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nLab > Latest Changes: locally presentable category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTobias Fritz
    • CommentTimeNov 9th 2012
    • (edited Nov 9th 2012)
    In the definition, the article states "every object in C is a small object (which follows from 2 and 3)". The bracketed remark doesn't seem quite right to me, since neither 2 nor 3 talk about smallness of objects. Presumably this should better be phrased as in A.1.1 of HTT, "assuming 3, this is equivalent to the assertion that every object in S is small".

    Am I right? I don't (yet) feel confident enough with my category theory to change this single-handedly.
    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 9th 2012

    Yes, you’re right. Someone should go through the article again and fix the mistakes (I may do so in the near future).

    • CommentRowNumber3.
    • CommentAuthorTobias Fritz
    • CommentTimeNov 10th 2012
    Thanks, Todd, I fixed this. (I feel a bit stupid for having my name there on the bottom although I've barely done anything...)
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 18th 2019

    Added well-poweredness and well-copoweredness to properties

    diff, v69, current

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 8th 2020

    Changed “presentable” to “locally presentable” in first paragraph, to reduce the chance that people think a second distinct notion is being introduced.

    diff, v71, current

    • CommentRowNumber6.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 8th 2020
    • (edited Feb 8th 2020)

    Why is Emily Riehl’s definition of “locally presentable” category in Categories in Context simpler than the nLab definition? Are they equivalent?

    The nLab says a categorry 𝒞\mathcal{C} is locally presentable iff

    1. 𝒞\mathcal{C} is a locally small category;

    2. 𝒞\mathcal{C} has all small colimits;

    3. there exists a small set S↪Obj(𝒞)S \hookrightarrow Obj(\mathcal{C}) of λ\lambda-small objects that generates 𝒞\mathcal{C} under λ\lambda-filtered colimits for some regular cardinal λ\lambda.

      (meaning that every object of 𝒞\mathcal{C} may be written as a colimit over a diagram with objects in SS);

    4. every object in 𝒞\mathcal{C} is a small object (assuming 3, this is equivalent to the assertion that every object in SS is small).

    Riehl’s definition is that 𝒞\mathcal{C} is locally presentable iff it is locally small, cocomplete, and for some regular cardinal λ\lambda it has a set SS of objects such that:

    1. Every object in 𝒞\mathcal{C} can be written as a colimit of a small diagram whose objects are in SS;

    2. For each object s∈Ss \in S, the functor preserves λ\mathcal{C{\lambda-filtered colimits.

    So, the nnLab definition seems to include two extra conditions. First, that every object in 𝒞\mathcal{C} can be written as a colimit of a λ\lambda-filtered small diagram whose objects are in SS. Second, condition 4, which seems redundant since it seems to be built into condition 3, at least if λ\lambda-small implies small.

    Surely there should be some way to simplify this nLab definition!

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 9th 2020

    The λ\lambda-filtered condition is in Adamek and Rosicky’s book, so either Riehl left it out by accident or somehow she noticed it could be safely dropped - I don’t see how.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 9th 2020

    Quick reaction: I’m not sure what 4. is doing there either, and I agree that Emily’s 1. needs to be fixed (my taste would be to have her 2. coming before 1., i.e. say what the objects in SS are doing before describing other objects in terms of SS).

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeNov 3rd 2020

    Added a corollary: locally presentable categories are complete.

    diff, v72, current

    • CommentRowNumber10.
    • CommentAuthorjdc
    • CommentTimeNov 19th 2021

    Item 4 of the definition of locally presentable category (Def 2.1) was there before Kevin Carlson added “λ\lambda-small” to item 3. So item 4 should be changed to a remark. I’ve just done this.

    I also think that “λ\lambda-small” should be changed to “λ\lambda-compact”, to be consistent with the rest of the page and linked pages, unless there is a subtle difference between the two that I’m not aware of. I’ve made this change as well.

    I also removed a parenthetical remark in item 3 that was no longer correct and wasn’t adding anything.

    diff, v77, current

    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeDec 17th 2021

    Clarified remark about “locally presentable category” versus “presentable category”.

    diff, v80, current

    • CommentRowNumber12.
    • CommentAuthorvarkor
    • CommentTimeJul 29th 2022
    • (edited Jul 29th 2022)

    Accidentally edited this page instead of creating a new one. I’m rectifying my mistake now.

    • CommentRowNumber13.
    • CommentAuthorvarkor
    • CommentTimeJul 29th 2022

    Revert accidental change.

    diff, v81, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    maded explicit the example of Grothendieck abelian categories (here)

    diff, v84, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023
    • (edited Apr 21st 2023)

    added (here) a proper reference (thanks to discussion here) for the fact that Func(small category,loc pres category)Func(\text{small category}, \text{loc pres category}) is locally presentable.

    Also re-arranged the list of examples slighty to be more systematic. But the whole arrangement remains a little odd.

    diff, v88, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2023

    added (here) statement of sufficient conditions for a Grothendieck construction to be locally presentable (from MO:a/102083)

    diff, v89, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023
    • (edited Apr 27th 2023)

    On closer inspection, I wonder that MO:a/102083 has a misprint:

    The condition on the pseudofunctor Φ:B op→Cat\Phi \,\colon\, B^{op} \to Cat seems to be to preserve 2-limits, not 2-co-limits — this according to Def. 5.3.1 (4) in Makkai & Paré 1989, who phrase it that Φ\Phi must take colimits in BB to pseudo-limits in CatCat.

    diff, v90, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023
    • (edited May 3rd 2023)

    added pointer to

    for discussion of local presentability in the context of enriched categories

    diff, v92, current

  19. Linked new page for Field, the category of fields.

    diff, v98, current

    • CommentRowNumber20.
    • CommentAuthorqschroed
    • CommentTimeAug 1st 2024

    updated reference from compact -> compact object

    diff, v100, current

    • CommentRowNumber21.
    • CommentAuthorchaitanyals
    • CommentTimeOct 23rd 2024

    In the table under the section Related Concepts, the localization theorem for locally presentable 1-categories is named "Adámek-Rosický’s theorem". This seems misleading, since not only is the linked theorem proved in Gabriel—Ulmer (1971), Theorem 8.5(c) and Remark 8.6(c), but there is also a definition of (Gabriel—Ulmer, paragraph 8.3), as well as description of the properties of (Gabriel—Ulmer, Lemma 8.4) the saturated class of morphisms that are inverted by a cocontinuous localization. Whereas (if memory serves) Adámek—Rosický’s book does not study the reflection as a cocontinuous localization, and so does not describe the properties of this class of maps.

    I would submit an edit, but I don’t know how to edit the table.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2024

    To edit the table, edit the entry locally presentable categories - table.

    • CommentRowNumber23.
    • CommentAuthorchaitanyals
    • CommentTimeOct 23rd 2024

    Thanks!

1 to 23 of 23

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