Timeline for Strøm model structures on the category of simplicial sets
Current License: CC BY-SA 4.0
13 events
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| Jan 17 at 13:47 | comment | added | David White | @AlexanderCampbell I knew you'd say that. In the edit, I was just trying to explain what I had in mind and what I think Rognes was saying. I think the structure provided is enough to get the Barthel-Riehl h-model structure. I'm not sure if this kind of weaker structure has a name. Anyway, this ever-expanding comment thread is a bad place for this discussion. I suggest we switch to email, or you can just ask "Is sSet actually a topological model structure?" Surely there are other ways to tensor and cotensor, but Rognes's remark said to use Sing. | |
| Jan 17 at 8:08 | comment | added | Alexander Campbell | Despite your edits, I still don't see how the theorem applies, as you haven't equipped sSet with a tensored and cotensored Top-enrichment. The universal properties of the tensors and cotensors need to be homeomorphisms, not just weak homotopy equivalences. | |
| Jan 17 at 0:30 | history | edited | David White | CC BY-SA 4.0 |
Identified the issue and added explanation around what I had in mind.
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| Jan 15 at 12:00 | comment | added | Tom Goodwillie | Assuming that the inclusion $\Delta^0\to \Delta^1$ is a cofibration and that the weak equivalences are the simplicial homotopy equivalences as defined in the question, I believe I can obtain a contradiction. | |
| Jan 15 at 5:48 | comment | added | Alexander Campbell | (And such homotopies agree with the simplicially defined homotopies in the question, since geometric realisation preserves connected components.) | |
| Jan 15 at 5:46 | comment | added | Alexander Campbell | Nevertheless one can of course still speak of homotopies between maps in this topologically enriched category (they’re just paths in the Hom spaces), but these aren’t representable by an interval object. | |
| Jan 15 at 5:44 | comment | added | Alexander Campbell | The topological enrichment of simplicial sets defined by change of base along the geometric realisation functor isn’t tensored or cotensored, so we don’t have an interval object to speak of. (It would have to be the value of a left adjoint to the geometric realisation functor at the unit interval, but that doesn’t exist.) So probably the theorem doesn’t apply. | |
| Jan 15 at 5:39 | comment | added | Tim Campion | We’re talking about a $Top$ tensoring here. So the notion of homotopy is just what you get from using the singular simplicial set of the interval as your interval object. In particular the interval object here is a Kan complex, so for example $\Delta^1$ is not contractible with respect to this notion of homotopy, whereas $\Delta^1$ is contractible via simplicial homotopies. | |
| Jan 15 at 5:15 | comment | added | Tyrone | I have to admit that I'm already confused by the tensoring and cotensoring in the topological structure you're proposing. Which definitions exactly are you proposing? It's great to see your answer, and I'll be happy to accept if I can resolve the issue that Tim has raised. | |
| Jan 14 at 21:18 | comment | added | David White | I think you get the same classes of maps whether you use $\Delta^1$ or $I$. Unfortunately, tomorrow is the first day of the spring semester, so I'm going to be busy with teaching stuff, but can return to this question on Tuesday if it's still not resolved. | |
| Jan 14 at 20:26 | comment | added | Tim Campion | The notion of h-cofibration and h-fibration in the definition of the notion of “h-model structure” use the tensoring and cotensoring by the topological interval, not the simplicial interval. I think the notion of homotopy equivalence does too (yes — see def 5.1). So I think the weak equivalences in the h-model structure are not the simplicial homotopy equivalences, which use the simplicial interval. | |
| Jan 14 at 19:24 | history | edited | David White | CC BY-SA 4.0 |
added 138 characters in body
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| Jan 14 at 19:11 | history | answered | David White | CC BY-SA 4.0 |