Timeline for Advantages of diffeological spaces over general sheaves
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Dec 15, 2010 at 10:11 | comment | added | Andrew Stacey | Thomas, let's do that. I've started us off on the nForum here: math.ntnu.no/~stacey/Mathforge/nForum/… | |
| Dec 14, 2010 at 22:33 | comment | added | Thomas Nikolaus | Andrew: Not bad for a typo ;) It would be cool to have a full statement of this type. If you come to some conclusion I would be highly interested. Maybe we could follow your advise and continue that discussion by mail or in a Forum!? | |
| Dec 14, 2010 at 19:18 | comment | added | Andrew Stacey | Thomas: Okay, I'll take a look at it when I get a moment (ought to be grading exams right now ...). I like the name: $CoCa_{co} t$! Sounds like a mix of cocoa and SohCahToa! By the way, the limits and colimits parts don't imply one another, they're both true but (slightly) independently. It's just that, as you say in your answer, limits are always preserved on this side of the tree so colimits are the interesting ones. I was just reminding us all of that fact. | |
| Dec 13, 2010 at 22:39 | comment | added | Thomas Nikolaus | Andrew: It could mean the following: Let $CoCa_{co}t$ be the (bi-)category of cocomplete categories together with cocontinous functors. There is the forgetful functor $CoCat_{co} \to Cat_{co}$ where $Cat_{co}$ denotes the (bi-)category of categories with cocontinous functors. Then it would be left adjoint to this functor... I am wondering why this also implies that functors and categories are complete and limit preserving... | |
| Dec 13, 2010 at 21:48 | comment | added | Andrew Stacey | Oh, and it preserves all limits as well. So it's the universal limit-and-colimit-preserving category (assuming there aren't any hidden hitches in the proof). | |
| Dec 13, 2010 at 21:47 | comment | added | Andrew Stacey | Thomas: Hausdorff Frolicher spaces (though there's very little difference between the two). I haven't written out the proof of that statement, but I'm pretty sure that it is correct - the main obstruction being my own understanding of what the statement actually means when written out in full! | |
| Dec 13, 2010 at 20:41 | comment | added | Thomas Nikolaus | @Andrew: Your edited stuff is great. Does this mean, that Fröhlicher Spaces are the universal cocompletion of $Man$ subject to the requirement that the inclusion preserves all colimits? | |
| Dec 13, 2010 at 20:33 | history | edited | Andrew Stacey | CC BY-SA 2.5 |
added stuff about colimits
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| Dec 8, 2010 at 16:00 | comment | added | Urs Schreiber | ... because whenever a big topos is cohesive it is in particular local and then there is canonically associated to it the sub-quasitopos of its concrete objects. So therefore we are now on the reverse path through those 45 years: after we have learned that the general ambient context is always an oo-topos, we should go back and check which of all the concrete smaller contexts that we might be interested in in special situations are canonically induced by that topos-context. For instance the quasitopos of diffeological space is canonically induced by the Cahiers topos. | |
| Dec 8, 2010 at 15:55 | comment | added | Urs Schreiber | There is a curious irony to this story: when John announced his article on the quasitopos of concrete smooth spaces on the category theory mailing list, Bill Lawvere complained harshly that after "the proliferation of such smooth categories 45 years ago" still not everyone has switched to working with genuine toposes. In his reply (archived at mta.ca/~cat-dist/archive/2008/08-8) he vaguely refers to his axioms for cohesive toposes (recalled here nlab.mathforge.org/nlab/show/cohesive (infinity,1)-topos). Interestingly, that's exactly where Dave's question above originates... | |
| Dec 8, 2010 at 11:02 | comment | added | Andrew Stacey | Todd, I'm pretty sure but I couldn't find the actual quote that I was looking for. Scanning through golem.ph.utexas.edu/category/2008/05/… I find some comments that hint at that view. | |
| Dec 8, 2010 at 1:06 | comment | added | Todd Trimble | Are you sure John is in the topos camp, Andrew? | |
| Dec 7, 2010 at 23:52 | comment | added | arsmath | Andrew, is there some natural case of "manifoldness" where having an underlying set is an obstacle? | |
| Dec 7, 2010 at 22:33 | comment | added | Urs Schreiber | Hi Andrew, there is some behing-the-scenes discussion between me and Dave. We are all on the same page as far as the pleasures of general abstract topos theory go. Here the question is about nice formal characterizations of a "filtering" of the category of all sheaves by tame/wild-degree. some intermediate steps are: representable, locally representable, concrete. Here the question is how to characterize abstractly what makes concrete sheaves nice. And then to say what makes concrete oo-stacks aka cohesive oo-groupoids so nice. Your statement about Isbell self-duality would be the sort needed. | |
| Dec 7, 2010 at 20:22 | comment | added | David Carchedi | Andrew, I'd like to believe that diffeological spaces have a use, besides being more "comfortable" than fully abstract sheaves. I mean, after all, there are many sheaves vastly far away from being manifolds. Somehow, by adding the property of concreteness, this should "tame the jungle" a little bit. I myself, am very comfortable with sheaves, but, I am not comfortable saying that they are manifolds. Somehow, I am more comfortable calling a diffeological space a manifold however... I wish I could be more precise, but if I could, I wouldn't have asked this question! | |
| Dec 7, 2010 at 20:12 | history | answered | Andrew Stacey | CC BY-SA 2.5 |