) yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\tfrac{1+2\alpha}{2(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
(A.46)
$\displaystyle\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\iint\limits_{Q_{T}}h^{\alpha+2(m-n+1)}\;dxdt.$

Adding

$\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt$

to both sides of (A.46) and using the inequality (A.42) yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\tfrac{(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
(A.47) $\displaystyle\hskip
14.45377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt\leq\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}$
$\displaystyle\hskip
14.45377pt+\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\iint\limits_{Q_{T}}h^{\alpha+2(m-n+1)}\;dxdt.$

Using (A.31) in (A.47) yields

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+\tfrac{a_{0}(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
$\displaystyle\hskip
14.45377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leq\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}$
$\displaystyle\hskip
14.45377pt+e_{1}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\;dt+e_{2}\;T$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+e_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}\;dt$
(A.48)

where

$e_{1}=\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\;b_{4},\
\
e_{2}=b_{5}\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\;M_{\varepsilon}^{\alpha+2(m-n+1)},$

and $e_{3}=e_{1}+e_{2}$.

Recall the bound (A.33):

$\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}{h^{2m-n}h^{2}_{x}\,dxdt}.$
(A.49)

As before, by the Cauchy-Schwartz inequality and the Cauchy inequality,

$\displaystyle\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}h^{2m-n}h_{x}^{2}\;dxdt\leqslant\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt$
(A.50) $\displaystyle\hskip
101.17755pt+\tfrac{9a_{1}^{4}}{a_{0}^{3}(1+2\alpha)(1-\alpha)}\iint\limits_{Q_{T}}h^{2(2m-n+1)-\alpha}\;dxdt.$

Using (A.50), and (A.35) in (A.49) yields

$\displaystyle\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
$\displaystyle\hskip
14.45377pt+e_{4}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\;dt+e_{5}\;T\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}$
(A.51) $\displaystyle\hskip
14.45377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}+e_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$

where

$e_{4}=\tfrac{9a_{1}^{4}}{a_{0}^{3}(1+2\alpha)(1-\alpha)}\;b_{4},\
e_{5}=b_{5}\tfrac{9a_{1}^{4}}{a_{0}^{3}(1+2\alpha)(1-\alpha)}\;M_{\varepsilon}^{2(2m-n+1)-\alpha},\
e_{6}=e_{4}+e_{5}.$

Add

$\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}$

to both sides of (A.51) and add

$\tfrac{a_{0}(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}h^{\alpha}h_{xx}^{2}\;dxdt$

to the right–hand side of the resulting inequality. Just as (A.32) and (A.33)
yielded (A.37), (A.48) combined with the above inequality yields

$\displaystyle\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
(A.52) $\displaystyle\hskip
14.45377pt\leq\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+e_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}$
$\displaystyle\hskip
28.90755pt+e_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+e_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{3}}\right\\}$

where $e_{7}=e_{3}+e_{6}$,
$\gamma_{3}=\max\\{\alpha/2+m-n+1,2m-n+1-\alpha/2\\}$.

The rest of the proof now continues as in the $0<\alpha<1$ case. Specifically,
one finds a bound

$\displaystyle\int\limits_{\Omega}(h_{x}^{2}(x,T)+G_{\varepsilon}^{(\alpha)}(h(x,T)))\;dx$
(A.53) $\displaystyle\hskip 28.90755pt\leqslant
2^{\tfrac{1}{\gamma_{3}-1}}\max\left\\{1,\int\limits_{\Omega}({h_{0,\varepsilon,x}}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right\\}=K_{\varepsilon}<\infty$

for all

$0\leqslant T\leqslant
T_{\varepsilon,loc}^{(\alpha)}:=\tfrac{1}{2e_{7}(\gamma_{3}-1)}\min\left\\{1,\left(\int\limits_{\Omega}({h_{0,\varepsilon,x}}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right)^{-(\gamma_{3}-1)}\right\\}.$

The time $T_{loc}^{(\alpha)}$ is defined as in (A.39) and the uniform bound
(A.53) used to bound the right hand side of (A.48) yields the desired bound

$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+\tfrac{a_{0}(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
$\displaystyle\hskip
86.72377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leqslant
K_{2}.$ (A.54)

∎

###### Sketch of Proof of Lemma 6.1.

In the following, we denote the positive, classical solution $h_{\varepsilon}$
constructed in Lemma 4.3 by $h$ (whenever there is no chance of confusion).

Recall the entropy function $G_{0}^{(\alpha)}(z)$ defined by (4.12). Using the
local entropy inequality (7.1) with $\zeta=\zeta(x)$ from Lemma 7.1, we obtain

$\int\limits_{\Omega}{\zeta^{4}(x)G^{(\alpha)}_{0}(h(x,T))\,dx}+C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant\int\limits_{\Omega}{\zeta^{4}(x)G^{(\alpha)}_{0}(h_{0})\,dx}+\\\
C_{2}\iint\limits_{Q_{T}}{h^{\alpha+2}(\zeta_{x}^{4}+\zeta^{2}\zeta_{xx}^{2})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{2(m-n+1)+\alpha}\zeta^{4}\,dxdt}.$
(A.55)

Due to $h\in L^{\infty}(0,T_{loc}^{(\alpha)};H^{1}(\Omega))$, we deduce from
(A.55) that

$\int\limits_{\Omega}{\zeta^{4}(x)G^{(\alpha)}_{0}(h(x,T))\,dx}\leqslant\int\limits_{\Omega}{\zeta^{4}(x)G_{0}(h_{0}(x))\,dx}+C_{4}T\leqslant
K<\infty$ (A.56)

for any $T\in[0,T_{loc}^{(\alpha)}]$. ∎

## B Proof of second moment estimate

###### Sketch of Proof of Lemma 8.1.

Let

$\widetilde{G}_{\varepsilon}(z)=\tfrac{z^{2-n}}{2-n}-\tfrac{\varepsilon\,z^{2-s}}{s-2},\
\widetilde{G}^{\prime}_{\varepsilon}(z)=\tfrac{z}{f_{\varepsilon}(z)},$ (B.1)
$\widetilde{G}^{\prime\prime}_{\varepsilon}(z)=\tfrac{1}{f_{\varepsilon}(z)}-\tfrac{zf^{\prime}_{\varepsilon}(z)}{f^{2}_{\varepsilon}(z)}=(1-n)\tfrac{1}{f_{\varepsilon}(z)}-\varepsilon(s-n)z^{-s}.$

Here we use the equality

$f^{\prime}_{\varepsilon}(z)=nz^{-1}f_{\varepsilon}(z)+\varepsilon(s-n)z^{-(s+1)}f_{\varepsilon}^{2}(z).$

Multiplying (4.1) for $\delta=0$ by
$x^{2}\widetilde{G}^{\prime}_{\varepsilon}(h)$, and integrating on $Q_{T}$,
yields

$\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h)\,dx}-\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h_{0\varepsilon})\,dx}=\\\
\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\bigl{(}2x\tfrac{h}{f_{\varepsilon}(h)}+x^{2}\widetilde{G}^{\prime\prime}_{\varepsilon}(h)h_{x}\bigr{)}\,dxdt}=\\\
2\iint\limits_{Q_{T}}{xh(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}+\\\
(1-n)\iint\limits_{Q_{T}}{x^{2}h_{x}(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}-\\\
\varepsilon(s-n)\iint\limits_{Q_{T}}{x^{2}h^{-s}f_{\varepsilon}(h)h_{x}(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}=:I_{1}+I_{2}+I_{3}.$
(B.2)

We now bound the terms $I_{1}$, $I_{2}$ and $I_{3}$. First,

$I_{1}=2a_{0}\int\limits_{0}^{T}{xhh_{xx}\biggl{|}_{\partial\Omega}\,dx}-2a_{0}\iint\limits_{Q_{T}}{\\{hh_{xx}+x\,h_{x}h_{xx}\\}\,dxdt}+\\\
2a_{1}\iint\limits_{Q_{T}}{xhD^{\prime\prime}_{\varepsilon}(h)h_{x}\,dxdt}=\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}h_{x}^{2}+2a_{1}L_{\varepsilon}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+\\\
3a_{0}\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-2a_{1}\iint\limits_{Q_{T}}{L_{\varepsilon}(h)\,dxdt},\
L_{\varepsilon}(z):=\int\limits_{0}^{z}{\tau\,D^{\prime\prime}_{\varepsilon}(\tau)\,d\tau},$
(B.3)

and

$I_{2}=a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h_{x}h_{xxx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=\\\
-2a_{0}(1-n)\iint\limits_{Q_{T}}{xh_{x}h_{xx}\,dxdt}-a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+\\\
a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=-a_{0}(1-n)\int\limits_{0}^{T}{xh_{x}^{2}\biggl{|}_{\partial\Omega}\,dt}+a_{0}(1-n)\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-\\\
a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}.$
(B.4)

Using the Hölder inequality and (4.9) with $\delta=0$, we find

$I_{3}\leqslant\varepsilon(s-n)\Bigl{(}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}\times\\\
\Bigl{(}\iint\limits_{Q_{T}}{x^{4}h^{-2s}f_{\varepsilon}(h)h_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}\leqslant\varepsilon\,C\Bigl{(}\iint\limits_{Q_{T}}{\tfrac{h_{x}^{2}}{(h^{s-n}+\varepsilon)^{2}}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}.$

Using the Young’s inequality $ab\leqslant\tfrac{a^{p}}{p}+\tfrac{b^{q}}{q},\
\tfrac{1}{p}+\tfrac{1}{q}=1$ ($\Rightarrow p\,ab\leqslant
a^{p}+\tfrac{p}{q}b^{q}=a^{p}+(p-1)b^{q}$) with $a=z^{\tfrac{s-n}{p}}$ and
$b=\bigl{(}\tfrac{\varepsilon}{p-1}\bigr{)}^{\tfrac{1}{q}}$, we deduce

$I_{3}\leqslant\varepsilon^{\tfrac{q-1}{q}}\,C\Bigl{(}\iint\limits_{Q_{T}}{h^{-\tfrac{2(s-n)}{p}}h_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}=\varepsilon^{\tfrac{q-1}{q}}\,\widetilde{C}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{p-s+n}{p}})_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}},$

choosing $p=-\tfrac{s-n}{\alpha}>1$ and $q=\tfrac{2(s-n)}{2(s-n)+\alpha}>1$
($\Rightarrow 0>\alpha>-2(s-n)$), due to (4.14), we find

