The Dold-Thom theorem for infinity categories?

It so happens that Emmanuel Dror Farjoun is visiting the EPFL this week. I figured I'd ask him about this problem at lunch today. What a coincidence! He proved exactly this statement using the exact same techniques. In fact, the construction of $SP^n$ as a homotopy colimit is the subject of Chapter 4 in "Cellular Spaces, Null Spaces, and Homotopy Localization," Lecture Notes in Mathematics, 1622.

It turns out, the idea works more generally so that we can always replace strict colimits with homotopy colimits: define an orbit on a category $\mathcal C$ to be a functor $O : \mathcal C \rightarrow \mathcal Set$ such that $\mathrm{colim}_{\mathcal C} O \cong *$. There is a category of such functors which we call the orbit category of $\mathcal C$, denoted $\mathcal O(\mathcal C)$. The Yoneda embedding factors through $\mathcal O(\mathcal C)$, and the right Kan extension along this inclusion always results in a "free" diagram.

I still want to play around with the construction a bit to see if there are any wrinkles with $n \rightarrow \infty$, and if I can use this to give easy calculations of homology in the $\infty$-category $\mathcal S$, but I think it's safe to say at this point that answer to my question is yes.

I just came across this old posting, and see that folks had correctly discovered presentations (dating back to the mid 1980's) by Emmanuel Dror about writing colimits as homotopy colimits.

The special case in hand -- symmetric powers of spaces -- is particularly elegant, as the orbits which arise are very limited and special. Kathryn Lesh and Greg Arone have a couple of lovely papers on this and related families of examples: Kathryn got this going with a 2000 paper in T.A.M.S., and then, with Greg, has a longer study in Crelle in 2007. (They then use these ideas in a 2010 paper in Fund. Math.) Kathryn's work was roughly contemporaneous with Emmanuel's - she has a 1997 Math. Zeit. paper where one can see the beginnings of the ideas.