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nLab > Latest Changes: well-generated triangulated category

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    • CommentRowNumber1.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 7th 2012
    I filled in a page that Urs created: well-generated triangulated category . I am really confused about the documentation of "small" and "compact" and so forth in the nlab. I made some links, but please, if somebody knows what should point to where, go ahead and fix it. I feel like maybe there should be pages just dedicated to "small object" and "compact object" as well "compactly generated category" but there appears to be a lot of stuff documenting such things already, just perhaps not explicitly said.
    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2012

    I can’t tell where the link to “compactly generated” should go because I don’t know what it’s referring to. Compactly generated what? (Pages on the nLab are generally named with nouns, not adjectives.)

    I made the link “small” point to compact object, which contains a section on this notion of smallness in additive categories. The page small object is basically a glorified redirect to compact object; I’m not sure why it needs to exist separately. Are there other pages about compact/small objects that you found confusing?

    • CommentRowNumber3.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 7th 2012
    Sorry. Feel free to remove the "compactly generated" link. I meant compactly generated category I guess. Yeah I mean that's fine. Thanks!
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2012

    No need to apologize; it takes some practice to learn the conventions of a community (have you read the naming conventions section of the HowTo?). What do you mean by a compactly generated category? I’m not familiar with that term either.

    • CommentRowNumber5.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 7th 2012
    • (edited Mar 7th 2012)
    I guess the reason it came up was because I was reading this (specifically just the introduction) and I took it to be a collection of generators of a category in the standard sense except that it was composed entirely of compact objects. I feel like maybe this also comes up in Hovey, Palmieri and Strickland's "Axiomatic Stable Homotopy Theory" (although maybe in a different sense)? In general I feel like I often encounter notions of categories being "accessible" in one way or another because of some nice set of "generators" of some kind.


    Later edit: And indeed in this paper Neeman defines that terminology, and I think it is as I said above (Definition 1.7).
    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2012

    Oh – did you mean to link to “compactly generated triangulated category”? The quote “the appropriate generalization of compactly generated for triangulated categories” at well-generated triangulated category sounded to me as though it was describing them as a generalization to triangulated categories of a notion of “compactly generated category” that applied to non- triangulated categories. But looking at the references, it does seem that Krause in that quote was referring to compactly generated triangulated categories, even though in that case I can’t make any sense of the definite article in “the appropriate generalization” (or, for that matter, the referent of “appropriate” – appropriate for what?).

    • CommentRowNumber7.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 7th 2012
    Yeah you're right. Should have said triangulated. Is the generalization related to the fact that well-generated allows for some choice of a cardinal in some way? I don't know.
    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 7th 2012
    • (edited Mar 7th 2012)

    Does “compactly generated” here mean “locally presentable”, i.e., does ’compact’ mean κ\kappa-compact for some regular infinite cardinal κ\kappa, and ’generated’ in the sense of every object being a κ\kappa-filtered colimit of κ\kappa-compact objects?

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 7th 2012

    I’ll let the question in #8 stand, even though it reflects that I had not been following the thread carefully starting from #1. In any case, it would be nice to see the relation to local presentability clarified.

    • CommentRowNumber10.
    • CommentAuthorjdc
    • CommentTimeJun 20th 2018

    Well-generated is weaker than compactly generated, not stronger. Correct the definition of alpha-small. Fix the link to Krause’s paper.

    diff, v8, current

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeAug 31st 2022

    Both references have spelling well generated rather than well-generated so I added redirect to the original version without a dash.

    diff, v9, current

1 to 11 of 11

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