There's a canonical functor $i: \Delta^{op} \to Fin_\ast$. For example, one uses the pullback $i^\ast$ to turn a $\Gamma$-space into a simplicial space, and then takes a geometric realization to obtain a delooping.

Question: Is the functor $i$ homotopy cofinal?

That is, can I compute the homotopy colimit of a $\Gamma$-space, viewed as a functor $Fin_\ast \to Top$, by precomposing with $i$ and taking the geometric realization of the resulting simplicial space? Equivalently (by the $\infty$-categorical Quillen's Theorem A), are the coslice categories $\langle n \rangle \downarrow i$ weakly contractible for each $\langle n \rangle \in Fin_\ast$?

I'm particularly fond of a description of $i$ that I learned from a paper of Ayala, Francis, and Tanaka. Think of $\Delta$ as a non-full subcategory of the category of 1-dimensional stratified spaces and stratified maps. Then $i$ is the functor which sends a CW complex $[n]$ to its set $\langle n \rangle$ of 1-dimensional strata, plus a disjoint basepoint. A morphism $f$ is sent to the map $f^\ast$ carrying a 1-dimensional stratum to the (unique if it exists) 1-dimensional stratum mapping to it under $f$, and the basepoint otherwise.

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    $\begingroup$ $\mathrm{Fin}_\ast$ has a terminal object, so the colimit is the evaluation at the terminal object. This implies that the map $\Delta^{op}→\mathrm{Fin}_\ast$ cannot be homotopy cofinal, since for any non-trivial $\Gamma$-space $X$ the map induced on colimits is $BX→\ast$. $\endgroup$ – Denis Nardin Sep 13 at 19:02
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    $\begingroup$ Oh right -- of course. I should probably get more sleep or something.. $\endgroup$ – Tim Campion Sep 13 at 19:03

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