$I_{3}\leqslant\varepsilon^{\tfrac{q-1}{q}}\,\widetilde{C}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{p-s+n}{p}})_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}=\\\
\varepsilon^{-\tfrac{\alpha}{s-n}}\,\widetilde{C}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}\leqslant
C_{1}\varepsilon^{-\tfrac{\alpha}{s-n}},$ (B.5)

where the constant $C_{1}>0$ is independent of $\varepsilon$. Hence,

$\lim_{\varepsilon\to 0}{I_{3}}\leqslant 0.$ (B.6)

From (B.2)–(B.4), we deduce that

$\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h)\,dx}=\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h_{0\varepsilon})\,dx}+\\\
\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}L_{\varepsilon}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+a_{0}(4-n)\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-\\\
a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}-\\\
2a_{1}\iint\limits_{Q_{T}}{L_{\varepsilon}(h)\,dxdt}+I_{3}.$ (B.7)

Letting $\varepsilon\to 0$ in (B.7), due to (B.6), we find

$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+a_{0}(4-n)\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-\\\
2a_{1}(m-n+1)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}+\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}L_{0}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+\\\
a_{0}(n-1)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{m-n}h_{x}^{2}\,dxdt}=\\\
\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+2(4-n)\int\limits_{0}^{T}{\mathcal{E}_{0}(t)\,dt}-2\,a_{1}(m-3)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}+\\\
\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}L_{0}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+\\\
a_{0}(n-1)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{m-n}h_{x}^{2}\,dxdt}.$
(B.8)

Due to

$\iint\limits_{Q_{T}}{x^{2}h^{m-n}h_{x}^{2}\,dxdt}=-2\int\limits_{0}^{T}{xD_{0}(h)\biggl{|}_{\partial\Omega}\,dt}+2\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}-\\\
\iint\limits_{Q_{T}}{x^{2}D^{\prime}_{0}(h)h_{xx}\,dxdt},$

from (B.8), in view of (3.6), we deduce that

$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}+2a_{1}(m+n-4)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\\\
k_{1}\mathcal{E}_{0}(0)T-(1-n)\iint\limits_{Q_{T}}{x^{2}h_{xx}(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))\,dxdt}+W(T)=\\\
\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T-\tfrac{1-n}{a_{0}}\iint\limits_{Q_{T}}{x^{2}(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))^{2}\,dxdt}+\\\
\tfrac{a_{1}(1-n)}{a_{0}}\iint\limits_{Q_{T}}{x^{2}D^{\prime}_{0}(h)(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))\,dxdt}+W(T),$
(B.9)

where $k_{1}=2(4-n)$, and $W(T)$ is from (8.2). From (B.9) we find that

$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}+2a_{1}(m+n-4)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}+\\\
\tfrac{1-n}{2a_{0}}\iint\limits_{Q_{T}}{x^{2}(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))^{2}\,dxdt}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T+\\\
\tfrac{a_{1}^{2}(1-n)}{2a_{0}}\iint\limits_{Q_{T}}{x^{2}(D^{\prime}_{0}(h))^{2}\,dxdt}+W(T)\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T+\\\
\int\limits_{0}^{T}{\Bigl{(}\widetilde{A}(t)\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}\Bigr{)}dt}+W(T),$
(B.10)

where $m\geqslant 4-n$, and

$\widetilde{A}(t):=\tfrac{a_{1}^{2}|1-n|(2-n)}{2a_{0}(m-n+1)^{2}}\|h(.,t)\|_{L^{\infty}(\Omega)}^{2m-n}.$

Applying the nonlinear Grönwall lemma [15] to

$v(T)\leqslant
v(0)+k_{1}\mathcal{E}_{0}(0)T+\int\limits_{0}^{T}{\widetilde{A}(s)v(s)\,dt}+W(T)$

with $v(T)=\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h(x,T))\,dx}$ yields

$v(T)\leqslant
e^{\widetilde{B}(T)}\Bigl{(}v(0)+\int\limits_{0}^{T}{(k_{1}\mathcal{E}_{0}(0)+W^{\prime}(t))e^{-\widetilde{B}(t)}dt}\Bigr{)}$

with $\widetilde{B}(T):=\int\limits_{0}^{T}{\widetilde{A}(\tau)\,d\tau}$. From
this we obtain (8.1) for $0<n\leqslant 1$.

In the case of $1<n<2$, from (B.9) we find that

$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}+2a_{1}(m+n-4)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T+\\\
\tfrac{3a_{0}(n-1)}{2}\iint\limits_{Q_{T}}{x^{2}h_{xx}^{2}\,dxdt}+\tfrac{a_{1}^{2}(n-1)}{2a_{0}}\iint\limits_{Q_{T}}{x^{2}(D^{\prime}_{0}(h))^{2}\,dxdt}+W(T)\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\\\
k_{1}\mathcal{E}_{0}(0)T+k_{2}\iint\limits_{Q_{T}}{x^{2}h_{xx}^{2}\,dxdt}+\int\limits_{0}^{T}{\Bigl{(}\widetilde{A}(t)\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}\Bigr{)}dt}+W(T),$
(B.11)

where $m\geqslant 4-n$, $k_{2}=\tfrac{3a_{0}(n-1)}{2}$, and $W(t)$ is from
(8.2). Applying the Grönwall lemma [15] to (B.11), we obtain the estimate
(8.1) for $1<n<2$.

∎

## Appendix C

######  Lemma D.1.

([32]) Suppose that $X,\ Y,$ and $Z$ are Banach spaces,
$X\\!\Subset\\!Y\subset\\!Z$, and $X$ and $Z$ are reflexive. Then the
imbedding $\\{u\in\\!L^{p_{0}}(0,T;$ $X):$ $\partial_{t}u\in
L^{p_{1}}(0,T;Z),1<p_{i}<\infty,i=0,1\\}\Subset L^{p_{0}}(0,T;Y)$ is compact.

######  Lemma D.2.

([36]) Suppose that $X,\ Y,$ and $Z$ are Banach spaces and
$X\\!\Subset\\!Y\subset\\!Z$. Then the imbedding $\\{u\in
L^{\infty}(0,T;X):\partial_{t}u\in L^{p}(0,T;Z)$, $p>1\\}\Subset C(0,T;Y)$ is
compact.

######  Lemma D.3.

([26, 6]) Let $\Omega\subset\mathbb{R}^{N},\ N<6$, be a bounded convex domain
with smooth boundary, and let
$n\in\bigl{(}2-\sqrt{1-\tfrac{N}{N+8}},3\bigr{)}$ for $N>1$, and
$\tfrac{1}{2}<n<3$ for $N=1$. Then the following estimates hold for any
strictly positive functions $v\in H^{2}(\Omega)$ such that $\nabla
v\cdot\vec{n}=0$ on $\partial\Omega$ and
$\int\limits_{\Omega}{v^{n}|\nabla\Delta v|^{2}}<\infty$:

$\int\limits_{\Omega}{\varphi^{6}\\{v^{n-4}|\nabla
v|^{6}+v^{n-2}|D^{2}v|^{2}|\nabla v|^{2}\\}}\leqslant
c\Bigl{\\{}\int\limits_{\Omega}{\varphi^{6}v^{n}|\nabla\Delta
v|^{2}}+\int\limits_{\\{\varphi>0\\}}{v^{n+2}|\nabla\varphi|^{6}}\Bigr{\\}},$
$\int\limits_{\Omega}{\varphi^{6}|\nabla\Delta
v^{\tfrac{n+2}{2}}|^{2}}\leqslant
c\Bigl{\\{}\int\limits_{\Omega}{\varphi^{6}v^{n}|\nabla\Delta v|^{2}}+\\\
+\int\limits_{\\{\varphi>0\\}}{v^{n+2}\\{|\nabla\varphi|^{6}+\varphi^{2}|D^{2}\varphi|^{2}|\nabla\varphi|^{2}+\varphi^{3}|\Delta\varphi|^{3}\\}}\Bigr{\\}},$

where $\varphi\in C^{2}(\Omega)$ is an arbitrary nonnegative function such
that the tangential component of $\nabla\varphi$ is equal to zero on
$\partial\Omega$, and the constant $c>0$ is independent of $v$.

######  Lemma D.4.

([33]) If $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with piecewise-
smooth boundary, $a>1$, $b\in(0,a),\ d>1,$ and $0\leqslant i<j,\
i,j\in\mathbb{N}$, then there exist positive constants $d_{1}$ and $d_{2}$
$(d_{2}=0\text{ if }\Omega$ is unbounded$)$ depending only on $\Omega,\ d,\
j,\ b,$ and $N$ such that the following inequality is valid for every $v(x)\in
W^{j,d}(\Omega)\cap L^{b}(\Omega)$:

$\left\|{D^{i}v}\right\|_{L^{a}(\Omega)}\leqslant
d_{1}\left\|{D^{j}v}\right\|_{L^{d}(\Omega)}^{\theta}\left\|v\right\|_{L^{b}(\Omega)}^{1-\theta}+d_{2}\left\|v\right\|_{L^{b}(\Omega)},\
\theta=\tfrac{{\tfrac{1}{b}+\tfrac{i}{N}-\tfrac{1}{a}}}{{\tfrac{1}{b}+\tfrac{j}{N}-\tfrac{1}{d}}}\in\left[{\tfrac{i}{j},1}\right)\\!\\!.$

######  Lemma D.5.

([24]) Let $(\beta_{1},\ldots,\beta_{m})\in\mathbb{R}^{m},m\geqslant 1$, and
let $\beta=\prod\limits_{j=1}^{m}\beta_{j},\
\overline{\beta_{i}}=\tfrac{\beta}{\beta_{i}}=\prod\limits_{j=1,j\neq
i}^{m}\beta_{j}$. Assume that $G_{i}(s)$ are nonnegative nonincreasing
functions satisfying the conditions

