How should I think of a pointed excisive functor from pointed finite spaces to spaces as a spectrum? Like, what's the intuition for taking a functor and producing my usual idea of a spectrum?

@JonathanBeardsley If F is such a functor the associated spectrum is $\mathrm{colim}Σ^{-n}Σ^∞F(S^n)$, i.e. the Ω-spectrum $\{F(S^n)\}$ where the equivalences $F(S^n)\cong ΩF(S^{n+1})$ comes from the pushout square $(S^n→*)→(*→S^{n+1})$ (I'm writing a square as an arrow of arrows)

Viceversa, if $E$ is a spectrum, the associated functor is just $Ω^∞(E∧Σ^∞-)$

Aha. It does seem like the Ω-spectrum you're describing has to be connective?

No, why?

Er. Hm. For the naive reason that it appears to be indexed by the natural numbers?

All spectra can be indexed only on the natural numbers: remember that the $n$-space sees all homotopy groups starting from the $(-n)$-th

(i.e. the numbering goes "backwards")

Ah yes. The magic of loop spaces.

Or something.

Well, dunno if it's magic. But it works. It's more magic that the restriction to the spheres determines the functor but that isn't super hard to prove (exercise! :P)

Hrm, because it's excisive?

Yeah, of course you've gotta use that. I was more referring to proving the equivalence $Ω^∞(E∧Σ^∞-)=F-$ where $E$ is the spectrum I described above

Well, I already did one of your "easy" exercises today (the thing about Cartesian edges in the tangent category), so I think I'll save thinking about this for tomorrow.

Don't worry, I'm mainly joking :)

I was actually feeding the baby, and getting read to go to sleep when the defining diagram for Cartesian morphisms kind of clicked into my brain and I realized the relevant horn filling properties were the same as the necessary thing for the associated pulled back morphism to be an equivalence.

As David Foster Wallace liked to say about mathematics (and then later about literature), "the click of a well made box."

I am a higher category theory noob, so pardon if the following question is really easy: Is the category of presheaves on connective commutative ring spectra presentable?

In HTT there is the comment that the presheaf category on any small infinity category is presentable, but it is not clear to me that connective commutative algebras form a small ∞-category.

@CWcx No, exactly for the problem you point out: connective commutative rings (heck, even discrete commutative rings) form a proper class, so their category is not small

Usually people use finitely presented connective commutative rings (i.e. compact objects in connective commutative rings) as a replacement

This is equivalent to restricting to those presheaves that preserve cofiltered limits (since the ∞-cat of connective commutative rings is compactly generated)

Ahh ok. The reason I ask is because I came across the following statement: Suppose that X is a presheaf on connective commutative ring spectra. Restriction to the heart defines a presheaf X_{cl} on regular old commutative rings. This defines a functor X |--> X_{cl} from presheaves on connective commutative rings to presheaves on classical commutative rings. I was trying to show that this functor admits a left adjoint.

I was thinking I would need to use the higher categorical adjoint functor theorem for this, but that requires that the categories involved are presentable (I think). So maybe the reference I am looking at is secretly calling presheaves which preserve cofiltered limits presheaves.

Well, you can do this via a trick

The inclusion of discrete rings inside connective rings has a left adjoint (taking π_0), and so restriction has a right adjoint (precomposition with π_0)

Also, I don't like working with presheaves over a big category... it is unsettling (this independently on whether you work with ∞-categories of just 1-categories)

ooohhh that's a nice trick! thanks a lot Denis!

I seem to have confused left and right again. Please be careful when using stuff I wrote...

What do you find unsettling about presheaves on big categories? I think one is led to think about presheaves on connective commutative ring spectra because one wants to try to define spectral schemes and stuff

I also confuse right and left all the time, I'll definitely be careful

@CWcx Set-theoretic issues start becoming really important (e.g. presheaves with values in which sets?) and I don't like worrying about set-theoretic issues

For example, the failure of the adjoint functor theorem is annoying because then you don't have useful stuff like sheafification of presheaves

That's why some people object to the fpqc topology

ahh I see, I definitely want things like sheafification!!

Hey @RuneHaugseng suppose I've got a cocartesian fibration E→B and B has a terminal object x. Then for the fiber over b∊B there's a cocartesian lift E_b→E_x. Can I use the fact that E is the oplax colimit of a functor B→Cat_∞ to conclude that these assemble into a functor E→E_x?

Like, is this something I get for free from the universal property?

(full disclosure, Denis has given me another argument for why I get such a functor, but I was wondering if there was something slick one could do with the oplax colimit stuff)

@DenisNardin Is there any example in which the fpqc topology helps to prove something in algebraic geometry which can't be done with fppf + epsilon?

@SaalHardali I'm the wrong person to ask, but I think Brian Conrad said somewhere on MO that he's not aware of any application that really needs it (I hope I'm not misrepresenting him)

I also vaguely remember reading this by him.

At least until i'm presented such an example i'm willing to tag the whole "for and against fpqc" debate as not particularly mathematically relevant.

I mean debating about size issues might be important and interesting (and in fact is many times). Its just that in this guise it seems pretty irrelevant and uninteresting