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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 $\kappa$-compact for some regular infinite cardinal $\kappa$, and ’generated’ in the sense of every object being a $\kappa$-filtered colimit of $\kappa$-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.