$G_{i}(s+\delta)\leqslant
c_{i}\Bigl{(}{\sum\limits_{i=1}^{m}\tfrac{G_{i}(s)}{\delta^{\alpha_{i}}}}\Bigr{)}^{\beta_{i}}\
\forall\,s>0,\ \delta>0,\ i=\overline{1,m}$

with real constants $c_{i}>0,\ \beta_{i}>1$, and $\alpha_{i}\geqslant 0$ for
$i=\overline{1,m}$, and $\alpha_{i}>0$ for $i=\overline{1,\ell}$. Let
$G(s)\\!=\\!\sum\limits_{i=1}^{m}\\!{\overline{(c_{i}^{\overline{\beta}_{i}})}\left({G_{i}(s)}\right)^{\overline{\beta}_{i}}}$,
and let the function
$H(s)=m^{\beta}\sum\limits_{i=\ell+1}^{m}\\!{c_{i}^{\overline{\beta}_{i}}\overline{(c_{i}^{\overline{\beta}_{i}})}^{1-\beta_{i}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\left({G_{i}(s)}\right)^{\beta_{i}-1}}$
be such that $H(s_{1})<1$ at a some $s_{1}\geqslant 0$. Then there exists a
positive constant $c>1$ depending on $m,\ \alpha_{i},\ \beta_{i},\ \ell$, and
$H(s_{1})$ such that $G_{i}(s_{0})\equiv 0$ for all $i=\overline{1,\ell}$,
where
$s_{0}=s_{1}+c\sum\limits_{i=1}^{\ell}{\bigl{(}{c_{i}^{\overline{\beta}_{i}}{\overline{(c_{i}^{\overline{\beta}_{i}})}}^{1-\beta_{i}}\left({G(s_{1})}\right)^{\beta_{i}-1}}\bigr{)}^{\tfrac{1}{\alpha_{i}\beta}}}$.

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# SAHAAYAK 2023 - the Multi Domain Bilingual Parallel Corpus of Sanskrit to
Hindi for Machine Translation

Vishvajitsinh Bakrola  Dr. Jitendra Nasariwala

###### Abstract

The data article presents the large bilingual parallel corpus of low–resourced
language pair Sanskrit – Hindi, named SAHAAYAK 2023. The corpus contains total
of 1.5M sentence pairs between Sanskrit and Hindi. To make the universal
usability of the corpus and to make it balanced, data from multiple domain has
been incorporated into the corpus that includes, News, Daily conversations,
Politics, History, Sport, and Ancient Indian Literature. The multifaceted
approach has been adapted to make a sizable multi-domain corpus of low-
resourced languages like Sanskrit. Our development approach is spanned from
creating a small hand-crafted dataset to applying a wide range of mining,
cleaning, and verification. We have used the three-fold process of mining:
mining from machine-readable sources, mining from non-machine readable
sources, and collation from existing corpora sources. Post mining, the
dedicated pipeline for normalization, alignment, and corpus cleaning is
developed and applied to the corpus to make it ready to use on machine
translation algorithms.

##  1 Introduction

Parallel corpus is a crucial component in the field of machine translation. It
refers to a collection of texts in two or more languages, where each sentence
in one language has a corresponding sentence in the other language. Building
parallel corpora can be a challenging task, especially for low-resource
languages. The effectiveness of machine translation systems relies heavily on
the quality and size of the corpus used for training. Therefore, building
high-quality parallel corpus is essential for achieving accurate and human-
like translation results. State-of-the-art Neural machine translation systems
are outperforming previous translation approaches. (Singh et al., 2019) One of
the significant limitations of these systems is they are data-hungry. NMT
models require a large amount of data to offer good translation accuracy, a
particular hindrance from the beginning of the neural machine translation
journey.

##  2 Need of the Corpus

Both languages belong to 22 scheduled languages of India under article 343 (1)
of the Indian constitution. (Article-343, 1) In addition, two of the Indian
states, i.e., Himachal Pradesh and Uttarakhand, have included Sanskrit as an
additional official language, and nine states have Hindi as their first
official language. As per the last census of India, 528M native speakers are
using Hindi as their mother tongue, and 24.8K native speakers are using
Sanskrit as their mother tongue. (Census, 2011) Sanskrit fall under the
category of Low Resourced Language (LRL). It becomes a real problem when it
comes to the creation of LRLs corpus. These categories of languages are less
studied, resource-scarce, less computerized, and less privileged for the time
being. (Singh, 2008) Most of today’s NLP research focuses on 20 of nearly 7000
spoken languages worldwide. This is because of their significantly less
digital availability. Sanskrit belongs to the same category in multiple
dimensions. Despite its rich grammatical tradition in the linguistic community
and being one of the oldest living languages, there was an absolute absence of
a general-purpose bilingual corpus with Sanskrit as one of the languages. We
cannot consider Sanskrit a resource-scarce and less-studied language, but it
is less computerized. Though some good classical Sanskrit texts from Vedas,
Ramayana, Mahabharata, and Upanishads are digitized chiefly, they have a
monolingual form or are in non-machine-readable formats.

We found that all the previous works contended against the scarcity of
sufficient parallel corpus with Sanskrit as one of the languages in the
translation pair. We observed that developing a fully-aligned general-purpose
parallel corpus keeping Sanskrit as the source language is the first need to
establish a state-of-the-art machine translation system. To our knowledge,
SAHAAYAK 2023 is the most extensive dataset released in the public domain,
with Sanskrit as one of its languages.

##  3 General Architecture of Corpus Creation

The task of corpus creation of LRL required a 360-degree approach for language
coverage. To enhance the universal usability the corpus should be balanced.
(ATKINS et al., 1992) We have applied our efforts to balance this corpus by
covering various domains. We initiated the parallel corpus development of
Sanskrit to Hindi with nearly 3.3K sentences in 2018, and these translations
included 2590 human-translated sentences of daily conversations in the Indian
context and 701 slokas from the Gita. Over a period, we expanded this further
with varieties of mining, human expert verification, and translation by human
experts. Our mining approach is divided into three–fold, mining from machine-
readable sources, mining from non-machine-readable sources, and Collation from
other existing sources of corpora. An overview of our pipeline is shown in
Figure – 1.

Figure 1: Pipeline of Sanskrit-Hindi Parallel Corpus Creation

We identified several web sources which publish content in multiple Indic
languages, including Sanskrit. Most of these sources contain comparable texts
rather than the direct translation of each other. We have created a source-
specific standalone procedure for fetching text data.

For Sanskrit, a considerable amount of physical and scanned documents are
available which are not machine-readable. We have developed separate
procedures for fetching and aligning text from these sources using OCR. In
this mining procedure, there are two varieties of seeds we have worked with.
One belongs to those sources which are publicly available data, i.e., scanned
copies of Sanskrit and its Hindi translations and physical books, scriptures,
and other documents which are in public domains. The dedicated OCR techniques
are developed for extraction from such public sources. In addition to these
pipelines for parallel text collection, samples were collected from the OPUS,
and NLLB. We have also applied parallel text extraction and post-processing
steps on these samples to remove the noise and to make the text more clear.
Text normalization, Text alignment, and Corpus cleaning techniques are
developed and applied keeping in mind the Devanagari script.

##  4 Analysis of SAHAAYAK 2023

Table – 1 provides a glimpse on our bilingual parallel Sanskrit-Hindi corpus.

Table 1: Analysis of SAHAAYAK 2023 Parameters | Sanskrit | Hindi
---|---|---
Number of samples | 1580701 | 1580701
Average Sentence Length | 64 | 66
Longest Sentence Length | 2005 | 2069

##  5 Conclusion

SAHAAYAK 2023 is an attempt to contribute to the community of machine
translation and computational linguistics. We presented a bilingual parallel
corpus of 1.5M parallelly aligned sentence pairs from Sanskrit to Hindi. Tens
of methods are applied to ensure the overall quality of the corpus in the
post–processing. We hope that SAHAAYAK 2023 will be proven helpful in
developing state-of-the-art machine translation systems for Indian languages.
The improvements in the translation quality of Sanskrit to other Indic
languages will also be proven beneficial for Education and different societal
needs.

##  6 Acknowledgment

We would like to thank the researchers of NLP and computational linguistic
community. We would also like to acknowledge the authors of NLLB and OPUS. We
would like to take this opportunity to express our gratitude towards our
Sanskrit language experts Shri Harshad Joshi and Shri Udit Sharma for their
significant contributions for translation and verification. The acknowledgment
is incomplete without showing deep gratitude towards Maharshi Panini, the
complete source of inspiration to work in the direction of Sanskrit based
computational linguistics and Padmabhusan Shripad Damodar Satwalekar, whose
books are proven corner stone for creating interest in the Sanskrit language.

##  7 Dataset Source

The corpus can be accessed at https://rb.gy/hf6bp

## References

  * Article-343 (1) Article-343(1).  Official languages of the union and eight schedule, constitution of republic of india, government of india.  24 June 2023.  URL http://mha.gov.in.
  * ATKINS et al. (1992) ATKINS, S., CLEAR, J., and OSTLER, N.  Corpus Design Criteria.  _Literary and Linguistic Computing_ , 7(1):1–16, 01 1992.  ISSN 0268-1145.  doi: 10.1093/llc/7.1.1.  URL https://doi.org/10.1093/llc/7.1.1.
  * Census (2011) Census2011.  Language – india, states, and union territories, office of registrar general, government of india.  24 June 2023.  URL https://censusindia.gov.in/.
  * Singh (2008) Singh, A. K.  Natural language processing for less privileged languages: Where do we come from? where are we going?  In _Proceedings of the IJCNLP-08 Workshop on NLP for Less Privileged Languages_ , 2008.  URL https://aclanthology.org/I08-3004.
  * Singh et al. (2019) Singh, M., Kumar, R., and Chana, I.  Neural-based machine translation system outperforming statistical phrase-based machine translation for low-resource languages.  In _2019 Twelfth International Conference on Contemporary Computing (IC3)_ , pp. 1–7, 2019.  doi: 10.1109/IC3.2019.8844915.


