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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2010

    Started bornological set. Some people call it a bornological space, but that conflicts with the terminology in functional analysis which refers to a locally convex TVS with a suitable “bounded = continuous” property. I quickly wrote that uniformly continuous maps between metric spaces induce bounded maps, but I’ll recheck when I have a free moment.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2014
    • (edited Jul 9th 2014)

    I have added pointers to

    and

    and created a stub for bornological group. Added the statements proven in in these articles that complex bornological vector spaces and also bornological abelian groups form a quasi-abelian category (and added this as examples there).

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2014

    Lawvere has spoken often about bornology, though I didn’t really get his point. I see I mentioned something of this here.

    In Volterra’s functionals and covariant cohesion of space, he writes

    Grothendieck remarked to me in 1981 that already around 1950 he had had great difficulty in persuading analysts that there can be more than one natural topology for the same family of linear spaces. This remark in effect presupposes that there is some more basic cohesive structure which can be the ‘same’ for various notions of neighborhood. Perhaps this situation could be schematically represented as [diagram], where the family F of basically-cohesive linear spaces is functorially parameterized by a category C of problems, and we consider liftings across the forgetful functor from basically-cohesive spaces which are moreover compatibly equipped with a given topology.

    But what could such a more basic notion of cohesiveness amount to? A very strong candidate is bornology.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2014

    Yakov Kremnitzer advocates the point of view that the good category to look at for all purposes of analysis is Ind(Ban)Ind(Ban) (ind-objects in Banach spaces). Everything else should nicely sit in there, in particular bornological spaces.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeJul 9th 2014

    Then for analysis generally, the corresponding category would seem to be Ind(Met)Ind(Met) (or perhaps Ind(QMet)Ind(Q Met) for full generality). This surprises me, since I know that a lot of things can be done with (quasi)-gauge spaces, which generalize to prometric spaces, but these are pro-objects rather than ind-objects.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2014
    • (edited Jul 9th 2014)

    I shouldn’t maybe be talking about this at this point. But since we got this far now I need to add:

    the suggestion is that Comm(Ind(Ban))Comm(Ind(Ban)) is the good context for analytic rings and then (Comm(Ind(Ban))) op(Comm(Ind(Ban)))^{op} the good site for analytic spaces. So this does match your pro-space feelings.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2014

    Is terminology standardised in this area? Bornological set has the category of bornological sets as a quasitopos, where Lawvere speaks in

    of

    …the topos of bornological sets (in which linear algebra becomes functional analysis).

    and in

    of

    my bornological topos (sheaves for finite coverings of countable sets)

    and in

    of

    bornology in its own sequential version has a very simple topos incarnation which could be considered parallel to Johnstone’s sequential-convergence topos.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 10th 2014

    I don’t know how standard the terminology is. What is in the nLab as bornological set is in some contexts better known as a bornological space, although that term could easily be confused with bornological topological vector space. I have a vague memory that this naming issue was discussed here in the nForum.

    I should probably take another look to see how this quasitopos compares with Lawvere’s topos. Maybe the two are closely connected.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 10th 2014

    Does Lawvere ever explain his construction? At CT2010, there was a talk by Montañez Puentes:

    Starting from the space of the natural numbers \mathbb{N} with the discrete bornology and imposing the extensive topology on its endomorphism monoid, we determined the Bornological topos that coincides with the proposed by F. W. Lawvere during an on published conference, in Bogota in 1983 and studied subsequently by L. Español, C. Minguez and L. Lamban professors of the Universidad de la Rioja (Spain ).

    Espanol had given a talk three years earlier defining the bornological topos as sheaves over a site.

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJul 10th 2014

    @Urs #6: What does CommComm mean here?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2014
    • (edited Jul 10th 2014)

    @Toby,

    CommComm” is meant to stand for the category of commutative monoids.

    My apologies, I realize that I made comments above without making the context from which i am speaking clear.

    So I have been chatting with people about what the right perspective on global analytic geometry would be, which is in parts about the question which analytic constructions one wants to take as fundamental, which as derived.

    There is the feeling that the existing definitions of rigid analytic space and Berkovich space and Huber space etc. are good in themselves, but should eventually be subsumed in something more general and maybe more natural. More general abstract.

    In this spirit for instance Bambozzi 14 “On a generalization of affinoid varieties” proposes to base the theory on bornological algebras.

    On the other hand, Ben-Bassat & Kremnitzer 13 had observed that traditional rigid/Berkovich/Huber analytic geometry is all neatly realized inside a topos over (Comm(Ban)) op(Comm(Ban))^{op} in the style of generalized algebraic geometry based on symmetric monoidal categories, here that of Banach spaces.

    An evident question then was: to get a more general picture, should one consider Comm(Born) opComm(Born)^{op} instead? And Yakov Kremnitzer’s suggestion – which I had tried to refer to, but clearly very insufficiently so – is that: no, the right perspective should be to consider Comm(Ind(Ban)) opComm(Ind(Ban))^{op}.

    • CommentRowNumber12.
    • CommentAuthorThomas Holder
    • CommentTimeJul 10th 2014

    @David#9: the richest discussion of bornology in Lawvere is probably in his 1989 treasure box ’Qualitative Distinctions between some toposes of generalized graphs’ -paper. I think he mentions his and the Johnstone topos as well in the last section of the 2008 Como lectures albeit briefly.

