We consider two differential categories π² {\mathcal{W}} π² β² {\mathcal{W}}^{\prime} ( E i , Ο i ) (E_{i},\tau_{i}) ( E i β² , Ο i β² ) (E^{\prime}_{i},\tau^{\prime}_{i}) π° {\mathcal{U}} i β { 1 , 2 } i\in\{1,2\} Ο : E 2 β E 1 β E 1 β E 2 \sigma:E_{2}E_{1}\to E_{1}E_{2} Ο β² : E 2 β² β E 1 β² β E 1 β² β E 2 β² \sigma^{\prime}:E^{\prime}_{2}E^{\prime}_{1}\to E^{\prime}_{1}E^{\prime}_{2} 4.3.1 Ο β² \sigma^{\prime}
Let Ξ¦ : π² β π² β² \Phi:{\mathcal{W}}\to{\mathcal{W}}^{\prime} Ο i : Ξ¦ β E i β βΌ E i β² β Ξ¦ \varphi_{i}:\Phi E_{i}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}_{i}\Phi ( Ξ¦ , Ο i ) (\Phi,\varphi_{i}) 2 2 i β { 1 , 2 } i\in\{1,2\}
(4.3.4) ( E 1 β² β Ο 2 ) β ( Ο 1 β E 2 ) β ( Ξ¦ β Ο ) = ( Ο β² β Ξ¦ ) β ( E 2 β² β Ο 1 ) β ( Ο 2 β E 1 ) : Ξ¦ β E 2 β E 1 β E 1 β² β E 2 β² β Ξ¦ . (E^{\prime}_{1}\varphi_{2})\circ(\varphi_{1}E_{2})\circ(\Phi\sigma)=(\sigma^{\prime}\Phi)\circ(E^{\prime}_{2}\varphi_{1})\circ(\varphi_{2}E_{1}):\Phi E_{2}E_{1}\to E^{\prime}_{1}E^{\prime}_{2}\Phi.
0P5Z
Proposition 4.3.9 . There is a differential functor Ξ β Ξ¦ : Ξ Ο β π² β Ξ Ο β² β π² β² \Delta\Phi:\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime} ( m , Ο ) β¦ ( Ξ¦ β‘ ( m ) , Ο 1 β ( m ) β Ξ¦ β‘ ( Ο ) β Ο 2 β ( m ) β 1 ) (m,\pi)\mapsto(\Phi(m),\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}(m)^{-1})
There is a closed isomorphism of functors
Ο = ( Ο 2 Ο 1 ) : Ξ β Ξ¦ β E β βΌ E β² β Ξ β Ξ¦ . \varphi=\left(\begin{matrix}\varphi_{2}\\
&\varphi_{1}\end{matrix}\right):\Delta\Phi E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\Delta\Phi.
