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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 $X^\bullet$.

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  • $\begingroup$ Google reveals notes by Bert Guillou about homotopy theory of cosimplicial spaces. There was some stuff much earlier by Bousfield but I cannot remember any precise references, sorry. In any case there is an enrichment of cosimplicial objects of a nice enough category into simplicial sets. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 28, 2022 at 9:17
  • $\begingroup$ The latter is a generalization of a special case of the fact that for any small category $\Delta$, one can define, for $F,G:\Delta\to\mathrm{Sets}$, the functor $\operatorname{Hom}(F,G):\Delta^{\mathrm{op}}\to\mathrm{Sets}$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 28, 2022 at 9:19
  • $\begingroup$ This goes as follows. First, for $X:\Delta^{\mathrm{op}}\to\mathrm{Sets}$ and $G:\Delta\to\mathrm{Sets}$, define $G^X:\Delta\to\mathrm{Sets}$ via $(G^X)(n)=G(n)^{X(n)}$, with $n\in\Delta$. Next, define the above $\operatorname{Hom}(F,G)$ in such a way that for any $X:\Delta^{\mathrm{op}}\to\mathrm{Sets}$, natural transformations $X\to\operatorname{Hom}(F,G)$ are in one-to-one correspondence with natural transformations $F\to G^X$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 28, 2022 at 9:22
  • $\begingroup$ More concisely, $\operatorname{Hom}(F,G)(m)$, for $m\in\Delta$, is the set of natural transformations $F\to G^m$, where $G^m(n)=G(n)^{\hom_\Delta(n,m)}$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 28, 2022 at 9:36
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    $\begingroup$ @მამუკაჯიბლაძე I think the homotopy theory of cosimplicial spaces goes in a different direction than what I have in mind: I'm looking for a notion of homotopy cogroups (or rather, "homotopy sets") of a "cospace"/$\infty$-cogroupoid, instead of the usual homotopy groups of (totalisations of) cosimplicial spaces. It seems to me that this might be a kind of "dual notion" of homotopy, obtained via subobjects rather than quotients, as happens at least for the $\pi_0$. $\endgroup$
    – Emily
    Commented Oct 28, 2022 at 20:14

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