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What methods are there to approximate an arbitrary homotopy type by an algebraic-geometric object in a concrete (read: computable) way?

I know this is an ambitious question, so maybe I should narrow it to a special case: what methods are there to approximate $K(\pi,1)$ spaces (i.e., spaces with non-vanishing homotopy groups only in degree one) by algrbraic-geometric objects?

I am familiar with Toen's schematic homotopy type and J.Pridham's pro-algebraic homotopy type; however, it seems all but impossible to directly compute nontrivial examples for interesting homotopy types. I am not familiar with any concrete examples of either of these constructions in the literature. Does anyone have a reference containing examples?

Perhaps another question tangential to the question above is the following: is there a notion of an algebra-geometric Eilenberg-Maclane space $K(G,1)$ for an arbitrary non-abelian group $G$?

Finally, is anyone familiar with any other methods in the literature, perhaps using pro-schemes?

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  • $\begingroup$ I'm aware of only one way of attaching homotopy types to algebraic objects. You can take the étale pro homotopy type a derived higher stack arxiv.org/abs/1506.07155 . Étale homotopy types were first defined by Artin-Mazur and refined by Frieddlander. Later Hoyois gave this better description. I don't know though what would be the correct site for derived schemes/stack. In the case of $K (G, 1)$, any field with absolute Galois group G will do the job. There's also Scholze-Kucharczyk paper arxiv.org/pdf/1609.04717.pdf constructing a space functorially for fields of char $0$ ... $\endgroup$
    – user40276
    Mar 3, 2018 at 2:07
  • $\begingroup$ ... that have the $G_F$ as a $\pi^1$ and cohomology on $Z/n$ agreeing with Galois cohomology. Btw, I don't know how étale homotopy theory should be on the pro-étale site instead of the étale. $\endgroup$
    – user40276
    Mar 3, 2018 at 2:07
  • $\begingroup$ Perhaps you should indicate what you mean by 'an arbitrary homotopy type' and also how you would like `'interesting' to be interpreted. Pro-finite homotopy theory is the obvious place to look. $\endgroup$
    – Tim Porter
    Mar 3, 2018 at 6:41
  • $\begingroup$ The issue with schematic homotopy types comes entirely from π1; for instance the pro-reductive completion of an abelian group is Spec of its character ring (usually huge). Smaller relative completions are more manageable; completion relative to 1 just gives Sullivan's rational homotopy types, regarded as stacks by passing to the associated cosimplicial algebra. $\endgroup$
    – Jon Pridham
    Mar 3, 2018 at 11:04
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    $\begingroup$ Of course, for any homotopy type $X$ you have an associated $\infty$-stack given by hypersheafifiying the constant presheaf $X$. $K(G,1)$ would then usually be called $BG$. [Etale homotopy types go in the opposite direction, producing homotopy types from stacks.] $\endgroup$
    – Jon Pridham
    Mar 3, 2018 at 15:10

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