*[HoTT] The Hodge structure of a type@ 2018-09-22 16:58 José Manuel Rodriguez Caballero2018-09-22 23:43 ` Ali Caglayan 2018-09-24 22:30 ` Ali Caglayan 0 siblings, 2 replies; 5+ messages in thread From: José Manuel Rodriguez Caballero @ 2018-09-22 16:58 UTC (permalink / raw) To: HomotopyTypeTheory [-- Attachment #1: Type: text/plain, Size: 745 bytes --] Recently, there was a post about the Euler characteristic of a type. In my case, I am interested in the Hodge structure of the Hilbert scheme of n points on a 2-dimensional torus. Does such a topological construction make sense in HoTT for an arbitrary type, under some general hypothesis? Kind Regards, Jose M References about the topological structure that I am studying: https://www.sciencedirect.com/science/article/pii/S0001870812004008 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #2: Type: text/html, Size: 1097 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: [HoTT] The Hodge structure of a type2018-09-22 16:58 [HoTT] The Hodge structure of a type José Manuel Rodriguez Caballero@ 2018-09-22 23:43 ` Ali Caglayan2018-09-24 22:30 ` Ali Caglayan 1 sibling, 0 replies; 5+ messages in thread From: Ali Caglayan @ 2018-09-22 23:43 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 2070 bytes --] I don't think HoTT is any position to be used in algebraic geometry as it stands currently. There may be a way in via a functor of points approach which is well suited for constructing Hilbert schemes. However I think it would be impossible to say anything concrete at this point. Homology and cohomology are usually seen as "easy" invariants about spaces to calculate however in HoTT it is all very new and nobody quite know the best way to go about reasoning with these things. Let alone thinking about something with extra structure like Hodge structure. However representation theory, as you have cited, may be more tractable. There are good formal properties of HoTT which may allow it to reason in representation theoretic terms quite concretely. So if you are interested in studying quiver varieties you may just be able to get away with studying quiver representations. However this is all speculative at this point. I don't think there are any researchers looking into any of these things yet as I believe HoTT just is not sophisticated to carry out such reasoning. But this is all my opinion. I would be very suprised if anybody says otherwise. On Saturday, 22 September 2018 17:59:03 UTC+1, José Manuel Rodriguez Caballero wrote: > > Recently, there was a post about the Euler characteristic of a type. In my > case, I am interested in the Hodge structure of the Hilbert scheme of n > points on a 2-dimensional torus. Does such a topological construction make > sense in HoTT for an arbitrary type, under some general hypothesis? > > Kind Regards, > Jose M > > References about the topological structure that I am studying: > https://www.sciencedirect.com/science/article/pii/S0001870812004008 > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 3063 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: [HoTT] The Hodge structure of a type2018-09-22 16:58 [HoTT] The Hodge structure of a type José Manuel Rodriguez Caballero 2018-09-22 23:43 ` Ali Caglayan@ 2018-09-24 22:30 ` Ali Caglayan2018-09-24 23:59 ` Steve Awodey 1 sibling, 1 reply; 5+ messages in thread From: Ali Caglayan @ 2018-09-24 22:30 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 776 bytes --] However I don't want to discourage you. One possible solution is (differential?) cohesive homotopy type theory (which is at the moment even more undeveloped). This may allow you to talk about manifolds and their structure "synthetically" which would allow for definitions of de Rham cohomology and possibly with care allow you to talk about hilbert schemes of some torus. Pessemistically I would add that it would be at least 10 years before any of this is considered. -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 974 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: [HoTT] The Hodge structure of a type2018-09-24 22:30 ` Ali Caglayan@ 2018-09-24 23:59 ` Steve Awodey2018-09-25 8:59 ` José Manuel Rodriguez Caballero 0 siblings, 1 reply; 5+ messages in thread From: Steve Awodey @ 2018-09-24 23:59 UTC (permalink / raw) To: Ali Caglayan;+Cc:Homotopy Type Theory > On Sep 24, 2018, at 6:30 PM, Ali Caglayan <alizter@gmail.com> wrote: > > However I don't want to discourage you. One possible solution is (differential?) cohesive homotopy type theory (which is at the moment even more undeveloped). This may allow you to talk about manifolds and their structure "synthetically" which would allow for definitions of de Rham cohomology and possibly with care allow you to talk about hilbert schemes of some torus. Pessemistically I would add that it would be at least 10 years before any of this is considered. just for perspective: - 10 years ago we had the (higher) homotopy group(oid)s, and not much more. - 5 years ago the HoTT book was just finished. - 1 year ago the Serre spectral sequence was finished. things are moving pretty fast - I would not be so pessimistic. Steve -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. ^ permalink raw reply [flat|nested] 5+ messages in thread

*2018-09-24 23:59 ` Steve AwodeyRe: [HoTT] The Hodge structure of a type@ 2018-09-25 8:59 ` José Manuel Rodriguez Caballero0 siblings, 0 replies; 5+ messages in thread From: José Manuel Rodriguez Caballero @ 2018-09-25 8:59 UTC (permalink / raw) To: HomotopyTypeTheory [-- Attachment #1: Type: text/plain, Size: 2227 bytes --] > > Michael Shulman wrote: > what "algebraic" information it uses as input. In the case of the Hilbert scheme of n points on X, the information comes from the n-th symmetric power of X: https://arxiv.org/pdf/math/0304302.pdf So, the input are X as an infinite-groupoid and the natural number n. The output is the Hilbert scheme of n points on X as an infinite groupoid. I do not know if there is some nice functoriality in this process which could be expressed in HoTT in a natural way. There are more results about the Hilbert schemes it in Goettsche's homepage: http://users.ictp.it/~gottsche/ Kind Regards, José M. El lun., 24 sept. 2018 a las 19:59, Steve Awodey (<awodey@cmu.edu>) escribió: > > On Sep 24, 2018, at 6:30 PM, Ali Caglayan <alizter@gmail.com> wrote: > > > > However I don't want to discourage you. One possible solution is > (differential?) cohesive homotopy type theory (which is at the moment even > more undeveloped). This may allow you to talk about manifolds and their > structure "synthetically" which would allow for definitions of de Rham > cohomology and possibly with care allow you to talk about hilbert schemes > of some torus. Pessemistically I would add that it would be at least 10 > years before any of this is considered. > > just for perspective: > > - 10 years ago we had the (higher) homotopy group(oid)s, and not much more. > - 5 years ago the HoTT book was just finished. > - 1 year ago the Serre spectral sequence was finished. > > things are moving pretty fast - I would not be so pessimistic. > > Steve > > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #2: Type: text/html, Size: 3424 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

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