Timeline for What's special about the Simplex category?
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Jun 17, 2014 at 16:50 | vote | accept | CommunityBot | ||
| Jun 16, 2014 at 18:07 | comment | added | Todd Trimble | produce any continua. How many (path-)connected topological Boolean algebras do you know? Probably not many. One funky example is the infinite-dimensional sphere, as explored in this MO answer mathoverflow.net/a/43047/2926. I don't know of any good compact connected topological Boolean algebras off the top of my head. So for ordinary algebraic topology purposes, there doesn't seem to be a good geometric realization functor for this presheaf category, as far as I know. | |
| Jun 16, 2014 at 18:03 | comment | added | Todd Trimble | So then: what's the deal if we use instead the category of (presumably nonempty) finite ordinals and functions between them? (Don't call it $\Gamma$; that's used for something else.) Well, you can, but to get a geometric realization functor that is a left exact left adjoint, you have to do something a little bit funky. The fact of the matter is that the presheaf topos in that case classifies Boolean algebras (bounded, with top and bottom). Now, how many topological Boolean algebras can you think of? There's the discrete two-element one, but the induced realization doesn't (cont.) | |
| Jun 16, 2014 at 17:49 | history | edited | Todd Trimble | CC BY-SA 3.0 |
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| Jun 16, 2014 at 17:33 | history | answered | Todd Trimble | CC BY-SA 3.0 |