Categorifying the Reals via von Neumann Algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:45:04Z http://mathoverflow.net/feeds/question/1556 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1556/categorifying-the-reals-via-von-neumann-algebras Categorifying the Reals via von Neumann Algebras? Chris Schommer-Pries 2009-10-21T02:46:02Z 2009-10-22T18:45:00Z <p>So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to graded vector spaces. Also a linear monoidal category (assuming some finiteness conditions) will lead to an algebra defined over the natural numbers and so can be viewed as a categorification of the this algebra. </p> <p>Now one thing that I found intriguing when I learned about factor von Neumann algebras is that the type II_1 factor has modules which have "dimensions" which land in the (positive) real numbers. Has anyone ever seen a categorification of some real number quantity by using these sorts of modules? It seems that graded modules (or complexes of modules) would then give all real numbers.</p> <p>Is there some sort of categorification of real algebras to a monoidal category enriched over II_1-modules? Or some other type of categorification of the reals which I am not even guessing at?</p> http://mathoverflow.net/questions/1556/categorifying-the-reals-via-von-neumann-algebras/1570#1570 Answer by Qiaochu Yuan for Categorifying the Reals via von Neumann Algebras? Qiaochu Yuan 2009-10-21T03:21:47Z 2009-10-21T03:21:47Z <p>One categorification of the reals that I know of is via groupoids; see Aleks Kissinger's comment <a href="http://mathoverflow.net/questions/1114/whats-a-groupoid-whats-a-good-example-of-a-groupoid/1409#1409" rel="nofollow">on the groupoid question</a> as well as the accompanying link.</p> http://mathoverflow.net/questions/1556/categorifying-the-reals-via-von-neumann-algebras/1851#1851 Answer by Urs Schreiber for Categorifying the Reals via von Neumann Algebras? Urs Schreiber 2009-10-22T09:14:57Z 2009-10-22T09:14:57Z <p>Hi Chris,</p> <p>that's a really good question, I think. When you find out anything stable, I'd be grateful if you could drop me a note somehow. </p> http://mathoverflow.net/questions/1556/categorifying-the-reals-via-von-neumann-algebras/1889#1889 Answer by Jamie Vicary for Categorifying the Reals via von Neumann Algebras? Jamie Vicary 2009-10-22T16:23:54Z 2009-10-22T16:23:54Z <p>In a sense we already have a great categorification of the complex numbers, and that's given by <strong>FdHilb</strong>, the *-category of finite-dimensional Hilbert spaces and continuous maps. Results like the Doplicher-Roberts theorem give us good reasons to believe this. So from this perspective, we don't need to go as far as looking at fancy von Neumann algebras to get what you want.</p> <p>One point I'm implicitly making here is that categorification isn't necessarily the inverse process to taking isomorphism classes!</p>