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Stream: community: general

Topic: (pre)sheaves on homotopy equivalent spaces

Naso (Jul 02 2023 at 02:08):

If two 'nice' topological spaces $(X,\mathcal{T})$ and $(X',\mathcal{T'})$ are (weakly) homotopy equivalent, what can we say about their categories of presheaves $\mathsf{Psh}(\mathcal{T})$ and $\mathsf{Psh}(\mathcal{T'})$? Are they equivalent categories? What about for their categories of sheaves? (Feel free to interpret 'nice' in the most convenient way.)

Mike Shulman (Jul 02 2023 at 03:57):

They're definitely not equivalent categories. The most that can be said is that their toposes of sheaves are "(weakly) homotopy equivalent" in an appropriate sense for toposes (as a generalization of spaces).

John Baez (Jul 04 2023 at 14:29):

Just think about the category of presheaves on a point versus presheaves on an interval. They're both toposes. The first one is $\mathsf{Set}$ so its subobject classifier is $\{T,F\}$; the second has a bigger subobject classifier so it's not equivalent. (There are lots of other ways to see that these two categories are not equivalent, but this is the first one I thought of.)

John Baez (Jul 04 2023 at 14:30):

The same sentences are true if we replace "presheaves" by "sheaves" everywhere.

John Baez (Jul 04 2023 at 14:53):

Basically, since the concepts of "presheaf" and "sheaf" on a topological space depend on the details of the topology, you should expect that homeomorphic spaces have equivalent presheaf or sheaf categories - but not homotopy equivalent spaces, since homotopy equivalence is a much coarser relation.

David Michael Roberts (Jul 06 2023 at 00:08):

Yes. The presheaves see the whole lattice of open subsets of the topological space, which is a homeomorphism invariant, and very much not a homotopy invariant. The thing you can say that's stronger is that two spaces with the same soberification (i.e. identify points that share all their open neighbourhoods) have equivalent toposes of presheaves. But if a a space is Hausdorff, it is its own soberification, so that's only interesting for rather unseparated spaces.