Timeline for Compact Hausdorff and C^*-algebra "objects" in a category.
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 13, 2010 at 5:44 | comment | added | Sridhar Ramesh | No, I don't think you've missed anything; at any rate, certainly not anything completely obvious. I'd answer your Question 1 (now, wiser than I started...) with "Yes". | |
| Feb 13, 2010 at 5:41 | history | edited | Sridhar Ramesh | CC BY-SA 2.5 | added 130 characters in body |
| Feb 10, 2010 at 20:48 | comment | added | Andrew Stacey | Thanks for the clarifications. I wondered if there was some issue with finite/infinite stuff that I was completely unaware of. I feel reassured that at least I didn't miss something completely obvious - should I be feeling that? Or is there still something I've missed in my setup? | |
| Feb 10, 2010 at 19:51 | comment | added | Sridhar Ramesh | Such morphisms will automatically satisfy the appropriate commutative diagrams (by virtue of the appropriate equations holding in each algebra Hom(c^k, c)). Thus, they can be combined into what I thought of as an M object. I haven't sat down and checked all the details, but I am quite confident now that it works and I was wrong.) | |
| Feb 10, 2010 at 19:50 | comment | added | Sridhar Ramesh | (Why? The latter amounts to just putting the structure of a Set-algebra for M on Hom(x, c) for each x, such that precomposition is a homomorphism of such algebras. For every element in M(k), thought of as a k-ary operation, we obtain a morphism from c^k to c by applying that operation to the k many projections in Hom(c^k, c), from which the result of that operation on arbitrary Hom(x, c) is determined... [continued in next comment] | |
| Feb 10, 2010 at 19:31 | comment | added | Sridhar Ramesh | Egads, no, you're right. The correspondence does go both ways. I failed to see it before, being so used to viewing things one way, but if M is a monad on Set, then product-preserving functors from the dual of M's Kleisli category to C (what I was thinking of as an M object) are in correspondence with tuples of the form <contravariant functor F from C to the category of Set-algebras of M, object c in C such that the product of any set of copies of c exists, and natural isomorphism between Hom_C(-, c) and UnderlyingSet(F(-))> (what you were thinking of as an M object). So, I retract my "No". | |
| Feb 10, 2010 at 11:43 | comment | added | Andrew Stacey | But it all works if I replace "compact Hausdorff" by "group", doesn't it? Maybe I'm missing something there as well. If so, perhaps I should ask the more basic question about the difference between finitary and infinitary theories first. | |
| Feb 10, 2010 at 10:57 | history | undeleted | Sridhar Ramesh | ||
| Feb 10, 2010 at 10:57 | history | edited | Sridhar Ramesh | CC BY-SA 2.5 | Minor rewording |
| Feb 10, 2010 at 10:47 | history | deleted | Sridhar Ramesh | ||
| Feb 10, 2010 at 10:42 | history | edited | Sridhar Ramesh | CC BY-SA 2.5 | Hesitation, considering deletion |
| Feb 10, 2010 at 10:41 | history | undeleted | Sridhar Ramesh | ||
| Feb 10, 2010 at 10:40 | history | deleted | Sridhar Ramesh | ||
| Feb 10, 2010 at 10:35 | history | answered | Sridhar Ramesh | CC BY-SA 2.5 |