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Atrium > Mathematics, Physics & Philosophy: locally presentable category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 7th 2010

    I dropped some query boxes on some basic questions and observations over at locally presentable category. I ask experts to help bring me up to speed.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 7th 2010
    • (edited Feb 7th 2010)

    Not that I've answered your questions, but I consolidated the discussion on terminology there. (Mike and Urs may want to note that Reid Barton responded in December to Mike's final point, so the discussion is still open.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2010
    • (edited Feb 8th 2010)

    Thanks, Todd, for these remarks. Yes, given what i understand I would think that the answers are Yes, notably to the first query box. Notably that would then harmonize with the statement for oo,1-categories.

    But I would like to cross-check this with exactly how this is stated in "Locally Presentable and Accessible Categories". I had a Google books copy of that book with some of the pages available, and what I did put into this entry I gotout of these pages that I saw on Google books. It was a bit of a pain looking at the the book that way, and I can't claim to have a good overview of the content of that book. Given how straightforwward the statements you suggest should be true are, one would think they'd be mentioned this way in the book, but I can't remember having seen them. So I would just like to cross check again. Possibly they have some extra funny condition of what counts as a sketch for them?

    Trouble is, now that I search around, I can't even seem to find the Google-books version of the book. (??)

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 8th 2010

    Thanks, Urs. Yes, I can't find the Google Book either. But I think I have some answers. There is in particular an important theorem found in the book on accessible categories by Makkai and Paré (which I don't have), to the effect that the 2-category of accessible categories and accessible functors admits weighted bilimits. This implies in particular that if $A$ is locally presentable and $C$ is small, then $A^C$ is locally presentable, which answers one of my questions. I'm not yet sure what the answer is to this in the locally finitely presentable case.

    Regarding the question about reflective subcategories, the impression I'm getting is that the answer is 'no' in general; the answer would be 'yes' if the inclusion is an accessible functor (which I think would boil down to the inclusion preserving $\kappa$-filtered colimits, if the containing category is $\kappa$-accessible). This 'yes' is actually another corollary of the Makkai-Paré weighted bilimit result; more generally, the Eilenberg-Moore category for an accessible monad (on an accessible category) is accessible by this result.

    I found all this in an online copy of Tibor Beke's thesis. I would still like to be further educated by experts around here.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2010

    Hi Todd,

    good. Would you have the time to put in the answers to your questions, as far as you have them now, into the entry? Or should I do it?

    By the way, to get math displayed here, it needs to be enclosed in a pair of double dollar signs. (And may not have any line breaks in between these.)

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 8th 2010

    No, I can do it Urs. Thanks for the offer.

    I knew about the double dollars here, but forgot because I was in a hurry. Thanks though.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 8th 2010

    Okay, removed the old query boxes, added some content, and also added some new query boxes (which are mostly notes to self) to locally presentable category.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 8th 2010

    Replied to Reid.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 9th 2010

    Todd,

    thanks, this is a major improvement to what we had. Nice. It's very good to have you around.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 9th 2010

    Asking a question here, because it is slightly quicker.

    Can the 'locally' in locally presentable category refer to properties of the slice categories C/a or a/C? Or more precisely, can a locally presentable category be defined as a category C such that the slice categories C/a (or a/C, as necessary) have such and such a property?

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 9th 2010

    @David: I don't think so.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 9th 2010

    I made a small fix - the κ in "locally κ-presentable category" doesn't mean that all objects are κ-compact, only that the generating set can be taken to be so.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 11th 2010

    Ok thanks

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2012

    for completeness, I have added to locally presentable category what should have been there but wasn’t: the remark after the statement about sheaf toposes being presentable that it’s of course the set of representables that serve as generators, being κ\kappa-small for κ\kappa bounded below by the cardinality of morphisms of the site (now prop. 6 there).

    • CommentRowNumber15.
    • CommentAuthorZhen Lin
    • CommentTimeAug 29th 2012
    • (edited Aug 29th 2012)

    @Todd (#4):

    I’ve added a proof that [C,A][C, A] is locally finitely presentable when CC is small and AA is a locally finitely-presentable category.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2012

    Thanks. I have added some links.

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 29th 2012
    • (edited Aug 29th 2012)

    Thanks, Zhen! (At the time of writing #4, I was evidently far less fluent in this part of category theory than I feel nowadays! (-: )

    Edit: And now I just tried to look and see the proof you wrote, but I can’t find it anywhere. I didn’t see it under Recently Revised.

    Edit: Ah, there it is – never mind. Thanks again.

    • CommentRowNumber18.
    • CommentAuthorZhen Lin
    • CommentTimeAug 29th 2012

    I changed “finitely-accessible” to link to accessible functor instead of accessible category. Basically, I’m trying to say that the inclusion C↪[B,Set]C \hookrightarrow [B, Set] is finitary… but the wording is ambiguous and should be improved.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2012
    • (edited Aug 29th 2012)

    Thanks, right. I wanted to link to something like accessibly embedded subcategory, but we don’t have such an entry. We should, though.

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