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There are several notions of weak homotopy equivalence for topological spaces. The standard one can be formulated as follows: a map of spaces $X\to Y$ is a homotopy equivalence if the map of simplicial sets $C_\Delta(X) \to C_\Delta(Y)$ of simplices is a weak homotopy equivalence of simplicial sets.

There is a dual notion, where we say that a map of locally contractible topological spaces is an equivalence if it induces equivalence of "bar" simplicial sets associated to sufficiently fine Cech resolutions. I'm interested in a notion like the latter for general sites, except in a profinite sense to account for the fact that they might not be locally contractible.

For example, I want the map $X\times \mathbb{A}^1 \to X$ to induce an equivalence in this sense on étale sites in characteristic $0$, and for this to imply equivalence of étale cohomology and other étale invariants.

Here are some approximations:

  1. A map of sites is a weak homotopy equivalence if the $\infty$-categorical global sections of the constant sheaf on $X$ of sets *in a condensed sense* (in the sense of Scholze, or Pyknotic in the sense of Barwick and Haine, etc.) are equivalent to the corresponding global sections of the constant sheaf on $Y$, via the natural map.
  2. A map of sites is a "fibrant" weak homotopy equivalence if every hollow simplex of a relative bar (simplicial) complex of $X/Y$ associated to a covering can be filled in, possibly after passing to a finer covering (exact meaning left open to interpretation).

The first definition should be a genuine notion of equivalence and the second one should not give a notion of equivalence but rather generate one (it looks like a trivial fibration in a model category).

Is there a standard notion that works here?

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    $\begingroup$ Small note: in positive characteristic, $\mathbf A^1$ is not 'contractible' or even simply connected. Are you thinking of varieties over a field of characteristic $0$? $\endgroup$ – R. van Dobben de Bruyn Aug 14 '19 at 23:44
  • $\begingroup$ @Remy, of course. Edited. $\endgroup$ – Dmitry Vaintrob Aug 14 '19 at 23:55
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This is basically a long comment, so I'm making it CW. I just wanted to advertise some related work, because I wasn't sure if you were aware of it or not. Back in 2004, Dugger, Hollander, and Isaksen published a paper that showed how to modify the model structure on simplicial presheaves so that equivalences are detected on hypercovers (including Cech resolutions), and this works for very general sites (including étale). You can also do $\mathbb{A}^1$-localization in these settings. I don't know what the words in your approximations (1) and (2) mean, but maybe this does what you want in your paragraph:

For example, I want the map $X\times \mathbb{A}^1 \to X$ to induce an equivalence in this sense on étale sites in characteristic $0$, and for this to imply equivalence of étale cohomology and other étale invariants.

Isaksen also has a paper on étale homotopy theory from the point of view of pro-spaces. And Kirsten Wickelgren has recent work on étale homotopy theory that you might not have seen yet: 1, 2, 3. Hope this helps!

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