Etale topological type of a simplicial presheaf $X$ on the etale site of a scheme is a pro-space. A simplicial presheaf can be written as a coequalizer of presheaves $P_n$, and each such presheaf is a colimit of representables $X_i$. Then $Et(X)$ is defined as a coequilizer of $Et(P_n)=\text{colim} Et(X_i)$ where $Et(X_i)$ is Friedlander's definition of etale topological type. This definition is due to Dan Isaksen. Given this definition is there a version of the comparison theorem for schemes over $\mathbb{C}$?
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$\begingroup$ I believe so, see recent work of David Carchedi. $\endgroup$– David Roberts ♦Commented Aug 2, 2016 at 6:29
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$\begingroup$ Thanks for your suggestion. I guess if one can replace the two colimits with hocolims then comparison theorem in this case should be a direct consequence of the classical one. $\endgroup$– user95770Commented Aug 3, 2016 at 5:50
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