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small in 3.5
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UlrikBuchholtz committed Sep 14, 2023
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11 changes: 5 additions & 6 deletions circle.tex
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Expand Up @@ -1483,17 +1483,16 @@ \section{A reinterpretation of the circle}\label{sec:S1isC}
The equivalence $\preim$ maps $\cst\base$ to
$(x:\Sc) \mapsto \sum_{\_:\bn 1}(x\eqto \base)$
which can be identified with $\uc\base$.
Now consider the following diagram
Now consider the following diagram:
\begin{equation}\label{eq:setbundle-Sc-univ-comp}
\begin{tikzcd}
& \Sc\ar[dl,"\cst{\blank}"']\ar[d,"\uc{\blank}"]\ar[dr,"c"] & \\
\conncomp{\SetBundle(\Sc)}{\cst\base} \ar[r,"\sim"]
& \Sc\ar[dl,"{(\bn 1,\cst{\blank})}"']\ar[d,"\uc{\blank}"]\ar[dr,"c"] & \\
\conncomp{\SetBundle(\Sc)}{\bn1,\cst\base} \ar[r,"\sim"]
& \conncomp{(\Sc\to\UU)}{\uc\base} \ar[r,"\sim","\ev_\UU"']
& \InfCyc,
& \InfCyc
\end{tikzcd}
\end{equation}
where indeed the components in the last row are corresponding.

%where the last row is indeed well defined.
Both the left and the right triangle represent identity types.
We have an identification for the left triangle
because the fiber $\sum_{\_:\bn 1}(x\eqto z)$ of $\cst z$ at $x:\Sc$ can be identified with $\uc{z}(x) \jdeq (z\eqto x)$, for any $z:\Sc$.
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