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Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.

As an example, the Polish Circle has the same sheaf cohomology with constant coefficients as $S^1$, and both spaces also have equivalent categories of covering spaces. So one could ask if in such a model structure, the polish circle would be weakly equivalent to an ordinary one.

Edit: I would be happy with an answer for any nice category of topological spaces which contains the Polish circle, e.g. compactly generated weak Hausdorff spaces. Please ignore the following paragraph.

Also, sheaf theory is defined on rather general topological spaces, so it would be nice if the underlying category of topological spaces would be much more general than, say, compactly generated weak Hausdorff spaces.

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    $\begingroup$ Hmm... the shape of algebraic varieties does not depend on the underlying topological space, unless I'm mistaken (rather, it is the shape of the étale topos), so I doubt a naive approach would work there. $\endgroup$
    – Denis Nardin
    Commented Nov 6, 2019 at 17:36
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    $\begingroup$ What is the precise definition of 'shape equivalences' (it is not in the wikipedia link) ? I'm not expert in shape theory but I believe there has been several non-equivalent definition... is it equivalent to the definition Lurie gave for infinity toposes ? $\endgroup$
    – Simon Henry
    Commented Nov 6, 2019 at 18:24
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    $\begingroup$ In Lurie's perspective shape theory is an adjunction between the $\infty$-category of $\infty$-toposes and the $\infty$-category of pro-spaces. I always had the feeling that this adjunction is idempotent ( not sure if this is really true though, is there someone who know ?) if it is the case "shapes" are a localization of the category of pro-spaces, as such it should be possible to construct a model structure representing them (maybe on pro-simplicial sets...). If this works one can start thinking about whether this transfer to a model structure on some category of topological spaces... $\endgroup$
    – Simon Henry
    Commented Nov 7, 2019 at 15:54
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    $\begingroup$ @SimonHenry It's not idempotent, but it becomes one if you consider the profinite shape instead (see the appendix to SAG and the Barwick-Glasman-Haine work on the stratified shape) $\endgroup$
    – Denis Nardin
    Commented Nov 7, 2019 at 17:02
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    $\begingroup$ @SebastianGoette Well, it is sort of the same definition, only you need to generalize it to a notion of the shape of a topos rather than just of topological spaces (you might have heard it under the name "étale homotopy type").The point is that the underlying space of an algebraic variety does not have much useful information (e.g. all curves over an alg. closed field are homeomorphic!),and you need more refined objects (e.g the étale topos)to recover the information present in the topology for manifolds. Still, your question is interesting, I would only remove the reference to alg. varieties. $\endgroup$
    – Denis Nardin
    Commented Nov 8, 2019 at 12:26

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