Loading [MathJax]/extensions/TeX/HTML.js

Wednesday, June 16, 2021

Certain preserved limits give left adjoints

WORK IN PROGRESS!

Here we use the fact that (the cone  \boxed{    \pi : \leftcat{1_{(l \downarrow \rightadj R)} }\lim \Rightarrow \leftcat{1_{(l \downarrow \rightadj R)} }: \leftcat{(l \downarrow \rightadj R)} \to \leftcat{(l \downarrow \rightadj R)}    }) is (a natural transformation). 
Hence (the following triangles) commute for all (\rightcat r \leftcat{[\lambda]}\in \leftcat{(l \downarrow \rightadj R)}) and (\rightcat f as shown):
  \boxed{  \begin{array}  {c|cccccc|cccccc}  \leftcat{1_{(l \downarrow \rightadj R)} } & \kern0em   &  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)} &  {} \rlap{   \kern-1em \xrightarrow[\smash{\kern15em}]{\textstyle \rlap{\pi_{\rightcat r[\lambda]}} }  }   &&&  \rightcat r \leftcat{[\lambda]} & \kern0em   &  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)} &  {} \rlap{   \rightcat{  \kern-1em \xrightarrow[\smash{\kern15em}]{\textstyle f }  }  }   &&&  \rightcat r  \leftcat{[\lambda]}     \\      \llap{\pi} \Big\Uparrow &  &&  \llap{ \pi_{\rightcat{ (\black Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }}  } \nwarrow  &  \pi_{\pi_{\rightcat r \leftcat{[\lambda]}}}  &  \nearrow \rlap{\pi_{\rightcat r[\lambda]}} &&  &&  \llap{ \pi_{\rightcat{ (\black Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }}  } \nwarrow  &  \pi_{\rightcat f}  &  \nearrow \rlap{ \pi_{\rightcat r[\lambda]} }  \\    \leftcat{   1_{(l \downarrow \rightadj R)} \lim_{(l \downarrow \rightadj R)  }   }  & & &&  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}  & & && &&    \leftcat{  1_{(l \downarrow \rightadj R)} \lim_{(l \downarrow \rightadj R)}  }    \\ \end{array}  } 

To reiterate, (the two triangles above) commute for all (\rightcat r \leftcat{[\lambda]}\in \leftcat{(l \downarrow \rightadj R)}) and (\rightcat f as shown) because (\rightcat\pi is a cone).

Using (naturality of \pi at \pi_{\rightcat r \leftcat{[\lambda]}}), i.e.  \pi_{\pi_{\rightcat r \leftcat{[\lambda]}}},  plus 
(the uniqueness clause in the universal property of \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}), 
we get that \pi_{\rightcat{ (\black Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }} = 1_{  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}  }.
This, plus (naturality of \pi at \rightcat f), i.e. \pi_{\rightcat f},
gives \rightcat f = \pi_{\rightcat r[\lambda]}.

This in turn gives us that 
\leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)} is 
(an initial object in {(l \downarrow \rightadj R)}},
which is equivalent to
\rightcat{ (Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] } being 
(a universal arrow from \leftcat l to \rightadj\calR
which is equivalent to
\rightcat{ (Q\lim_\calR) } being (a left adjoint \leftcat  l \leftadj L for \leftcat  l) with (\leftcat{ \overline{\lambda} } its unit).

<hr />

\boxed{ \begin{array} {c|c|c} \kern2em & \kern2em & \kern2em \\ xx & yy & zz \\ \end{array} }

\boxed{ \begin{array} {} && r \\ & \llap{\pi_r} \nearrow & \pi_{\pi_r} & \nwarrow \rlap{\pi_r} \\ \leftcat l\leftadj L & {} \rlap{ \kern-1em \xrightarrow[\pi_{\leftcat l\leftadj L}] {\kern8em} } &&& \leftcat l\leftadj L \\ \end{array} }

No comments:

Post a Comment

MathJax 2.7.9