@DylanWilson so i was basically just getting this from the nlab. haven't thought deeply about it. but i think the two terms have ended up getting conflated in the past few years? anyway, i just take it to mean the stuff from HTT that more or less says you can build objects from colimits of smallish objects

To be specific: ncatlab.org/nlab/show/…

Let R be a commutative S-algebra and let $x\in \pi_n(R)$. I naïvely thought that the commutative R-algebra $R[x^{-1}]$ could be constructed as the usual telescope. But I'm not sure this works anymore: reading EKMM VIII.4 makes me doubt this. It seems like the telescope only constructs $R[x^{-1}]$ as an R-ring spectrum? I don't get it.

@BrunoStonek you can define this telescope but the map are not maps of commutative algebras, so it is not clear a priori that the result should have a commutative algebra structure. It turns out that it does in an essentially unique way but this is a non-trivial theorem.

@JonBeardsley "Locally presentable" (which I think goes back to Gabriel & Ullman) is the term used for 1-categories. Lurie chose to call the infinity analogue "presentable" (and I think he uses the same term "presentable" for the 1-category version as well). In other words, as far as I know, they are just two terms for the same thing. Where have you seen otherwise?

@GeoffroyHorel thanks!

is there an easy description of this commutative algebra structure?

@BrunoStonek You can describe it by the universal property it satisfies. Namely $R[x^{-1}]$ is the initial commutative algebra under $R$ such that the element $x$ is sent to an invertible element.

@CharlesRezk Hey Charles, so I hadn't really officially seen this anywhere. I think I had just seen them both used, and I recalled at some point someone saying "What we call locally presentable, Lurie calls presentable." The nlab has some vague comments about what the "locally" in "locally presentable" is supposed to refer to, but that's about it.

I would probably have just stuck with saying "locally presentable" had Dylan not pointed out that there's a bit of confusion around the difference between this and "presentable."

@JonBeardsley I think the confusion is that "presentable" refers to different things in the two terms. C is "locally presentable" when it is cocomplete and every object in C is "presentable", i.e., every object is a colimit of a small diagram of objects from some fixed small subcategory of C. Thus, "locally" is a synonym for "objectwise" ...

Ah, I see.

Right, so a "presentable category" could potentially be a colimit of small diagram in categories

In Lurie's language, it's the category C itself that is presentable. That is, C is "presented" (in the category of cocomplete categories and colimit preserving functors) by a set of generators and relations.

Right.

Nlab wants to say that "presentable" means a C presented by generators and relations in Cat (categories and functors), rather than (cocomplete categories and colim preserving functors).

Yeah. I mean, ultimately, were I to want to write down an actual proof of something using this language, I'd pick a reference and use the terminology there.

The nlab terminology seems idiosyncratic to me. At least in the infty case, everybody follows Lurie on this.

Yeah.

Does anyone know if it's written down anywhere (or just false?) that a t-structure on a quasicategory C lifts to a t-structure on a quasicategory of K-comodules for K a coassociative coalgebra in C?

Or, more generally, just ANY reference discussing t-structures on categories of comodules?

@Jon I'd guess that it's false in general (e.g. in the dual case, you only get an easily induced t-structure on the category of modules of the ring is connective -- so you'd need something opposite to that condition)

Hm, yeah. I'd certainly be fine with restricting myself to a coalgebra which is connective in the underlying category, but it might be that coconnective or something is necessary.