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In an $\infty$-topos, suppose we have two cartesian diagrams of the form $$ \require{AMScd} \begin{CD} \overline{A} @>>> \overline{B} \\ @VVV @VVV \\ A @>>> B . \end{CD} $$ Let $$ \begin{align} \varinjlim \left( \overline{A} \rightrightarrows \overline{B} \right) \stackrel{\sim}{\longrightarrow}\overline{C}, && \varinjlim \left( A \rightrightarrows B \right)\stackrel{\sim}{\longrightarrow} C \end{align} $$ be the resulting coequalizers. Is the diagram $$\require{amscd}\begin{CD} \overline{B} @>>> \overline{C} \\ @VVV @VVV \\ B @>>> C \end{CD}$$ cartesian?

I think this is a formal consequence of Mather's First Cube Theorem (or Magic Cube Lemma), but I also don't know if I can use this theorem in any $\infty$-topos.

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    $\begingroup$ As a general principle, since $\infty$-topoi are left-exact localizations of presheaf categories, this sort of fact will hold in all $\infty$-topoi if and only if holds it holds in the special case of presheaf topoi. Since limits and colimits in presheaf topoi are calculated pointwise, you can then reduce further to whether or not its true for spaces. $\endgroup$
    – daniel gratzer
    Commented Sep 28, 2023 at 14:27
  • $\begingroup$ Ah, that's helpful. I see how to proceed. Thank you. $\endgroup$
    – grass man
    Commented Sep 28, 2023 at 14:31

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