TITLE: The Dold-Thom theorem for infinity categories?
QUESTION [24 upvotes]: Let $\mathcal{M}$ denote the category of finite sets and monomorphisms, and let $\mathcal T$ denote the category of based spaces.  For a based space $X \in \mathcal T$, one has a canonical funtor $S_X : \mathcal M \rightarrow \mathcal T$ defined by $\{n\} \mapsto X^n$.  The definition on morphisms is to insert basepoints on the factors which are not in the image of a given monomorphism.
As is well know, the homotopy groups of $\mathrm{colim} S_X = SP^\infty X$ give the homology of $X$ (this is the Dold-Thom theorem), and the homotopy groups of $\mathrm{hocolim} S_X = SP^\infty_h X$ given the stable homotopy of $X$.
Is there a model for $SP^\infty X$, the ordinary infinite symmetric product, as a homotopy colimit as opposed to a categorical colimit?
The motivation for this question comes from thinking about $\infty$-categories.  In an $\infty$-category, one does not really have a good notion (at least not one that I am aware of) of strict categorical colimits.  So I'm wondering if there is, nonetheless, some easily defined functor on the $\infty$-category of spaces which will let us calculate ordinary homology.  In short, is there any $\infty$-categorical analog of the Dold-Thom theorem?
Update: Following up on André's remark it seems using the orbit category is heading in the right direction, at least for the $n$-th approximations.  I'll just quickly sketch what I have so far:
Let $\mathcal O(\Sigma_n)$ denote the orbit category.  The objects are the homogeneous (discrete) spaces $\Sigma_n/H$ (with left actions) as $H$ runs over all the subgroups of $\Sigma_n$, and the morphisms are the $\Sigma_n$-equivariant maps.  There is a canonical functor $$\Sigma_n \rightarrow \mathcal O(\Sigma_n)^{op}$$ where we regard $\Sigma_n$ as a category with one object as usual.
Given a $\Sigma_n$ space $X$, right Kan extension along this inclusion produces a $\mathcal O(\Sigma_n)^{op}$ diagram $\tilde X$ defined by $$\tilde X(\Sigma_n/H) = X^H$$ It turns out that the above inclusion is final so that it induces an isomorphism of colimits.  Hence $\mathrm{colim}_{\mathcal O(\Sigma_n)} \tilde X \cong X_{\Sigma_n}$, i.e., the coinvariants.  It's also not hard to see that the undercategories are copies of $B\Sigma_n$, hence not contractible, so we don't expect an equivalence of homotopy colimits, which is good.
On the other hand, I can now show that when $X$ is discrete, the canonical map $$\mathrm{hocolim} \tilde X \rightarrow \mathrm{colim} \tilde X$$ is an equivalence.  My methods here do not generalize to all spaces, so if someone has a reference for why this is true in general, that would be much appreciated.  (I think something like this must appear in May's book on equivariant homotopy theory if it's true, but I did not have it available this weekend.)
The remaining part would be to let $n \rightarrow \infty$, but somehow this seems like it should not be too bad.  (Something like: make a functor $\mathcal M \rightarrow \mathcal Cat$ by $n \mapsto \mathcal O(\Sigma_n)$.  Take the Grothendieck construction.  Some natural diagram on this category might give the right answer.)

REPLY [5 votes]: 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.