If Ο \sigma Ο β² \sigma^{\prime} ( Ξ β Ξ¦ , Ο ) (\Delta\Phi,\varphi) 2 2 Ξ Ο β π² β Ξ Ο β² β π² β² \Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}
0P60
Proof. Let ( m , Ο ) (m,\pi) Ξ Ο β π² \Delta_{\sigma}{\mathcal{W}} Ο β² = Ο 1 β ( m ) β Ξ¦ β‘ ( Ο ) β Ο 2 β 1 β ( m ) \pi^{\prime}=\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}^{-1}(m) Z β Hom π² β² Β― i β‘ ( E 2 β² β Ξ¦ β ( m ) , E 1 β² β Ξ¦ β ( m ) ) Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}^{\prime}}^{i}}(E^{\prime}_{2}\Phi(m),E^{\prime}_{1}\Phi(m))
We have
( E 1 β² β Ο β² ) β ( Ο β² β Ξ¦ β ( m ) ) β ( E 2 β² β Ο β² ) = (E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime})=
= ( E 1 β² β Ο 1 β ( m ) ) β ( E 1 β² β Ξ¦ β Ο ) β ( E 1 β² β Ο 2 β 1 β ( m ) ) β ( Ο β² β Ξ¦ β ( m ) ) β ( E 2 β² β Ο 1 β ( m ) ) β ( E 2 β² β Ξ¦ β Ο ) β ( E 2 β² β Ο 2 β 1 β ( m ) ) \displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(E^{\prime}_{1}\Phi\pi)\circ(E^{\prime}_{1}\varphi_{2}^{-1}(m))\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\varphi_{1}(m))\circ(E^{\prime}_{2}\Phi\pi)\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))
= ( E 1 β² β Ο 1 β ( m ) ) β ( E 1 β² β Ξ¦ β Ο ) β ( Ο 1 β ( E 2 β ( m ) ) ) β ( Ξ¦ β Ο β ( m ) ) β ( Ο 2 β 1 β ( E 1 β ( m ) ) ) β ( E 2 β² β Ξ¦ β Ο ) β ( E 2 β² β Ο 2 β 1 β ( m ) ) \displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(E^{\prime}_{1}\Phi\pi)\circ(\varphi_{1}(E_{2}(m)))\circ(\Phi\sigma(m))\circ(\varphi_{2}^{-1}(E_{1}(m)))\circ(E^{\prime}_{2}\Phi\pi)\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))
= ( E 1 β² β Ο 1 β ( m ) ) β ( Ο 1 β ( E 1 β ( m ) ) ) β Ξ¦ β‘ ( ( E 1 β Ο ) β Ο β‘ ( m ) β ( E 2 β Ο ) ) β ( Ο 2 β 1 β ( E 2 β ( m ) ) ) β ( E 2 β² β Ο 2 β 1 β ( m ) ) \displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(\varphi_{1}(E_{1}(m)))\circ\Phi\bigl((E_{1}\pi)\circ\sigma(m)\circ(E_{2}\pi)\bigr)\circ(\varphi_{2}^{-1}(E_{2}(m)))\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))
It follows that
( E 1 β² β Ο β² ) β ( Ο β² β Ξ¦ β ( m ) ) β ( E 2 β² β Ο β² ) β ( Ο 2 β² β Ξ¦ β ( m ) ) = ( Ο 1 β² β Ξ¦ β ( m ) ) β ( E 1 β² β Ο β² ) β ( Ο β² β Ξ¦ β ( m ) ) β ( E 2 β² β Ο β² ) , (E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime})\circ(\tau^{\prime}_{2}\Phi(m))=(\tau^{\prime}_{1}\Phi(m))\circ(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime}),
hence ( Ξ¦ β‘ ( m ) , Ο β² ) (\Phi(m),\pi^{\prime}) Ξ Ο β² β π² β² \Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime} Ξ β Ξ¦ β ( m , Ο ) = ( Ξ¦ β‘ ( m ) , Ο β² ) \Delta\Phi(m,\pi)=(\Phi(m),\pi^{\prime})
