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Madeleine Birchfield (Dec 07 2024 at 17:51):What model structures, if any, exist on the category of locales?
Notification Bot (Dec 07 2024 at 18:53):A message was moved here from #learning: questions > What Are Categories of Spaces? by Madeleine Birchfield.
John Onstead (Dec 07 2024 at 19:16):Madeleine Birchfield said:
What model structures, if any, exist on the category of locales?
That's an interesting thing to consider! To start, there's a way to transfer model structures along adjoint functors. Since and share an adjunction maybe you can transfer some model structure (like the classical or Strom model structure) from to along it. But I have no idea if those model structures actually would transfer along this adjunction!
Mike Shulman (Dec 07 2024 at 19:57):One problem with putting model structures on Loc is that the most general method for constructing model structures uses the small object argument to build factorizations, but I am skeptical that there are very many small objects in Loc, because it is the dual of an algebraic category. In particular, it's certainly not locally presentable.
Mike Shulman (Dec 07 2024 at 19:57):But you might have more luck with a Strom-style model structure.
John Onstead (Dec 15 2024 at 16:28):Well as it turns out, there's a Quillen-like model structure on locales after all, and in fact it's part of a whole class of model structures that one can put on any bicomplete category that shares a special sub-category with Top containing the unit interval, point, and a few related objects!
Mike Shulman (Dec 16 2024 at 06:46):Nice!
James E Hanson (Dec 16 2024 at 17:31):Mike Shulman said:
One problem with putting model structures on Loc is that the most general method for constructing model structures uses the small object argument to build factorizations, but I am skeptical that there are very many small objects in Loc, because it is the dual of an algebraic category. In particular, it's certainly not locally presentable.
Is there some general statement about it being hard for co-algebraic categories to be locally presentable? Set is co-algebraic, so it's not literally impossible.
Mike Shulman (Dec 16 2024 at 18:43):If a category and its dual are both locally presentable, then it is a preorder. So Set is not co-algebraic in the sense of being co-locally-presentable.