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nLab > Latest Changes: nonstandard analysis

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)

    I made explicit a subsection on Topos-theoretic models at nonstandard analysis.

    Not much there yet, though. I don't really fully understand this yet, but I thought I'd start recording some aspects.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)

    One could be more bold and talk about topos theoretic generalizations of nonstandard analysis. The Russian edition of Johnstone's Topos theory has an appendix devoted solely to that, but I do not have English translation for others.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010

    Thanks, Zoran. Is that the Elephant, of the previous one?

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 12th 2010

    No, Topos theory is the title of the old book. The new book is called Skectches of the elephant: A topos theory compendium and it is not even finished (third volume is missing), so I do not expect so soon a Russian translation. Besides it is general opinion that 1977 Topos theory is better written and more useful than the elephant which is of course more encyclopaedic and conatins many newer results.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)

    in the clause that defined infinitesimal numbers at nonstandard analysis the formula

    r lowerthan delta lowerthan r

    was displayed. I fixed that to

    minus r lowerthan delta lowerthan r

    Also edited the formatting and wording slightly, to make it more pronounced.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)

    Zoran, it may be your opinion that the 1977 Topos theory is better written and more useful, but I think calling it "general opinion" is unjustifiable.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)

    Out of about 5-6 people from that subject I talked personally so far all were strongly of that opinion. But being that true or not it is not only my opinion that this book is called "Topos theory" and the other is not, the reason why I emphasised that it is NOT default the opposite: that once one says Johnstone's topos theory that one assumes Elephant. If one assumes elephant, one says elephant, isn't this a general opinion ?

    Of course, to people who go as far into subject as you and had already new textbooks like MacLane Moerdijk and lectures at the university, the Johnstone's Topos theory might be not detailed enough for an average using day. Most of the people I talked about were there even in 1970s. If the book is much bigger of course it has more material and for modern usage of an expert is more useful. But if you use it as a first book, you will be lost in the pedestrian and slow style of elephant while the slick and quick style of old book will be usually more attractive. If you are personally very pedantic and do not need fire from the book, but have your own motivation which is strong, than you will pass through long introductions on things like regular categories first. It is difficult to find a student with such a motivation.

    The old Topos theory first circulated as a preprint, before it was actually published. This infuenced much activity in some seminars around the world before peple had the published version.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 13th 2010

    I'm not saying I necessarily disagree with you. What I'm saying is that I think it comes across as abrasive to state something as a fact rather than as an opinion. I would have said "Most people I've talked to are of the opinion that..." instead of "it is general opinion that..."

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJan 13th 2010
    • (edited Jan 13th 2010)

    I agree with you, but I am usually in a hurry working in nlab, and many times this implies some unfortunately rough edges.

    By the way, I may misremember, but I think you gave once a hint relating homological algebra in semiabelian category to some model categorical framework ? Is there a way to look at complexes in semiabelian category in the sense of Janelidze, Manes and others and do cohomology as derived functors for some model category structure ? I would be quite interested to know if there is a sketch of such an argument available.

    I suspect that one should use the Dold-Kan in semiabelian setup and that do some standard sort of model structure on simplicial objects in semiabelian category, but do not know what theorems could be applied to get as far as derived functors.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJan 13th 2010

    Hmm... I don't think that was me. Unless I was just asking the same question that you just asked. I certainly don't know the answer. (-:

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 13th 2010
    • (edited Jan 13th 2010)
    @Zoran

    Tim van der Linden has I think done some stuff on this - he has a paper with two others on the model stricture on the category of internal categories, the ambient category being semiabelian was the prime example, and something more recently. In a hurry, can get the references later.

    Later....

    T. Van der Linden, Simplicial homotopy in semi-abelian categories, J. K-Theory 4 (2009), no. 2, 379-390.
    Preprint arXiv:math/0607076v1

    T. Everaert, R. W. Kieboom and T. Van der Linden, Model structures for homotopy of internal categories, Theory Appl. Categ. 15 (2005), no. 3, 66-94.

    and also, for good measure,

    T. Van der Linden, Homology and homotopy in semi-abelian categories, Ph.D. Thesis, Vrije Universiteit Brussel, 2006, arXiv:math/0607100.
    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJan 14th 2010

    Good, thanks! I actually seen his paper quite a while ago (and was excited then about it) and completely forgotten by now of its existence. I'll need to delve into that real soon (though so many new things appeared in last few days on my plate!).

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeSep 18th 2016
    • (edited Sep 18th 2016)

    I added the following new reference at history of mathematics and nonstandard analysis.

    The relation of the techniques of the pioneers of infinitesimal calculus and the modern nonstandard analysis is discussed in

    • Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, David Sherry, Toward a history of mathematics focused on procedures, arxiv/1609.04531

    At history of mathematics I also added the J. Gray’s book some ideas from which are criticised in the article above.

    • CommentRowNumber14.
    • CommentAuthorgregprice
    • CommentTimeJul 31st 2022

    Added redirect from non-standard analysis

    diff, v37, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2023

    added pointer to:

    • Jerome Keisler, Foundations of Infinitesimal Calculus, Prindle Weber & Schmidt (1976, 2022) [pdf]

    diff, v38, current

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