# Computability models over categories

Iosif Petrakis
Mathematics Institute, Ludwig-Maximilians-Universität München
<EMAIL_ADDRESS>
###### Abstract

Generalising slightly the notions of a strict computability model and of a
simulation between them, which were elaborated by Longley and Normann in [8],
we define canonical computability models over categories and appropriate
$\mathrm{\mathbf{Set}}$-valued functors on them. We study the canonical total
computability model over a category, and the partial one over a category with
pullbacks. Our notions and results are generalised to categories with a base
of computability, connecting, unexpectedly, Rosolini’s theory of dominions
with the theory of computability models.
Keywords: computability models, category theory, dominions

##  1 Introduction

In their book [8] Longley and Normann not only give a comprehensive
introduction to Higher-Order Computability (HOC) presenting the various
approaches to HOC, but they also fit them into a coherent and unifying
framework, making their comparison possible. Their main tool is a general
notion of computability model111This notion is rooted in previous work of
Longley in [5]-[7], and is influenced by the work of Cockett and Hofstra in
[3] and [4] (see [8], p. 52). and of an appropriate concept of simulation of
one computability model in another. Turing machines, programming languages,
$\lambda$-calculus etc., are shown to be computational models, not necessarily
in a unique way.

As it is remarked in [2], p. 3, computability theory has “still not received
the level of categorical attention it deserves”. It seems though, that the
“categorical spirit” of the notions of a computability model and a simulation
between them is behind the remarkable success of the framework of Longley and
Normann. E.g., their, completely categorical in nature, notion of equivalence
of computability models (see Definition 2.2) is deeper than the one suggested
by the standard presentations of computability theory, according to which
different notions of computability are equivalent if they generate the same
class of partial computable functions from ${\mathbb{N}}$ to ${\mathbb{N}}$
(see [8], Section 1.1.5). Longley and Normann associated in a canonical way to
a computability model222In [8] this category is defined for lax computability
models, but, as it is remarked in [8], p. 91, the definition makes sense for
an arbitrary computability model. $\bm{C}$ its category of assemblies
$\mathcal{A}sm(\bm{C})$, and showed that the computability models $\bm{C}$ and
$\bm{D}$ are equivalent if and only if the categories of assemblies
$\mathcal{A}sm(\bm{C})$ and $\mathcal{A}sm(\bm{D})$ are equivalent. The
category of computability models and the corresponding functor
$\bm{C}\mapsto\mathcal{A}sm(\bm{C})$ are studied extensively in [7].

In this paper we associate in a canonical way a computability model to a
category $\mathcal{C}$, given an appropriate $\mathrm{\mathbf{Set}}$-valued
functor on $\mathcal{C}$. For that we slightly generalise the definition of a
computability model, allowing the classes of type names $T$ and of partial
functions $C[\sigma,\tau]$ in a computability model to be proper classes. The
computability models considered in [8] are called small in Definition 2.1, and
working solely with them we can only define the computability model of a small
category. The notion of a simulation between computability models is also
slightly generalised, as the function between the corresponding classes of
type names is, generally, a class-function. Although Longley and Normann
usually work with lax computability models, to our needs the so-called strict
computability models suit best333For a discussion on “strict vs. lax” see [7],
section 2.1..

In section 2 we include all basic notions and facts necessary to the rest of
this paper. In section 3 we study the canonical total computability model over
a category $\mathcal{C}$ and a functor
$S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$. In section 4 we study the
canonical (partial) computability model over a category $\mathcal{C}$ with
pullbacks and a functor $S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ that
preserves pullbacks. In section 5 we define the notion of a base of
computability for a category $\mathcal{C}$, and the canonical (partial)
computability model over a category $\mathcal{C}$ with a base of computability
for $\mathcal{C}$ and a functor $S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$
that preserves pullbacks. The first two constructions are shown to be special
cases of the last one. All computability models defined here have their dual
form e.g., the computability model over a category $\mathcal{C}$ with a cobase
of computability for $\mathcal{C}$ and a functor
$S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ that sends pushouts to pullbacks
is also defined.

For all categorical notions and facts mentioned here we refer to [1].

##  2 Computability models

######  Definition 2.1.

Let $T$ be a class, the elements of which are called type names. A
computability model $\bm{C}$ over $T$ is a pair

$\bm{C}=\big{(}\big{(}\bm{C}(\tau)\big{)}_{\tau\in
T},\big{(}\bm{C}[\sigma,\tau]\big{)}_{(\sigma,\tau)\in T\times T}\big{)},$

where $|\bm{C}|=\big{(}\bm{C}(\tau)\big{)}_{\tau\in T}$ is a family of sets
$\bm{C}(\tau)$ over $T$, called the datatypes of $\bm{C}$, and
$\big{(}\bm{C}[\sigma,\tau]\big{)}_{(\sigma,\tau)\in T\times T}$ is a family
of classes $\bm{C}[\sigma,\tau]$ of partial functions
$\bm{C}(\sigma)\rightharpoondown\bm{C}(\tau)$ over $T\times T$, such that the
following hold:
$(\textnormal{{CM}}_{1})$ The identity function
$\mathrm{id}_{\bm{C}(\tau)}\colon\bm{C}(\tau)\to\bm{C}(\tau)$ is in
$\bm{C}[\tau,\tau]$, for every $\tau\in T$,
$(\textnormal{{CM}}_{2})$ For every $f\in\bm{C}[\rho,\sigma]$ and
$g\in\bm{C}[\sigma,\tau]$, the composite partial function $g\circ f$ is in
$\bm{C}[\rho,\tau]$, for every $\rho,\sigma,\tau\in T$.
If the classes $\bm{C}[\sigma,\tau]$ are sets, for every $\sigma,\tau\in T$,
we call $\bm{C}$ locally small. If $T$ is also a set, we call $\bm{C}$ small.
If every element of $\bm{C}[\sigma,\tau]$ is a total function, for every
$\sigma,\tau\in T$, then $\bm{C}$ is called total.

For the rest of this paper $T,U,W$ are classes, and $\bm{C},\bm{D},\bm{E}$ are
computability models over $T,U,W$, respectively.

######  Definition 2.2.

A simulation $\bm{\gamma}$ of $\bm{C}$ in $\bm{D}$, in symbols
$\bm{\gamma}\colon\bm{C}\rightarrowtriangle\bm{D}$, is a pair

$\bm{\gamma}=\big{(}\gamma,\big{(}\Vdash^{\bm{\gamma}}_{\tau}\big{)}_{\tau\in
T}\big{)},$

where $\gamma\colon T\to U$ is a class function, and
$\Vdash^{\bm{\gamma}}_{\tau}\subseteq\bm{D}(\gamma(\tau))\times\bm{C}(\tau)$,
for every $\tau\in T$, such that the following hold:

$(\textnormal{{Siml}}_{1})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \forall_{\tau\in
T}\forall_{x\in\bm{C}(\tau)}\exists_{x{{}^{\prime}}\in\bm{D}(\gamma(\tau))}\big{(}x{{}^{\prime}}\Vdash^{\bm{\gamma}}_{\tau}x\big{)},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$(\textnormal{{Siml}}_{2})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \forall_{\sigma,\tau\in
T}\forall_{f\in\bm{C}[\sigma,\tau]}\exists_{f{{}^{\prime}}\in\bm{D}[\gamma(\sigma),\gamma(\tau)]}\big{(}f{{}^{\prime}}\Vdash^{\bm{\gamma}}_{(\sigma,\tau)}f\big{)},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $

where the relation “$f$ is tracked by $f{{}^{\prime}}$ through $\bm{\gamma}$”
is defined by

$f{{}^{\prime}}\Vdash^{\bm{\gamma}}_{(\sigma,\tau)}f:\Leftrightarrow\forall_{x\in\bm{C}(\sigma)}\big{(}x\in\mathrm{dom}(f)\Rightarrow$
$\forall_{x{{}^{\prime}}\in\bm{D}(\gamma(\sigma))}\big{(}x{{}^{\prime}}\Vdash^{\bm{\gamma}}_{\sigma}x\Rightarrow
x{{}^{\prime}}\in\mathrm{dom}(f{{}^{\prime}})\ \&\
f{{}^{\prime}}(x{{}^{\prime}})\Vdash^{\bm{\gamma}}_{\tau}f(x)\big{)}\big{)}.$

The identity simulation
$\bm{\iota}_{\bm{C}}\colon\bm{C}\rightarrowtriangle\bm{C}$ is the pair
$\big{(}\mathrm{id}_{T},(\Vdash^{\bm{\iota}_{\bm{C}}}_{\tau})_{\tau\in
T}\big{)}$, where
$x{{}^{\prime}}\Vdash^{\bm{\iota}_{\bm{C}}}_{\tau}\Leftrightarrow
x{{}^{\prime}}=x$, for every $x{{}^{\prime}},x\in\bm{C}(\tau)$. If
$\bm{\delta}\colon\bm{D}\rightarrowtriangle\bm{E}$, the composite simulation
$\bm{\delta}\circ\bm{\gamma}\colon\bm{C}\rightarrowtriangle\bm{E}$ is the pair
$\big{(}\delta\circ\gamma,(\Vdash^{\bm{\delta}\circ\bm{\gamma}}_{\tau})_{\tau\in
T}\big{)}$, where the relation
$\Vdash^{\bm{\delta}\circ\bm{\gamma}}_{\tau}\subseteq\bm{E}\big{(}\delta(\gamma(\tau))\big{)}\times\bm{C}(\tau)$
is defined by

$z\Vdash^{\bm{\delta}\circ\bm{\gamma}}_{\tau}x\Leftrightarrow\exists_{y\in\bm{D}(\gamma(\tau))}\big{(}z\Vdash^{\bm{\delta}}_{\gamma(\tau)}y\
\&\ y\Vdash^{\bm{\gamma}}_{\tau}x\big{)}.$

We denote by CompMod the category of computability models with simulations. If
$\bm{\gamma},\bm{\delta}\colon\bm{C}\rightarrowtriangle\bm{D}$, then
$\bm{\gamma}$ is transformable to $\bm{\delta}$, in symbols
$\bm{\gamma}\preceq\bm{\delta}$, if

$\forall_{\tau\in
T}\exists_{f\in\bm{D}[\gamma(\tau),\delta(\tau)]}\forall_{x\in\bm{C}(\tau)}\forall_{x{{}^{\prime}}\in\bm{D}(\gamma(\tau))}\big{(}x{{}^{\prime}}\Vdash^{\gamma}_{\tau}x\Rightarrow
x{{}^{\prime}}\in\mathrm{dom}(f)\ \&\
f(x{{}^{\prime}})\Vdash^{\bm{\delta}}_{\tau}x\big{)}.$

The simulations $\bm{\gamma},\bm{\delta}$ are equivalent, in symbols
$\bm{\gamma}\sim\bm{\delta}$, if $\bm{\gamma}\preceq\bm{\delta}$ and
$\bm{\delta}\preceq\bm{\gamma}$. The computational models $\bm{C}$ and
$\bm{D}$ are equivalent, if there are simulations
$\bm{\gamma}\colon\bm{C}\rightarrowtriangle\bm{D}$ and
$\bm{\delta}\colon\bm{D}\rightarrowtriangle\bm{C}$ such that
$\bm{\delta}\circ\bm{\gamma}\sim\bm{\iota_{\bm{C}}}$ and
$\bm{\gamma}\circ\bm{\delta}\sim\bm{\iota_{\bm{D}}}$.

