Is there a good homotopy theory for cospaces, where a cospace (or $\infty$-cogroupoid) would be a cosimplicial set satisfying some appropriate dual version of the Kan condition?

One point I'm curious about is on whether or not finding a good model structure on cosimplicial sets would suffice: one possible subtlety might be that the homotopy (co?)groups of a cosimplicial set might now be subobjects instead of a quotient by an equivalence relation, as done for homotopy classes in model categories. This is at least the case for $\pi_0$, where we have $$\pi_0(X^\bullet)\cong\mathrm{lim}(X^\bullet)\cong\mathrm{Eq}(X^0\rightrightarrows X^1)\subset X^0$$ for a cosimplicial set $X^\bullet$.