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nLab > Latest Changes: pretopos

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMar 21st 2012

    I have incorporated Jonas’ comment into the text at pretopos, changing the definition to “a category that is both exact and extensive”, as this is sufficient to imply that it is both regular and coherent.

    • CommentRowNumber2.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 14th 2017
    • (edited Jul 14th 2017)

    Just a terminological comment for the benefit of future readers:

    in the Elephant, (pretopos)==(coherent,effective,positive). In pretopos, (pretopos)==(exact,extensive)⇔\Leftrightarrow(ditto, with “coherent” added).

    The discrepancy between “effective” and “exact” is terminological: “effective” is just Johnstone’s synonym for “exact” (cf. Elephant p.24)

    This leaves the question whether (coherent,exact,positive)⇔\Leftrightarrow(coherent,exact,extensive); this is true for the maximally-strong, terminological reason that (positive)==definitionally==(extensive). (cf e.g. here for more)

    [ I shied away from making a footnote on this in pretopos, since to achieve a maximally-clear comment like “This is sometimes synonymously stated as “effective and positive” one has to know whether it is true in general that (effective,positive)⇒\Rightarrow(coherent), as is true when “effective,positive” is replaced with “exact,extensive”; the latter I did not stop to to try to ascertain, in particular since “positive” seems to usually only be discussed for categories which are assumed to be coherent in the first place ]

    • CommentRowNumber3.
    • CommentAuthorkevin.watkins
    • CommentTimeMay 13th 2018
    When I view the pretopos page, in some places I see "∞-pretopos" where it ought to be "$\Pi$-pretopos" and "?W-pretopos" where it ought to be "$\Pi$-$W$-pretopos".

    I wonder if the history of the page in the database actually got corrupted? In particular, some related concepts links at the bottom are affected, and I don't remember them looking like that before.
    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2018

    It’s a bug we know about, see here. Hopefully Richard Williamson will be able to fix it.

  5. Thank you very much for reporting this, Kevin. I have now fixed ΠW\Pi W-pretopos and hopefully Π\Pi-pretopos (if there were any occurrences of ’∞\infty-pretopos’ which genuinely referred to ∞\infty-categories, these will now wrongly be Π\Pi-pretoposes as well, but I could not find any such occurrences). Please report any other occurrences of question marks and seemingly erroneous ∞\infty-symbols that you come across!

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 14th 2018

    Thanks for doing these fixes Richard!

    The term “∞\infty-pretopos” is also used in the literature (at least, in the Elephant) for what we on the nLab call an infinitary (1-)pretopos. Apparently we didn’t mention that anywhere though! I’ve now added a note to pretopos.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2019

    Added conceptual completeness to Related concepts.

    diff, v37, current

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 23rd 2020

    Added note in section on colimits that a pretopos that admits pullback-stable countable unions is called a σ\sigma-pretopos in the Elephant.

    diff, v38, current

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2020

    At infinitary coherent category we have remarked on the κ\kappa-ary version. Maybe we should use our terminology, and move the remark about the Elephant’s terminology to that page?

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 24th 2020

    Sure, that’s fine. I feel that σ\sigma-topos is still a better name, though, than ℵ 1\aleph_1-ary pretopos (or ℵ 1\aleph_1-ary regular pretopos), even though it’s not systematic. It’s a special enough case that a special name (with echoes of σ\sigma-algebras) for it is useful.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 24th 2020

    We might have to agree to disagree on that one. (-: Are there interesting examples of σ\sigma-pretoposes that are not infinitary pretoposes?

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 24th 2020

    The category of countable sets (in the presence of AC)? Probably something like countably-presented sets without AC

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJul 25th 2020

    Is that category interesting for a reason other than being a σ\sigma-pretopos?

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 25th 2020

    Not sure (-: I was wondering if there was some kind of computability-related category, or one that a particular type of quasi-finitist might be interested in. Or an arithmetic universe?

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 25th 2020
    • (edited Jul 25th 2020)

    I note that Simpson and Streicher, Constructive toposes with countable sums as models of constructive set theory, https://doi.org/10.1016/j.apal.2012.01.013 prove a nontrivial result about σ\sigma-Π\Pi-pretoposes, relating them closely to a variant of CZF.

    In particular they give the example of the ex/reg completion of the category of modest sets over the second Kleene algebra K 2K_2 as having countable but not small sums (and in fact, is essentially small).

