Timeline for (infinity,1)-categories directly from model categories
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2015 at 1:25 | comment | added | user62675 | Just a remark: the slogan is Proposition A.3.7.6. of Higher Topos Theory. | |
| Dec 14, 2009 at 4:23 | comment | added | Mike Shulman | That's an unfortunate use of "continuous," because "continuous" is well-established in ordinary category theory to mean "(small) limit-preserving," while "cocontinuous" means "(small) colimit-preserving." Also, I would have thought the natural class of functors between locally presentable (∞-)categories would be the accessible functors, i.e. those preserving k-filtered colimits for some k. I can think of lots of natural functors on locally presentable (∞-)categories that are accessible but don't preserve arbitrary colimits. But anyway, I was thinking about arbitrary functors. | |
| Dec 13, 2009 at 4:32 | comment | added | David Ben-Zvi | By continuous I mean colimit preserving (which seems to be the natural class of functors between presentable categories), and Lurie describes a presentable oo-category of continuous functors between presentable oo-categories -- was your comment an assertion regarding arbitrary rather than colimit preserving functors? | |
| Dec 13, 2009 at 3:38 | comment | added | Mike Shulman | @David: I think probably "continuous" is the problem, although I'm not sure what exactly you mean by it. | |
| Dec 13, 2009 at 3:31 | comment | added | Mike Shulman | The problem with "presentable" is that there is a general notion of a "presentable object" in a category, namely one such that hom(x,-) preserves k-filtered colimits for some k. So "presentable category" obviously means "presentable object in Cat." The term "locally presentable category" indicates that it is not the category itself, but its objects, which are presentable. | |
| Dec 12, 2009 at 22:02 | comment | added | Charles Rezk | @Reid: I presume so. Category theorists came up with the name "locally presentable" a while ago - they must have had a reason for distinguishing this from plain "presentable", no? | |
| Dec 12, 2009 at 21:44 | comment | added | Reid Barton | (This is getting a little off-topic, but are there people who use "presentable" and "locally presentable" non-synonymously?) | |
| Dec 12, 2009 at 20:51 | comment | added | David Ben-Zvi | Aren't categories of continuous functors between presentable \infty,1-categories presentable? (is "continuous" the problem? or is "locally" the point you were making?) | |
| Dec 12, 2009 at 20:45 | comment | added | Mike Shulman | Another reason to generalize past the locally presentable case is that you'd like an $(\infty,1)$-category of functors between two given $(\infty,1)$-categories. | |
| Dec 12, 2009 at 20:44 | comment | added | Mike Shulman | Ordinary category theory would be pretty limited if it could only talk about locally presentable categories! How do you even say something like "C is complete" i.e. "C has limits indexed by small categories" if you don't know what a "small category" is? The same applies in the $(\infty,1)$-world. | |
| Dec 12, 2009 at 20:22 | comment | added | David Ben-Zvi | My favorite small $(\infty,1)$-categories are those of perfect complexes on a scheme say, or more generally compact objects in a stable presentable $\oo$-category. Anytime you want to put finiteness conditions on your modules/representations/sheaves/spaces etc you leave the presentable world (though you can usually study them using presentable tools by say passing to Ind-categories). | |
| Dec 12, 2009 at 16:20 | comment | added | Charles Rezk | I dunno. Take a Lie group G. This is a topological group, so it is an ($\infty$,1)-category (use the "model" of ($\infty$,1)-categories as topologically enriched categories). It is small, and is not presentable. Lie groups are interesting, and one might want to think about "representations", which in some general formulation are functors from G to some fixed ($\infty$,1)-category C. You can think about this using model categories, if C is a topologically enriched model category (there's a model category of enriched functors $G\to C$). But it's a bit awkward as a general formalism. | |
| Dec 12, 2009 at 15:58 | comment | added | Harry Gindi | We can't use the usual universes trick is what you're getting at? Or do you mean small in the sense of not being complete and cocomplete? Why would we want to look at small $(\infty ,1)$ categories (this question may be dumb)? If they're not small, we can't define a suitable $(\infty,2)$ category of $(infty,1)$ categories? | |
| Dec 12, 2009 at 15:50 | comment | added | Charles Rezk | Well, if you're interested in small ($\infty$,1)-categories, you need something more general than model categories. Small ($\infty$,1)-categories are (almost) never presentable. | |
| Dec 12, 2009 at 15:46 | comment | added | Harry Gindi | Is there any reason to generalize past presentable $(\infty,1)$ categories other than for the sake of generalization? I presume so, but I don't have the background to know why we'd want to. | |
| Dec 12, 2009 at 15:39 | vote | accept | Harry Gindi | ||
| Dec 12, 2009 at 15:25 | history | answered | Charles Rezk | CC BY-SA 2.5 |