If $f{{}^{\prime}}\Vdash^{\bm{\gamma}}_{(\rho,\sigma)}f$ and
$g{{}^{\prime}}\Vdash^{\bm{\gamma}}_{(\sigma,\tau)}g$, then
$g{{}^{\prime}}\circ f{{}^{\prime}}\Vdash^{\bm{\gamma}}_{(\rho,\tau)}g\circ
f$. Similarly, if $\bm{\gamma}\preceq\bm{\gamma{{}^{\prime}}}$ and
$\bm{\delta}\preceq\bm{\delta{{}^{\prime}}}$, then
$\bm{\delta}\circ\bm{\gamma}\preceq\bm{\delta{{}^{\prime}}}\circ\bm{\gamma{{}^{\prime}}}$,
where
$\bm{\gamma},\bm{\gamma{{}^{\prime}}}\colon\bm{C}\rightarrowtriangle\bm{D}$
and
$\bm{\delta},\bm{\delta{{}^{\prime}}}\colon\bm{D}\rightarrowtriangle\bm{E}$.

######  Definition 2.3.

The category $\mathcal{A}sm(\bm{C})$ of assemblies over $\bm{C}$ has objects
triplets $(X,\tau_{X},\Vdash_{X})$, where $X$ is a set, $\tau_{X}\in T$, and
$\Vdash_{X}\subseteq\bm{C}(\tau_{X})\times X$ such that

$\forall_{x\in
X}\exists_{x{{}^{\prime}}\in\bm{C}(\tau_{X})}\big{(}x{{}^{\prime}}\Vdash_{X}x\big{)}.$

A morphism $(X,\tau_{X},\Vdash_{X})\to(Y,\tau_{Y},\Vdash_{Y})$ is a function
$f\colon X\to Y$, such that there is $\bar{f}\in\bm{C}[\tau_{X},\tau_{Y}]$
that “tracks” $f$, in symbols $\bar{f}\Vdash^{Y}_{X}f$, where

$\bar{f}\Vdash^{Y}_{X}f:\Leftrightarrow\forall_{x\in
X}\forall_{y\in\bm{C}(\tau_{X})}\big{(}y\Vdash_{X}x\Rightarrow
y\in\mathrm{dom}(\bar{f})\ \&\ \bar{f}(y)\Vdash_{Y}f(x)\big{)}.$

The forgetful functor
$\mathrm{Frg}^{\bm{C}}\colon\mathcal{A}sm(\bm{C})\to\mathrm{\mathbf{Set}}$ is
defined, as usual, by $\mathrm{Frg}^{\bm{C}}_{0}(X,\tau_{X},\Vdash_{X})=X$ and
$\mathrm{Frg}^{\bm{C}}_{1}(f)=f$.

Clearly, the forgetful functor $\mathrm{Frg}^{\bm{C}}$ is injective on arrows.

##  3 Total computability models over categories

For the rest of this paper $\mathcal{C},\mathcal{D}$ are categories,
$\mathrm{\mathbf{Cat}}$ is the category of categories, $\mathrm{\mathbf{Set}}$
is the category of sets, $[\mathcal{C},\mathrm{\mathbf{Set}}]$ is the category
of functors from $\mathcal{C}$ to $\mathrm{\mathbf{Set}}$, and
$\mathrm{\mathbf{Cat}}/\mathrm{\mathbf{Set}}$ is the slice category of
$\mathrm{\mathbf{Cat}}$ over $\mathrm{\mathbf{Set}}$. Let $C_{0},C_{1}$ be the
classes of objects and morphisms in $\mathcal{C}$, respectively. If $a,b\in
C_{0}$, let $C_{1}(a,b)$ be the class of morphisms $f$ in $C_{1}$ with
$\mathrm{dom}(f)=a$ and $\mathrm{cod}(f)=b$.

######  Definition 3.1.

Let $S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ a $($covariant$)$ functor.
The total computability model $\bm{\mathrm{CM}}^{t}(\mathcal{C};S)$ over
$\mathcal{C}$ and $S$ is the pair

$\bm{\mathrm{CM}}^{t}(\mathcal{C};S)=\big{(}\big{(}S_{0}(a)\big{)}_{a\in
C_{0}},\big{(}S[a,b]\big{)}_{(a,b)\in C_{0}\times C_{0}}\big{)},$
$S[a,b]=\\{S_{1}(f)\mid f\in C_{1}(a,b)\\}.$

If $T\colon\mathcal{C}^{\mathrm{op}}\to\mathrm{\mathbf{Set}}$ is
contravariant, the total computability model
$\bm{\mathrm{CM}}^{t}(\mathcal{C};T)$ over $\mathcal{C}$ and $T$ is defined
dually i.e., $T[a,b]=\\{T_{1}(f)\mid f\in C_{1}(b,a)\\}.$

Clearly, $\bm{\mathrm{CM}}^{t}(\mathcal{C};S)$ satisfies conditions
$(\mathrm{CM}_{1})$ and $(\mathrm{CM}_{2})$. With replacement, if
$\mathcal{C}$ is locally small, then $\bm{\mathrm{CM}}^{t}(\mathcal{C};S)$ is
locally small, and if $\mathcal{C}$ is small, then
$\bm{\mathrm{CM}}^{t}(\mathcal{C};S)$ is small. Next we describe an “external”
and an “internal” functor in the full subcategory TotCompMod of total
computability models. The corresponding proofs are omitted, as these are
similar to the included proofs of Propositions 4.3 and 4.4.

######  Proposition 3.2.

There is a functor
$\mathrm{CM}^{t}\colon\mathrm{\mathbf{Cat}}/\mathrm{\mathbf{Set}}\to\bm{\textnormal{{TotCompMod}}}$,
defined by

$\mathrm{CM}^{t}_{0}(\mathcal{C},S)=\bm{\mathrm{CM}}^{t}(\mathcal{C};S),$
$\mathrm{CM}^{t}_{1}\big{(}F\colon(\mathcal{C},S)\to(\mathcal{D},T)\big{)}:\bm{\mathrm{CM}}^{t}(C;S)\rightarrowtriangle\bm{\mathrm{CM}}^{t}(\mathcal{D};T),$
$\mathrm{CM}^{t}_{1}(F)=\bm{\gamma}_{F}=\big{(}F_{0},(\Vdash^{\bm{\gamma}_{F}}_{a})_{a\in
C_{0}}\big{)},$ $\Vdash^{\bm{\gamma}_{F}}_{a}\subseteq T_{0}(F_{0}(a))\times
S_{0}(a),\ \ \ \ y\Vdash^{\bm{\gamma}_{F}}_{a}x\Leftrightarrow y=x.$

######  Proposition 3.3.

There is a faithful functor
${}^{t}\mathrm{CM}^{\mathcal{C}}\colon[\mathcal{C},\mathrm{\mathbf{Set}}]\to\bm{\textnormal{{TotCompMod}}}$,
defined by

${}^{t}\mathrm{CM}_{0}^{\mathcal{C}}(S)=\bm{\mathrm{CM}}^{t}(\mathcal{C};S),$
${}^{t}\mathrm{CM}_{1}^{\mathcal{C}}\big{(}\eta\colon S\Rightarrow
T\big{)}:\bm{\mathrm{CM}}^{t}(C;S)\rightarrowtriangle\bm{\mathrm{CM}}^{t}(\mathcal{D};T),$
${}^{t}\mathrm{CM}_{1}^{\mathcal{C}}(\eta)=\bm{\gamma}_{\eta}=\big{(}\mathrm{id}_{C_{0}},(\Vdash^{\bm{\gamma}_{\eta}}_{a})_{a\in
C_{0}}\big{)},$ $\Vdash^{\bm{\gamma}_{\eta}}_{a}\subseteq T_{0}(a)\times
S_{0}(a),\ \ \ \ y\Vdash^{\bm{\gamma}_{\eta}}_{a}x\Leftrightarrow
y=\eta_{a}(x).$

By Definition 2.3 the category
$\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{t}(\mathcal{C};S)\big{)}$ of assemblies
of the total computability model $\bm{\mathrm{CM}}^{t}(\mathcal{C};S)$ has
objects triplets $(X,a_{X},\Vdash_{X})$, where $X$ is a set, $a_{X}\in C_{0}$,
and $\Vdash_{X}\subseteq S_{0}(a_{X})\times X$ such that $\forall_{x\in
X}\exists_{x{{}^{\prime}}\in
S_{0}(a_{X})}\big{(}x{{}^{\prime}}\Vdash_{X}x\big{)}$. A morphism
$(X,a_{X},\Vdash_{X})\to(Y,a_{Y},\Vdash_{Y})$ is a function $f\colon X\to Y$,
such that there is $\bar{f}\in S[a_{X},a_{Y}]$ i.e., $g\colon a_{X}\to
a_{Y}\in C_{1}$ and $\overline{f}=S_{1}(g)\colon S_{0}(a_{X})\to S_{0}(a_{Y})$
tracks $f$, where in this case $\bar{f}\Vdash^{Y}_{X}f$ if and only if
$\forall_{x\in X}\forall_{y\in
S_{0}(a_{X})}\big{(}y\Vdash_{X}x\Rightarrow[S_{1}(g)](y)\Vdash_{Y}f(x)\big{)}$.

