# Is the inclusion $\Delta^{op} \to \Gamma^{op} = Fin_\ast$ homotopy cofinal?

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.

• $\mathrm{Fin}_\ast$ 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 $BX→\ast$. – Denis Nardin Sep 13 at 19:02
• Oh right -- of course. I should probably get more sleep or something.. – Tim Campion Sep 13 at 19:03