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Geometry of the second barycentric subdivision (and Thomason-fibrant replacement)
Is
$\mathrm{sd}^2 (\Delta^n) = \mathrm{sd}^2(\partial \Delta^n) \times \Delta^1 \cup_{\mathrm{sd}^2(\partial \Delta^n) \times \{1\}} Cone(\mathrm{sd}^2(\partial \Delta^n))$
? Here $\mathrm{sd}^2$ ...
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