######  Proposition 3.4.

Let
$F^{t}\colon\mathcal{C}\to\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{t}(\mathcal{C};S)\big{)}$
be defined by

$F^{t}_{0}(a)=\big{(}S_{0}(a),a,\Vdash_{S_{0}(a)}\big{)};\ \ \ \ a\in C_{0},$
$y\Vdash_{S_{0}(a)}x\Leftrightarrow y=x;\ \ \ \ y,x\in S_{0}(a),$
$F^{t}_{1}(f\colon a\to
b)\colon\big{(}S_{0}(a),a\Vdash_{S_{0}(a)}\big{)}\to\big{(}S_{0}(b),b\Vdash_{S_{0}(b)}\big{)},\
\ \ \ F^{t}_{1}(f)=S_{1}(f).$

(i) $F^{t}$ is a full functor, injective on objects.
(ii) If $S$ is injective on arrows, then $F^{t}$ is an embedding.

###### Proof.

(i) Clearly, $S_{1}(f)$ tracks itself, and hence $S_{1}(f)$ is a morphism in
$\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{t}(\mathcal{C};S)\big{)}$. Clearly,
$F^{t}$ is a functor, injective on objects. Let
$g\colon\big{(}S_{0}(a),a\Vdash_{S_{0}(a)}\big{)}\to\big{(}S_{0}(b),b\Vdash_{S_{0}(b)}\big{)}$
a morphism in $\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{t}(\mathcal{C};S)\big{)}$
i.e., a function $g\colon S_{0}(a)\to S_{0}(b)$ such that there is
$\overline{g}=S_{1}(h)$, for some $h\colon a\to b$ in $C_{1}$, with
$S_{1}(h)\Vdash^{S_{0}(b)}_{S_{0}(a)}g$ i.e., $\forall_{x\in
S_{0}(a)}\forall_{y\in
S_{0}(a)}\big{(}y=x\Rightarrow[S_{1}(h)](y)=g(x)\big{)}$, hence $\forall_{x\in
S_{0}(a)}\big{(}[S_{1}(h)](x)=g(x)\big{)}$ i.e., $F_{1}^{+}(h)=S_{1}(h)=g$ and
$F^{t}$ is full.
(ii) If $S$ is injective on arrows, then $F^{t}$ is also injective on arrows,
hence faithful, and since it is also full and injective on objects, it is an
embedding. ∎

Let $\bm{C}$ be a small, and $\mathrm{Frg}^{\bm{C}}$ the forgetful functor on
$\mathcal{A}sm(\bm{C})$. If $\overline{X}=(X,\tau_{X},\Vdash_{X})$ and
$\overline{Y}=(Y,\tau_{Y},\Vdash_{Y})$, the computability model
$\bm{\mathrm{CM}}^{t}(\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}})$ over
$\mathcal{A}sm(\bm{C})$ and $\mathrm{Frg}^{\bm{C}}$ is the pair

$\bm{\mathrm{CM}}^{t}(\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}})=\bigg{(}\big{(}X\big{)}_{\overline{X}\in\mathcal{A}sm(\bm{C})_{0}},\big{(}\mathrm{Frg}^{\bm{C}}[\overline{X},\overline{Y}]\big{)}_{\overline{X},\overline{Y}\in\mathcal{A}sm(\bm{C})_{0}}\bigg{)},$
$\mathrm{Frg}^{\bm{C}}[\overline{X},\overline{Y}]=\big{\\{}f\colon X\to
Y\mid\exists_{\overline{f}\colon C(\tau_{x})\rightharpoondown
C(\tau_{Y})}(\overline{f}\Vdash^{Y}_{X}f)\big{\\}}.$

I.e., from a small, and, in general, partial computability model $\bm{C}$, we
get the locally small and total computability model
$\bm{\mathrm{CM}}^{t}\big{(}\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}}\big{)}$.
Due to lack of space, the proof of the following proposition is omitted.

######  Proposition 3.5.

(i) If $\bm{C}$ is small, there is a simulation

$\bm{\delta}^{t}\colon\bm{\mathrm{CM}}^{t}\big{(}\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}}\big{)}\rightarrowtriangle\bm{C},$

where
$\bm{\delta}^{t}=\big{(}\delta^{t},\big{(}\Vdash^{\bm{\delta}^{t}}_{\overline{X}}\big{)}_{\overline{X}\in\mathcal{A}sm(\bm{C})_{0}}\big{)},$
with $\delta^{t}\colon\mathcal{A}sm(\bm{C})_{0}\to T$, defined by
$\delta^{t}(\overline{X})=\tau_{X}$, and
$\Vdash^{\bm{\delta}^{t}}_{\overline{X}}\subseteq C(\tau_{X})\times X$ defined
by $\Vdash^{\bm{\delta}^{t}}_{\overline{X}}\ =\ \Vdash_{X}$.
(ii) If $\bm{C}$ is total and small, there is a simulation

$\bm{\gamma}^{t}\colon\bm{C}\rightarrowtriangle\bm{\mathrm{CM}}^{t}\big{(}\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}}\big{)},$

where
$\bm{\gamma}^{t}=\big{(}\gamma^{t},\big{(}\Vdash^{\bm{\gamma}^{t}}_{\tau}\big{)}_{\tau\in
T}\big{)},$ with $\gamma^{t}\colon T\to\mathcal{A}sm(\bm{C})_{0}$, defined by
$\gamma^{t}(\tau)=(C(\tau),\tau,\Vdash_{C(\tau)})$ and
$\Vdash_{C(\tau)}\subseteq C(\tau)\times C(\tau)$ is the equality on
$C(\tau)$, and $\Vdash^{\bm{\gamma}^{t}}_{\tau}\subseteq C(\tau)\times
C(\tau)$ is also the equality on $C(\tau)$.
(iii) If $\bm{C}$ is total and small, then
$\bm{\delta}^{t}\circ\bm{\gamma}^{t}=\bm{\iota}_{\bm{C}}$.

If $\bm{C}$ is total and small, then
$\bm{\gamma}^{t}\circ\bm{\delta}^{t}\sim\bm{\iota}_{\bm{\mathrm{CM}}^{t}(\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}})}$.
Similarly, if
$\mathcal{A}^{t}=\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{t}(\mathcal{C};S)\big{)}$
and $\mathrm{Frg}^{\mathcal{A}^{t}}$ is the forgetful functor on
$\mathcal{A}^{t}$, the computability models ${\mathrm{CM}}^{t}(\mathcal{C};S)$
and
${\mathrm{CM}}^{t}\big{(}\mathcal{A}^{t},\mathrm{Frg}^{\mathcal{A}^{t}}\big{)}$
are expected to be equivalent.

##  4 Computability models over categories with pullbacks

######  Definition 4.1.

Let $C_{1}(a\hookrightarrow b)$ be the subclass of monos in $C_{1}(a,b)$, let
$C_{1}(a\hookrightarrow)$ be the subclass of monos in $C_{1}$ with domain $a$,
and $C_{1}(\hookrightarrow a)$ the subclass of monos in $C_{1}$ with codomain
$a$. If $\mathcal{C}$ has pullbacks, a partial morphism444As here we don’t
define categories of partial morphisms, we avoid the equivalence relation
between them. A partial function between sets is a partial morphism in
$\mathrm{\mathbf{Set}}$. This is in accordance with the notion of partial
function between sets in Bishop set theory see [9] and [10]. $(i,f)\colon
a\rightharpoondown b$ is a pair $(i,f)$, where $i\in C_{1}(\hookrightarrow a)$
and $f\in C_{1}(\mathrm{dom}(i),b)$

$s$$a$$b$$S_{0}(s)$$S_{0}(a)$$S_{0}(b)$.$i$$f\ $$S_{1}(i)$$S_{1}(f)\
$$(i,f)$$S_{1}(i,f)$

If $(j,g)\colon b\rightharpoondown c$, where $j\colon t\hookrightarrow b$ and
$g\colon t\to c$, their composition is the partial morphism $(i\circ
i{{}^{\prime}},g\circ f{{}^{\prime}})\colon a\rightharpoondown c$, where
$i{{}^{\prime}}\colon s\times_{b}t\hookrightarrow s$ and $f{{}^{\prime}}\colon
s\times_{b}t\to t$ is determined by the corresponding pullback diagram

$s$$a$$b$$s\times_{b}t$$t\ \ $$c$$i$$\ f$$g\ $$f{{}^{\prime}}\
$$j$$i{{}^{\prime}}$$(i,f)$$(j,g)$$(i\circ i{{}^{\prime}},g\circ
f{{}^{\prime}})$

If $S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ is a functor that preserves
pullbacks, hence monos, and if $(i,f)\colon a\rightharpoondown b$, let
$S_{1}(i,f)=(S_{1}(i),S_{1}(f))\colon S_{0}(a)\rightharpoondown S_{0}(b)$. Let

$\mathrm{dom}(S_{1}(i,f))=\big{\\{}[S_{1}(i)](x)\mid x\in S_{0}(s)\big{\\}},$
$[S_{1}(i,f)](y)=[S_{1}(f)](x);\ \ \ \
y=[S_{1}(i)](x)\in\mathrm{dom}(S_{1}(i,f)).$

######  Definition 4.2.

If $\mathcal{C}$ has pullbacks, and
$S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ preserves pullbacks, the
$($partial$)$ computability model $\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$ over
$\mathcal{C}$ and $S$ is the pair

$\bm{\mathrm{CM}}^{p}(\mathcal{C};S)=\big{(}\big{(}S_{0}(a)\big{)}_{a\in
C_{0}},\big{(}S[a,b]\big{)}_{(a,b)\in C_{0}\times C_{0}}\big{)},$
$S[a,b]=\\{S_{1}(i,f)\mid i\in C_{1}(\hookrightarrow a)\ \&\ f\in
C_{1}(\mathrm{dom}(i),b)\\}.$

If $\mathcal{D}$ has pushouts, and
$T\colon\mathcal{C}^{\mathrm{op}}\to\mathrm{\mathbf{Set}}$ sends pushouts to
pullbacks, the $($partial$)$ computability model
$\bm{\mathrm{CM}}^{p}(\mathcal{D};T)$ over $\mathcal{D}$ and $T$ is defined
dually i.e., $T[a,b]=\\{T_{1}(i,f)\mid i\in C_{1}(a\hookrightarrow)\ \&\ f\in
C_{1}(b,\mathrm{cod}(i))\\}$.

The fact that $\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$ is a computability model
depends crucially on the preservation of pullbacks by $S$. If $\mathcal{C}$ is
small, then $\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$ is small, but if
$\mathcal{C}$ is locally small, $\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$ need not
be locally small. Next we describe the corresponding “external” and “internal”
functors.

