Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: well-generated triangulated category

Bottom of Page

1 to 11 of 11

  1.  
    • 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

  1.