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2020

    Just added a link to our page.

    diff, v39, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 27th 2021

    Added an example: compact Hausdorff spaces.

    diff, v40, current

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 30th 2021

    Added a reference to Marra–Reggio.

    diff, v41, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2021

    Hi Dmitri,

    I have replaced the non-supported bibitem code with code that works here.

    I get the impression that you add non-supported code not only by accident, but also intentionally?

    (Maybe with the idea that it ought to be supported, and that one day it will be supported, and maybe with the idea of nudging its implementation by intentionally producing pages that remain broken otherwise? )

    While I can see that this may make sense for internal development purposes, it seems wrong to do this on live pages, no?

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 31st 2021

    Re #19: I only use the new syntax when I need to reference bibliography items within the page.

    This is the only syntax I know, and I distinctly remember reading here on the nForum that it will be implemented. Indeed, \ref/\cite have been implemented for a long time, and \cite/\bibitem were mentioned in the same thread. \cite already appears to be partially implemented.

    Is it so difficult to add a similar line of code for \bibitem?

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeFeb 1st 2021

    Is it so difficult to add a similar line of code for \bibitem?

    I don’t know. But it seems wrong to take innocent readers of our pages as hostages in this debate. All they see is a broken page.

    This is the only syntax I know

    The working syntax is explained at HowTo. I guess most people pick it up by a glance at source code examples.

    To provide a reference with an anchor name do

      * {#AuthorYear} Author, _Title_, Journal, Year (Links)

    and then to cite it type

      See [Author Year](#AuthorYear)
    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 1st 2021
    • (edited Feb 1st 2021)
    So it would see \bibitem{...} should be made an abbreviation for * {#...} and \cite{...} should be made an abbreviation for [...](#...).

    Once the syntax is implemented, existing \bibitem's would automatically produce correct results.
    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeFeb 1st 2021
    • (edited Feb 1st 2021)

    If you insist on proceeding this way, maybe you could add a note to the pages of the following content:

    This page is broken intentionally, using syntax in its source code that seems desireable but is not currently supported by our software. We are desperately looking for more volunteers to help work on our code base. If you are bothered by this page not rendering properly, yet persuaded that it ought to and knowledgeable about programming and willing to lend a hand, then please contact our admin Richard Williamson, who will direct you on how to proceed.

    • CommentRowNumber24.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 1st 2021
    • (edited Feb 1st 2021)

    If we could add these two lines (* {#…} and […](#…).) at least to the “Markdown+itex2MML formatting tips”, which show up on the right side when one is editing a page, this would go a long way in helping users remember them.

    I am talking about this file: https://github.com/ncatlab/nlab/blob/master/app/views/markdown_table.html.erb

    This does not require any programming.

    • CommentRowNumber25.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 1st 2021
    • (edited Feb 1st 2021)

    Just a quick note (in haste!) that I worked on this yesterday, intending to complete the work on citations/referencing that I began some time ago. Something intervened which prevented me from finishing yesterday, but I should be able to do so over the next few days. The coming functionality is more comprehensive than just the bibitem aspect :-).

  26. Re #19 etc: as announced here, this is now implemented.

    • CommentRowNumber27.
    • CommentAuthorGuest
    • CommentTimeOct 15th 2022

    The article states that

    A Π\Pi-pretopos is a pretopos which is also a locally cartesian closed category. (A Π\Pi-pretopos is automatically a Heyting pretopos.)

    Is every locally cartesian closed coherent category a Heyting category?

    • CommentRowNumber28.
    • CommentAuthorJonasFrey
    • CommentTimeOct 15th 2022
    Yes! Simply because Pi-functors preserve monomorphisms.

  29. added examples of certain universes from dependent type theory as pretoposes

    Anonymous

    diff, v45, current

    • CommentRowNumber30.
    • CommentAuthorThomas Holder
    • CommentTimeJun 7th 2023

    Added the category of at most countable sets as an example of a Boolean pretopos that is not a topos.

    diff, v47, current

    • CommentRowNumber31.
    • CommentAuthorBryceClarke
    • CommentTimeOct 13th 2023

    Added an example of Ho(Top).

    diff, v49, current

    • CommentRowNumber32.
    • CommentAuthorvarkor
    • CommentTimeOct 2nd 2024

    Added a characterisation of the infinitary pretoposes.

    diff, v52, current

1 to 32 of 32

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