    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeJul 11th 2014

    Ah, I see, so Comm(Ind(Ban)) opComm(Ind(Ban))^op is a category of spaces much as Comm(Ab) opComm(Ab)^op is (being the category of affine schemes). I'm going to have to think about this to see how this all works out.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 11th 2014

    @Thomas#12: Is that paper only available through the book ’Categories in computer science and logic’? My institution does give me access to it.

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 11th 2014

    Re #11, it seems there’s a relationship between BanBan and bornological spaces

    Every bornological space EE is the inductive limit of a family of normed spaces (and Banach spaces if EE is quasi-complete).

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 11th 2014

    And from here

    Proposition 2.3. An inductive limit of Banach spaces (and in particular a strong algebra) is bornological and barrelled.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2014
    • (edited Jul 11th 2014)

    Yes, that’s the statement that one should consider Ind(Ban)Ind(Ban).

    • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 11th 2014

    Of course this suggestion to use Ind(Ban)Ind(Ban) is somewhat reminiscent of the category of LF spaces.

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 11th 2014

    Does LF stand for limit of Fréchet?

    I see

    An LF space is barrelled and bornological (and thus ultrabornological).

    generalizing #16, since Fréchet generalizes Banach.

    What a complicated set of concepts!

    And from the Lawvere paper (#14) that Thomas kindly sent me

    the category ab(bor) of bornological abelian groups contains the usual categories of Fréchet nuclear spaces, Banach spaces, etc., with continous maps as full subcategories.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2014

    What a complicated set of concepts!

    Yes!

    Would be nice to have a cleanup. The claim seems to be that everything is nicely cleaned up by organizing it inside Ind(Ban)Ind(Ban) and more generally Comm(Ind(Ban)) opComm(Ind(Ban))^{op}.

    • CommentRowNumber21.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 11th 2014

    So what’s this ’nuclear’ property? From Wikipedia, it seems ’orthogonal’ to Banach:

    There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a “naturally occurring” topological vector space is not a Banach space, then there is a good chance that it is nuclear.

    That was Grothendieck’s main area before algebraic geometry:

    Produits tensoriels topologiques et espaces nucléaires.

    • CommentRowNumber22.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 11th 2014

    Yes, LF means an inductive limit of Frechet spaces as computed in locally convex TVS. The inductive limits might be over countable diagrams; I’m not sure what is absolutely standard in this area. A somewhat typical example is smooth functions with compact support defined over an open set of Euclidean space, and there are many others.

    TVS theory is, I believe, notoriously complicated (a mathematical menagerie if you will). I’m all for a clean-up, as long as there are good solid supporting theorems in place.

    Where’s Andrew Stacey in this discussion? :-)

    • CommentRowNumber23.
    • CommentAuthorAndrew Stacey
    • CommentTimeJul 13th 2014

    Minor additions:

    1. Hopefully the diagram of LCTVS properties is useful in sorting the menagerie
    2. LF and LB spaces are limits of strict inductive sequences. This means that the family is countable, and the trace topology on each factor is the given topology. These are rather restrictive conditions, but you get strong inheritance properties as a trade.
    3. For me, nuclear spaces were always the preferred family. They have great properties wrt tensor products, and thus multilinear mappings. That study is what led to the key insight in my construction of the Dirac operator.
    • CommentRowNumber24.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2016

    ’Anonymous’ has posted a query box after

    X is any topological space, there is a bornology consisting of all precompact subsets of X (subsets whose closure is compact). Any continuous map is bounded with respect to this choice of bornology.

    there is now

    +– {: .query} This can’t be right. Consider the left order topology on the reals. This has no nonempty compact closed sets. =–

    • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 30th 2016

    It would have been less fuss just to add the obvious missing condition: that the space should be T 1T_1. I have added that and removed the query box (bornological set).

    • CommentRowNumber26.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 19th 2017
    • (edited Mar 19th 2017)

    Re #6, I see that the Comm(Ind(Ban))Comm(Ind(Ban)) approach appeared in

    • Federico Bambozzi, Oren Ben-Bassat, Dagger Geometry As Banach Algebraic Geometry (arXiv:1502.01401)

    and in a talk

    • Oren Ben-Bassat, A Perspective On The Foundations Of Derived Analytic Geometry, video

    Is there a suitable page to add these?

    • CommentRowNumber27.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 20th 2017

    And how does this perspective relate to the perfectoid geometry of Scholze et al?

    • CommentRowNumber28.
    • CommentAuthortrent
    • CommentTimeOct 13th 2017
    • (edited Oct 13th 2017)

    Same question as David, and unfortunately I’m too much of a noob in both fields to ask anything more specific: how does this relate to Perfectoid geometry?

    (I see in the notes for this talk that there is a version of the fargues-fontaine curve in global analytic geometry http://www-personal.umich.edu/~snkitche/Conference/notes/Kremnitzer-2.pdf + audio… so I’m going to try to understand that and the perfectoid geometry fargues fontaine curve to get an idea of how these two fields are comparable/different)

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