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:
- 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.
- 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?