######  Proposition 4.3.

Let $\mathrm{\mathbf{Cat}}^{\mathrm{pull}}$ be the category of categories with
pullbacks with morphisms the functors preserving pullbacks, and let
$\mathrm{\mathbf{Cat}}^{\mathrm{pull}}/\mathrm{\mathbf{Set}}$ the slice
category over $\mathrm{\mathbf{Set}}$. If
$\mathrm{\mathbf{Cat}}^{\mathrm{pull}}_{\hookrightarrow}{/}\mathrm{\mathbf{Set}}$
is the subcategory of
$\mathrm{\mathbf{Cat}}^{\mathrm{pull}}/\mathrm{\mathbf{Set}}$ with morphisms
the morphisms in $\mathrm{\mathbf{Cat}}^{\mathrm{pull}}/\mathrm{\mathbf{Set}}$
that preserve monos555If $F\colon\mathcal{C}\to\mathcal{D}$ such that $T\circ
F=S$, where $S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ and
$T\colon\mathcal{D}\to\mathrm{\mathbf{Set}}$, we cannot show, in general, that
$F$ preserves monos. It does, if, e.g., $T$ is injective on arrows., then

$\mathrm{CM}^{s}\colon\mathrm{\mathbf{Cat}}^{\mathrm{pull}}_{\hookrightarrow}/\mathrm{\mathbf{Set}}\to\bm{\textnormal{{CompMod}}}$

is a functor, where
$\mathrm{CM}^{p}_{0}(\mathcal{C},S)=\bm{\mathrm{CM}}^{p}(\mathcal{C};S),$ and
$\mathrm{CM}^{p}_{1}(F):\bm{\mathrm{CM}}^{p}(C;S)\rightarrowtriangle\bm{\mathrm{CM}}^{p}(\mathcal{D};T)$
is defined as the simulation $\mathrm{CM}_{1}^{t}(F)=\bm{\gamma}_{F}$ in
Proposition 3.2.

###### Proof.

First we show that
$\bm{\gamma}_{F}=\big{(}F_{0},(\Vdash^{\bm{\gamma}_{F}}_{a})_{a\in
C_{0}}\big{)},$ where $\Vdash^{\bm{\gamma}_{F}}_{a}\subseteq
T_{0}(F_{0}(a))\times S_{0}(a)$ is defined by
$y\Vdash^{\bm{\gamma}_{F}}_{a}x\Leftrightarrow y=x$, is a simulation
$\bm{\mathrm{CM}}^{p}(C;S)\rightarrowtriangle\bm{\mathrm{CM}}^{p}(\mathcal{D};T)$.
To show $(\textnormal{{Siml}}_{1})$, if $a\in C_{0}$ and $x\in S_{0}(a)$, then
$S_{0}(a)=T_{0}(F_{0}(a))\ni x=\Vdash^{\bm{\gamma}_{F}}_{a}x$. To show
$(\textnormal{{Siml}}_{2})$, if $a,b\in C_{0}$ and $(S_{1}(i),S_{1}(f))\colon
S_{0}(a)\rightharpoondown S_{0}(b)\in S[a,b]$, for some $s\in C_{0}$, we find
an element of $T[F_{0}(a),F_{0}(b)]$ that tracks $(S_{1}(i),S_{1}(f))$. Let
$(T_{1}\circ F_{1})(i,f)=\big{(}T_{1}(F_{1}(i)),T_{1}(F_{1}(f))\big{)}\colon
T_{0}(F_{0}(a))\rightharpoondown T_{0}(F_{0}(b))$, as $F$ preserves monos. If
$x\in\mathrm{dom}(S_{1}(i,f))$ i.e., there is a (unique) $x{{}^{\prime}}\in
S_{0}(s)$ with $x=[S_{1}(i)](x{{}^{\prime}})$, then if $y\in
T_{0}(F_{0}(a)=S_{0}(a)$ such that $y\Vdash^{\bm{\gamma}_{F}}_{a}x$, then
$y=x$. We show that $x\in\mathrm{dom}\big{(}(T_{1}\circ F_{1})(i,f)\big{)}$,
which by Definition 4.1 means that
$x=[T_{1}(F_{1}(i))](x{{}^{\prime\prime}})$, for some $x{{}^{\prime}}\in
T_{0}(F_{0}(s))=S_{0}(s)$. Clearly, $x{{}^{\prime\prime}}=x{{}^{\prime}}$
works. By Definition 4.1 the equality obtained unfolding the relation
$[(T_{1}\circ F_{1}(i,f)](x)\Vdash^{\bm{\gamma}_{F}}_{b}[S_{1}(i,ff)](x)$
follows immediately. Clearly,
$\mathrm{CM}_{1}^{p}(1_{(\mathcal{C},S)})=\bm{\iota}_{\bm{\mathrm{CM}}^{p}(\mathcal{C};S)}$,
and if $F\colon\mathcal{C}\to\mathcal{D}$,
$G\colon\mathcal{D}\Rightarrow\mathcal{E}$, then $\bm{\gamma}_{G\circ
F}=\bm{\gamma}_{G}\circ\bm{\gamma}_{F}$. ∎

######  Proposition 4.4.

If $\mathcal{C}$ has pullbacks and
$[\mathcal{C},\mathrm{\mathbf{Set}}]^{\mathrm{pull}}$ is the category of
functors from $\mathcal{C}$ to $\mathrm{\mathbf{Set}}$ preserving pullbacks,
then

${}^{p}\mathrm{CM}^{\mathcal{C}}\colon[\mathcal{C},\mathrm{\mathbf{Set}}]^{\mathrm{pull}}\to\bm{\textnormal{{CompMod}}}$

is a faithful functor, where
${}^{p}\mathrm{CM}_{0}^{\mathcal{C}}(S)=\bm{\mathrm{CM}}^{p}(\mathcal{C};S),$
and the simulation
${}^{p}\mathrm{CM}_{1}^{\mathcal{C}}(\eta):\bm{\mathrm{CM}}^{p}(C;S)\rightarrowtriangle\bm{\mathrm{CM}}^{p}(\mathcal{C};T)$
is defined as ${}^{t}\mathrm{CM}_{1}^{\mathcal{C}}(\eta)=\bm{\gamma}_{\eta}$
in Proposition 3.3.

###### Proof.

First we show that
$\bm{\gamma}_{\eta}=\big{(}\mathrm{id}_{C_{0}},(\Vdash^{\bm{\gamma}_{\eta}}_{a})_{a\in
C_{0}}\big{)},$ where $\Vdash^{\bm{\gamma}_{\eta}}_{a}\subseteq T_{0}(a)\times
S_{0}(a)$ is defined by $y\Vdash^{\bm{\gamma}_{\eta}}_{a}x\Leftrightarrow
y=\eta_{a}(x)$, is a simulation
$\bm{\mathrm{CM}}^{p}(C;S)\rightarrowtriangle\bm{\mathrm{CM}}^{p}(\mathcal{C};T)$.
To show $(\textnormal{{Siml}}_{1})$, if $a\in C_{0}$ and $x\in S_{0}(a)$, then
$T_{0}(a)\ni y=\eta_{a}(x)\Vdash^{\bm{\gamma}_{\eta}}_{a}x$. To show
$(\textnormal{{Siml}}_{2})$, if $a,b\in C_{0}$ and $(S_{1}(i),S_{1}(f))\colon
S_{0}(a)\rightharpoondown S_{0}(b)\in S[a,b]$, for some $s\in C_{0}$, we find
an element of $T[a,b]$ that tracks $(S_{1}(i),S_{1}(f))$. Let
$T_{1}(i,f)=\big{(}T_{1}(i),T_{1}(f)\big{)}\colon T_{0}(a)\rightharpoondown
T_{0}(b)$. We show that

$\big{(}T_{1}(i),T_{1}(f)\big{)}\Vdash^{\bm{\gamma}_{\eta}}_{(a,b)}\big{(}S_{1}(i),S_{1}(f)\big{)}.$

Let $x\in S_{0}(a)$ such that $x\in\mathrm{dom}(S_{1}(i,f))$ i.e., there is a
(unique) $x{{}^{\prime}}\in S_{0}(s)$ with $x=[S_{1}(i)](x{{}^{\prime}})$. Let
$y\in T_{0}(a)$ such that $y\Vdash^{\bm{\gamma}_{\eta}}_{a}x\Leftrightarrow
y=\eta_{a}(x)$. First we show that $y\in\mathrm{dom}(T_{1}(i,f))$ i.e.,
$y=[T_{1}(i)](y{{}^{\prime}})$, for some $y{{}^{\prime}}\in T_{0}(s)$. If
$y{{}^{\prime}}=\eta_{s}(x{{}^{\prime}})$, then by the commutativity of the
following left diagram we have that

$S_{0}(s)$$S_{0}(a)$$T_{0}(s)$$T_{0}(a)$$S_{0}(s)$$T_{0}(s)$$S_{0}(b)$$T_{0}(b)$$\eta_{a}$$S_{1}(i)$$\eta_{s}$$T_{1}(i)$
$\eta_{b}$$S_{1}(f)$$\eta_{s}$$T_{1}(f)$
$[T_{1}(i)](y{{}^{\prime}})=[T_{1}(i)](\eta_{s}(x{{}^{\prime}}))=\eta_{a}\big{(}[S_{1}(i)](x{{}^{\prime}})\big{)}=\eta_{a}(x)=y.$

Next we show that
$[T_{1}(i,f)](y)\Vdash^{\bm{\gamma}_{\eta}}_{b}[S_{1}(i,f)](x)$ i.e.,
$[T_{1}(f)](y{{}^{\prime}})\Vdash^{\bm{\gamma}_{\eta}}_{b}[S_{1}(f)](x{{}^{\prime}})$.
By the commutativity of the above right diagram we have that

$\eta_{b}\big{(}[S_{1}(f)](x{{}^{\prime}})\big{)}=[T_{1}(f)]\big{(}\eta_{s}(x{{}^{\prime}})\big{)}=[T_{1}(f)](y{{}^{\prime}}).$