Let f β Hom Ξ Ο β π² β‘ ( ( m , Ο ) , ( m ~ , Ο ~ ) ) f\in\operatorname{Hom}\nolimits_{\Delta_{\sigma}{\mathcal{W}}}((m,\pi),(\tilde{m},\tilde{\pi}))
E 2 β² β Ξ¦ β ( m ) \textstyle{E^{\prime}_{2}\Phi(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Ο 2 β 1 β ( m ) \scriptstyle{\varphi_{2}^{-1}(m)} E 2 β² β Ξ¦ β ( f ) \scriptstyle{E^{\prime}_{2}\Phi(f)} Ξ¦ β E 2 β ( m ) \textstyle{\Phi E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Ξ¦ β Ο \scriptstyle{\Phi\pi} Ξ¦ β E 2 β ( f ) \scriptstyle{\Phi E_{2}(f)} Ξ¦ β E 1 β ( m ) \textstyle{\Phi E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Ο 1 β ( m ) \scriptstyle{\varphi_{1}(m)} Ξ¦ β E 1 β ( f ) \scriptstyle{\Phi E_{1}(f)} E 1 β² β Ξ¦ β ( m ) \textstyle{E^{\prime}_{1}\Phi(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 β² β Ξ¦ β ( f ) \scriptstyle{E^{\prime}_{1}\Phi(f)} E 2 β² β Ξ¦ β ( m ~ ) \textstyle{E^{\prime}_{2}\Phi(\tilde{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Ο 2 β 1 β ( m ~ ) \scriptstyle{\varphi_{2}^{-1}(\tilde{m})} Ξ¦ β E 2 β ( m ~ ) \textstyle{\Phi E_{2}(\tilde{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Ξ¦ β Ο ~ \scriptstyle{\Phi\tilde{\pi}} Ξ¦ β E 1 β ( m ~ ) \textstyle{\Phi E_{1}(\tilde{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Ο 1 β ( m ~ ) \scriptstyle{\varphi_{1}(\tilde{m})} E 1 β² β Ξ¦ β ( m ~ ) \textstyle{E^{\prime}_{1}\Phi(\tilde{m})}
and it follows that Ξ¦ β‘ ( f ) β Hom Ξ Ο β² β π² β² β‘ ( Ξ β Ξ¦ β ( m , Ο ) , Ξ β Ξ¦ β ( m ~ , Ο ~ ) ) \Phi(f)\in\operatorname{Hom}\nolimits_{\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}}(\Delta\Phi(m,\pi),\Delta\Phi(\tilde{m},\tilde{\pi})) ( Ξ β Ξ¦ ) β ( f ) = Ξ¦ β ( f ) (\Delta\Phi)(f)=\Phi(f) Ξ β Ξ¦ \Delta\Phi Ξ Ο β π² β Ξ Ο β² β π² β² \Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}
We have
( Ξ β Ξ¦ ) β ( E β‘ ( m , Ο ) ) = ( Β Β Β Β Ξ¦ β‘ ( E 2 β ( m ) ) β Ξ¦ β‘ ( E 1 β ( m ) ) Β Β Ξ¦ β‘ ( Ο ) Β Β Β Β Β , Ξ² ) , (\Delta\Phi)(E(m,\pi))=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.11348pt\hbox{{\hbox{\kern-50.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{\Phi(E_{2}(m))\oplus\Phi(E_{1}(m))}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-11.31735pt\raise 21.03578pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\Phi(\pi)}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 19.91682pt\raise 8.53578pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}},\beta),
Ξ² = ( Ο 1 β E 2 β Ξ¦ β Ο β Ξ¦ β E 2 β Ο β Ξ¦ β Ο 2 β Ο 2 β 1 β E 2 Ο 1 β E 2 β Ξ¦ β Ο β Ο 2 β 1 β E 1 0 Ο 1 β E 1 β Ξ¦ β Ο 1 β Ξ¦ β E 1 β Ο β Ξ¦ β Ο β Ο 2 β 1 β E 1 ) \beta=\left(\begin{matrix}\varphi_{1}E_{2}\circ\Phi\sigma\circ\Phi E_{2}\pi\circ\Phi\tau_{2}\circ\varphi_{2}^{-1}E_{2}&\varphi_{1}E_{2}\circ\Phi\sigma\circ\varphi_{2}^{-1}E_{1}\\
