···

       
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Constructing a section to the comparison map \[ C^{\sigma_X}\colon (\Lift(X)\to C)\to (X_\bot\to C)\] is considerably more sensitive, and we will need to first re-analyse several notions. We recall a few preliminaries from \citet{pugh-sterling-2025}.

     

       
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-
       
\begin{proposition}[{\cite{pugh-sterling-2025}}]

     

       
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+
       
\begin{proposition}[{\citet{pugh-sterling-2025}}]

     

       
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  We have a canonical equivalence $\mathsf{SD}_X(C) \cong C^{X_\bot}$ where $\mathsf{SD}_X(C)$ is the following type of ``Sierpi\'nski cone data'' over $C$:

     

       
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  \[ 

     

       
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    \mathsf{SD}_X(C) :\equiv