Timeline for Are all Grothendieck topologies on Set equivalent?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 14, 2017 at 18:44 | comment | added | Tim Campion | @skd You should also do Tom Goodwillie's related exercise of showing that there are nine model structures on $\mathsf{Set}$. | |
| Feb 18, 2017 at 15:46 | vote | accept | Qfwfq | ||
| Feb 18, 2017 at 15:18 | comment | added | Tom Goodwillie | I don't have a reference. I noticed it myself years ago, but I've never written down a proof. I like to recommend it as an exercise. | |
| Feb 18, 2017 at 6:14 | comment | added | skd | @TomGoodwillie Why is this true? | |
| Feb 18, 2017 at 3:54 | answer | added | Tom Goodwillie | timeline score: 13 | |
| Feb 18, 2017 at 1:20 | comment | added | Tom Goodwillie | There are only six weak factorization systems on Set. | |
| Feb 18, 2017 at 0:58 | comment | added | fosco | Maybe this is a source of counterexamples? arxiv.org/abs/0902.1130 there might be many topologizing factorization systems on $\bf Set$... | |
| Feb 17, 2017 at 22:26 | comment | added | skd | @TomGoodwillie Yeah, when I said the trivial topology I meant the minimal topology. | |
| Feb 17, 2017 at 21:00 | comment | added | მამუკა ჯიბლაძე | I think there must also be nontrivial subcanonical examples - take the representables together with some non-representable one, like e. g. $F(X)=2^{2^{2^X}}$; I believe there is a proper subtopos containing all of them. | |
| Feb 17, 2017 at 20:27 | comment | added | მამუკა ჯიბლაძე | @TomGoodwillie There must be lots of similar examples - presheaves on any full subcategory of sets give a reflective subcategory in all presheaves. The reflector is restriction, and the embedding is given by the right Kan extension along the inclusion of the subcategory. Your last example corresponds to the full subcategory with the singleton as the only object. | |
| Feb 17, 2017 at 20:23 | comment | added | Tom Goodwillie | (I'm not sure which of those is called the trivial topology. I would guess the former.) There is also the slightly less than maximal topology in which the covering sieves are the nonempty sieves. For this the sheaves are the constant presheaves. | |
| Feb 17, 2017 at 20:21 | comment | added | Tom Goodwillie | The minimal topology is the one where the only covering sieve of an object X is the maximal sieve of X. For this topology, every presheaf is a sheaf. The maximal topology is the one where every sieve on every object is a covering sieve. For this topology the only way a presheaf F can be a sheaf is if F(X) is a singleton for every X. | |
| Feb 17, 2017 at 20:04 | comment | added | Qfwfq | Ok. Maybe I should make some assumptions in order to make the question more meaningful. What if I require the topology to be subcanonical? (which the trivial topology is not, if I'm not mistaken?) | |
| Feb 17, 2017 at 19:58 | comment | added | skd | The trivial topology? | |
| Feb 17, 2017 at 19:43 | history | asked | Qfwfq | CC BY-SA 3.0 |