It is straightforward to show that ${}^{p}\mathrm{CM}_{1}(1_{S}\colon
S\Rightarrow S)=\bm{\iota}_{\bm{\mathrm{CM}}^{p}(\mathcal{C};S)}$, and if
$\tau\colon S\Rightarrow T$, $\eta\colon T\Rightarrow U$, then
$\bm{\gamma}_{\eta\circ\tau}=\bm{\gamma}_{\eta}\circ\bm{\gamma}_{\tau}$. To
show that ${}^{p}\mathrm{CM}^{\mathcal{C}}$ is faithful, let
$\eta,\theta\colon S\Rightarrow T$ such that
$\bm{\gamma}_{\eta}=\bm{\gamma}_{\theta}$ i.e.,
$\Vdash^{\bm{\gamma}_{\eta}}_{a}=\Vdash^{\bm{\gamma}_{\theta}}_{a}$, for every
$a\in C_{0}$. If we fix some $a\in C_{0}$, let $y\in T_{0}(a)$ and $x\in
S_{0}(a)$. Then $y\Vdash^{\bm{\gamma}_{\eta}}_{a}x\Leftrightarrow
y\Vdash^{\bm{\gamma}_{\theta}}_{a}x$ i.e., $y=\eta_{a}(x)\Leftrightarrow
y=\theta_{a}(x)$, hence $\eta_{a}=\theta_{a}$, and since $a\in C_{0}$ is
arbitrary, we conclude that $\eta=\theta$. ∎

The category $\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{p}(\mathcal{C};S)\big{)}$
of assemblies of $\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$ has objects the
standard triplets $(X,a_{X},\Vdash_{X})$, and a morphism
$(X,a_{X},\Vdash_{X})\to(Y,a_{Y},\Vdash_{Y})$ is a function $f\colon X\to Y$,
such that there is $\bar{f}\in S[a_{X},a_{Y}]$ i.e., there is $s\in C_{0}$ and
$i\colon s\hookrightarrow a_{X},f\colon s\to a_{Y}$ such that
$\bar{f}=\big{(}S_{1}(i),S_{1}(f)\big{)}$ and $\bar{f}\Vdash^{Y}_{X}f$. One
can define a functor
$F^{p}\colon\mathcal{C}\to\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{p}(\mathcal{C};S)\big{)}$,
as in Proposition 3.4. In this case the function $F^{p}_{1}(f\colon a\to
b)=S_{1}(f)$ is tracked by the partial function
$\overline{S_{1}(f)}=\big{(}S_{1}(1_{a}),S_{1}(f)\big{)}$. This functor
$F^{p}$ though, is not, in general, full. If
$\mathcal{A}^{p}=\mathcal{A}sm\big{(}\bm{\mathrm{CM}}^{p}(\mathcal{C};S)\big{)}$
and $\mathrm{Frg}^{\mathcal{A}^{p}}$ is the corresponding forgetful functor,
one can study the total computability model
$\bm{\mathrm{CM}}^{t}\big{(}\mathcal{A}^{p};\mathrm{Frg}^{\mathcal{A}^{p}}\big{)}$
and the partial computability model
$\bm{\mathrm{CM}}^{p}\big{(}\mathcal{A}^{p};\mathrm{Frg}^{\mathcal{A}^{p}}\big{)}$,
and relate the latter to $\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$.

##  5 Categories with a base of computability

######  Definition 5.1.

A base of computability666Having defined this notion independently, it was a
nice surprise, and a posteriori not that surprising, to find out that a base
of computability is Rosolini’s notion of dominion, defined in [11], section
2.1. for $\mathcal{C}$ is a family $B=(B(a))_{a\in C_{0}}$, where $B(a)$ is a
subclass of the class $C_{1}(\hookrightarrow a)$ of monos with codomain $a$,
for every $a\in C_{0}$, such that the following conditions are satisfied:
$(\bm{\textnormal{{Base}}}_{1})$ For every $a\in C_{0}$, we have that
$1_{a}\in B(a)$.
$(\bm{\textnormal{{Base}}}_{2})$ For every $i\colon s\hookrightarrow a\in
B(a)$, for every $b\in C_{0}$, for every $f\colon s\to b$ and for every
$j\colon t\hookrightarrow b\in B(b)$ a pullback $s\times_{b}t$ exists and
$i\circ i{{}^{\prime}}\colon s\times_{b}t\hookrightarrow a\in B(a)$

$s\times_{b}t$$t$$s$$b$.$a$$j$$f{{}^{\prime}}$$i{{}^{\prime}}$$f$ $i$

A cobase of computability for $\mathcal{C}$ is a family $C=(C(a))_{a\in
C_{0}}$, where $C(a)$ is a subclass of the class $C_{1}(a\hookrightarrow)$ of
monos with domain $a$ such that $(\bm{\textnormal{{Base}}}_{1})$ and the dual
of $(\bm{\textnormal{{Base}}}_{2})$ are satisfied.

Notice that the interesction of two bases need not be a base, if different
pullbacks are considered in them.

######  Proposition 5.2.

(i) If $I(a)=\\{1_{a}\colon a\hookrightarrow a\\}$, for every $a\in C_{0}$,
then $I=(I(a))_{a\in C_{0}}$ is a base and a cobase of computability for
$\mathcal{C}$.
(ii) If $\mathrm{Iso}(a)=\\{i\in C_{1}(\hookrightarrow a)\mid i\ \mbox{is an
iso}\\}$, for every $a\in C_{0}$, then $\mathrm{Iso}=(\mathrm{Iso}(a))_{a\in
C_{0}}$ is a base of computability for $\mathcal{C}$.
(iii) If $\mathcal{C}$ has pullbacks, and $B(a)=C_{1}(\hookrightarrow a)$, for
every $a\in C_{0}$, then $B^{\hookrightarrow}=(B(a))_{a\in C_{0}}$ is a base
of computability for $\mathcal{C}$.
(iv) If $\mathcal{C}$ has pushouts, and $C(a)=C_{1}(a\hookrightarrow)$, for
every $a\in C_{0}$, then $C^{\hookleftarrow}=(C(a))_{a\in C_{0}}$ is a cobase
of computability for $\mathcal{C}$.

###### Proof.

(i) We fix $a\in C_{0}$, $b\in C_{0}$ and $f\colon a\to b$. The only element
of $I(b)$ is $1_{b}\colon b\hookrightarrow b$, and the following is a pullback
square with $1_{a}\circ 1_{a}=1_{a}\in I(a)$

$a$$b$$a$$b$.$a$$1_{b}$$f$$1_{a}$$f$$1_{a}$

(ii) In this case $a=s\times_{b}t$. Cases (iii) and (iv) are trivial. ∎

######  Definition 5.3.

If $B=(B(a))_{a\in C_{0}}$ is a base of computability for $\mathcal{C}$ and
$S\colon\mathcal{C}\to\mathrm{\mathbf{Set}}$ preserves pullbacks, the
computability model $\bm{\mathrm{CM}}^{B}(\mathcal{C};S)$ over $\mathcal{C}$
and $S$ with respect to $B$ is the pair

$\bm{\mathrm{CM}}^{B}(\mathcal{C};S)=\big{(}\big{(}S_{0}(a)\big{)}_{a\in
C_{0}},\big{(}S[a,b]\big{)}_{(a,b)\in C_{0}\times C_{0}}\big{)},$
$S[a,b]=\\{S_{1}(i,f)\mid i\in B(a)\ \&\ f\in C_{1}(\mathrm{dom}(i),b)\\}.$

If $C=(C(a))_{a\in C_{0}}$ is a cobase of computability for $\mathcal{C}$ and
$T\colon\mathcal{C}^{\mathrm{op}}\to\mathrm{\mathbf{Set}}$ sends pushouts to
pullbacks, the computability model $\bm{\mathrm{CM}}^{B}(\mathcal{C};T)$ over
$\mathcal{D}$ and $T$ is defined dually i.e., $T[a,b]=\\{T_{1}(i,f)\mid i\in
C(a)\ \&\ f\in C_{1}(b,\mathrm{cod}(i))\\}$.

If $B(a)$ is a set, for every $a\in C_{0}$, and $\mathcal{C}$ is locally
small, then $\bm{\mathrm{CM}}^{B}(\mathcal{C};S)$ is locally small. By
Proposition 5.2, if $B=I$, then
$\bm{\mathrm{CM}}^{I}(\mathcal{C};S)=\bm{\mathrm{CM}}^{t}(\mathcal{C};S)$.
Actually, any functor preserves the pullbacks related to the base $I$. If
$B=B^{\hookrightarrow}$, then
$\bm{\mathrm{CM}}^{B^{\hookrightarrow}}(\mathcal{C};S)=\bm{\mathrm{CM}}^{p}(\mathcal{C};S)$.
Consequently, the results shown in the previous two sections can be seen as
special cases of the corresponding results for categories with a base of
computability.

##  6 Concluding comments

The notions and results presented here form the very first steps in the study
of canonical computability models over categories and
$\mathrm{\mathbf{Set}}$-valued functors on them. Next we include some topics
that could be investigated in the future.

To study more examples of computability models over categories with a given
base of computability coming from Rosolini’s theory of dominions. To study the
computability models over a locally small category with pullbacks and the
pullback preserving, covariant representable functors $\mathrm{Hom}(a,-)$. To
find interesting conditions on a computability model $\bm{C}$ such that its
category of assemblies $\mathcal{A}sm(\bm{C})$ has a base of computability
$B$, and relate $\bm{C}$ with
$\bm{\mathrm{CM}}^{B}\big{(}\mathcal{A}sm(\bm{C});\mathrm{Frg}^{\bm{C}}\big{)}$.
To study the computability models over cartesian closed categories, as these
are computability models over a type world, in the sense of [8], p. 66. To
relate the various constructions in categories to their canonical
computability models. To examine the impact of a Grothendieck topology on
$\mathcal{C}$ to some computability model over $\mathcal{C}$. To generalise
the notions of a computability model and of a simulation between them so that
the datatypes $C(\tau)$ are not necesserily in $\mathrm{\mathbf{Set}}$, but
elements of some topos $\mathcal{S}$, and then study
$\mathcal{S}$-computability models over a category $\mathcal{C}$ and an
$\mathcal{S}$-valued functor on $\mathcal{C}$. But most importantly, to relate
the theory of $\mathrm{\mathbf{Set}}$-computability models over categories,
introduced here, to concrete case studies from the theory of Higher-Order
Computability, as this is developed in [8].

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