0&\varphi_{1}E_{1}\circ\Phi\tau_{1}\circ\Phi E_{1}\pi\circ\Phi\sigma\circ\varphi_{2}^{-1}E_{1}\end{matrix}\right)
and
E β² β ( ( Ξ β Ξ¦ ) β ( m , Ο ) ) = ( Β Β Β Β E 2 β² β ( Ξ¦ β‘ ( m ) ) β E 1 β² β ( Ξ¦ β‘ ( m ) ) Β Β Ο 1 β ( m ) β Ξ¦ β‘ ( Ο ) β Ο 2 β ( m ) β 1 Β Β Β Β Β , Ξ² β² ) , E^{\prime}((\Delta\Phi)(m,\pi))=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.11348pt\hbox{{\hbox{\kern-50.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.50891pt\hbox{$\textstyle{E^{\prime}_{2}(\Phi(m))\oplus E^{\prime}_{1}(\Phi(m))}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-41.0553pt\raise 21.53079pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.24501pt\hbox{$\scriptstyle{\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}(m)^{-1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 19.91682pt\raise 8.53578pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}},\beta^{\prime}),
OPEN Ξ² β² = ( Ο β² β Ξ¦ β E 2 β² β ( Ο 1 β Ξ¦ β Ο β Ο 2 β 1 ) β Ο 2 β² β Ξ¦ Ο β² β Ξ¦ 0 Ο 1 β² β Ξ¦ β E 1 β² β ( Ο 1 β Ξ¦ β Ο β Ο 2 β 1 ) β Ο β² β Ξ¦ ) ) \beta^{\prime}=\left(\begin{matrix}\sigma^{\prime}\Phi\circ E^{\prime}_{2}(\varphi_{1}\circ\Phi\pi\circ\varphi_{2}^{-1})\circ\tau^{\prime}_{2}\Phi&\sigma^{\prime}\Phi\\
0&\tau^{\prime}_{1}\Phi\circ E^{\prime}_{1}(\varphi_{1}\circ\Phi\pi\circ\varphi_{2}^{-1})\circ\sigma^{\prime}\Phi\end{matrix}\right))
We have
Ξ² β² β ( E 2 β² β Ο 2 0 0 E 2 β² β Ο 1 ) = ( E 1 β² β Ο 2 0 0 E 1 β² β Ο 1 ) β Ξ² , \beta^{\prime}\left(\begin{matrix}E^{\prime}_{2}\varphi_{2}&0\\
0&E^{\prime}_{2}\varphi_{1}\end{matrix}\right)=\left(\begin{matrix}E^{\prime}_{1}\varphi_{2}&0\\
0&E^{\prime}_{1}\varphi_{1}\end{matrix}\right)\beta,
hence
( Ο 2 β ( m ) Ο 1 β ( m ) ) \left(\begin{matrix}\varphi_{2}(m)\\
&\varphi_{1}(m)\end{matrix}\right) Ξ β Ξ¦ β ( E β‘ ( m , Ο ) ) β βΌ E β² β ( Ξ β Ξ¦ β ( m , Ο ) ) \Delta\Phi(E(m,\pi))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}(\Delta\Phi(m,\pi)) Ο 1 \varphi_{1} Ο 2 \varphi_{2} Ο \varphi
We have Ο i β² β Ξ¦ β E i β² β Ο i β Ο i β E i = E i β² β Ο i β Ο i β E i β Ξ¦ β Ο i \tau^{\prime}_{i}\Phi\circ E^{\prime}_{i}\varphi_{i}\circ\varphi_{i}E_{i}=E^{\prime}_{i}\varphi_{i}\circ\varphi_{i}E_{i}\circ\Phi\tau_{i} i β { 1 , 2 } i\in\{1,2\} 4.3.4 Ο β² β ( Ξ β Ξ¦ ) β E β² β Ο β Ο β E = E β² β Ο β Ο β E β ( Ξ β Ξ¦ ) β Ο \tau^{\prime}(\Delta\Phi)\circ E^{\prime}\varphi\circ\varphi E=E^{\prime}\varphi\circ\varphi E\circ(\Delta\Phi)\tau ( Ξ β Ξ¦ , Ο ) (\Delta\Phi,\varphi) 2 2
![[Uncaptioned image]](/assets/author-raster/W000055/8abda3633c5b9aca787d08af5abe830294146b302d2bf6ad474da51b6af417f2.png)