@JonathanBeardsley you've got a natural transformation to the constant diagram at E_x, so you get a map E--->B \times E_x by naturality of oplax colimits, then you can project to E_x. Of course, saying you have that natural transformation is just as hard as producing that functor in the end... so this seems like cheating. Probably a better way is to pullback along the equivalence B_{/x}-->B and then use the adjunction with the free cocartesian fibration to get a lil zig-zag that does it

@DylanWilson what about using the left adjoint to the inclusion of the fiber over x?

@JonathanBeardsley good point! I guess it takes a little argument to know that adjoint exists, but it’s still nicer than what I had above

(wasn't my idea!)

But yeah, Tom and Denis eventually concluded that that was probably the best way to do it, with a short argument that the adjoint exists.

Although Denis and I were trying to prove something related (mostly Denis obviously), that maybe you can see a nice solution to. Given the tangent fibration Exc(S_*^{fin},S)→S (where S is the ∞-cat of spaces), it should be true that the inclusion of ANY fiber, Fun(X,Sp)→Exc(S_*^{fin},S) is oplax monoidal.

Where the fiber here is equipped with the pointwise monoidal structure.

(and I think, if the space X is also an E_\infty-algebra, that inclusion should also be lax monoidal)

is Exc(S_*^{fin}, S) given the Day convolution monoidal structure?

(using... smash product on the source? and... product on the target?)

also isn't the tangent fibration supposed to land in Fun([1], Spaces)?

Well yeah, actually I'm slightly confused about what precisely the monoidal structure on Exc(S_*^{fin}, S) is here. It's defined in HA, but not explicitly, and not as Day convolution (as far as I can tell). And yeah, the tangent fibration I guess is Exc(S_*^{fin},S)→Fun(Δ¹,S), but I'm further composing with the map down to S, to think of it as being the "bundle"of categories of parameterized spectra over spaces.

@Jon Sleep seems to have brought at least some counsel. Since X∊S is a coalgebra, S_{/X}→S is oplax symmetric monoidal. Moreover $Exc(S^{fin}_\ast,S_{/X})→Exc(S^{fin}_\ast,S)$ is oplax symmetric monoidal, since it is the pullback of the above functor along the symmetric monoidal functor $Exc(S^{fin}_\ast,S)→S$. Finally the inclusion of the fiber over [X=X] in S_{/X} is oplax symmetric monoidal for the argument Tom explained

Does anyone know a reference spelling out the relationship between finitely presented idempotent algebras in a rigid-compactly generated sm stable category and q-compact open subsets of the Balmer spectrum? (I'd expect in fact it being an isomorphism of posets)

Possibly not the correct place to ask but is anyone aware of a good description of the integral cohomology of $\Omega^2 S^{2n+1}$?

@Niall This is definitely the right place to ask! I would try to attack the problem using the Snaith splitting to see that the cohomology of $Ω²S^{2n+1}$ splits as the sum of the cohomology of $(Σ^{(2n-1)j}E_2(j))_{h\Sigma_j}$, where $E_2(j)$ is the $j$-th space of the $E_2$-operad (i.e. the configuration space of $j$ points in $\mathbb{R}^2$). Maybe people more operadic than me can help you from there

@Niall you can also be a bit more old-schooled and use the serre spectral sequence (+ some other machinery), see section 19 here neil-strickland.staff.shef.ac.uk/courses/bestiary/bestiary.pdf

@Niall yeah fwiw this is extremely the right place to ask that question

@Niall the snaith summands of double-loops of an odd sphere turn into brown-gitler spectra after p-localization. you can extract the integral homology from that, though it's likely to be an eyesore and not whatever you intended by "good description"

What are some cool examples of weight structures? In particular are there interesting weight structures on spectra (or some interesting subcategories like K(n) or E_n-local) besides the one coming from cell decomposition?

@DenisNardin maybe section 2 of this survey has what you want? arxiv.org/abs/1601.03595

@AaronMazel-Gee That seems to do it. Thanks!

Actually, nevermind. They do not talk about finitely presented objects (and they cannot, since I'm not sure that notion is well defined in a tt category(, which was the part I was having difficulties proving

@EricPeterson Really? That's very interesting. Do you happen to know a reference for this result off the top of your head?

@DenisNardin haha yeah, sorry i was glossing over that in your question. can you be more precise about your expectation?

@AaronMazel-Gee If $C$ is a rigidly compactly generated stably symmetric monoidal ∞-category, there is a bijection between Thomason subsets of $\mathrm{Spc} C^\omega$ and idempotent algebras in $C$ (as detailed in section 2 of the reference you gave). I expect that under this bijection the complements of q-compact open subsets correspond to finitely presented idempotent algebras (where a commutative algebra is finitely presented if it is a compact object in $CAlg(C)$)

@CWcx I think a reference for this is Brown and Peterson's "On the stable decomposition of Omega^2 S^{r+2}".

Incidentally I'm going to have to invent a name for rigidly compactly generated stably symmetric monoidal ∞-categories because that's just too much to type

@DenisNardin i see, i would have perhaps naively guessed that it should be idempotent commutative algebras whose underlying object in C is compact. do you have a toy example where we can see what's going on?

@AaronMazel-Gee Take an affine scheme R. Then Spc Perf(R) = Spec R, and under the correspondence V(f) corresponds to R[1/f]

This works for any qcqs scheme and every q-compact open, although describing the corresponding idempotent algebra is slightly tricky in general

@DenisNardin great idea; from skimming the tt-geometry literature, it seems like this is the "good" situation anyways, so following the "good name for the good thing" philosophy, i'd vote that you should just choose something nice that isn't necessarily so descriptive (as opposed to e.g. "rcgssm category")

@DenisNardin okay, great. ah, so maybe you can equivalently characterize the kernel (an idempotent cocommutative coalgebra) as being compact

@AaronMazel-Gee The fiber of R→R_f is very much not a compact object. Or did you mean as a coalgebra?

What is the cleanest way to construct the infinity category of spaces? I know about the coherent nerve approach, and Dwyer-Kan localization

Oh god, we've had this conversation before, and I recall it getting a little heated.

Since it has a universal property you can "construct" it in like one line, if I recall correctly.

Something like... free presentable cocomplete ∞-category containing an object? Or something?

@Dedalus for an explicit quasicategory, I like the approach in Cisinski's book (Definition 5.2.3 in mathematik.uni-regensburg.de/cisinski/CatLR.pdf).

@DenisNardin to prove that the functor S_{/X}→S is oplax symmetric monoidal, are you identifying S_{/X} with Comod_X, and this with the left adjoint to the free-comodule functor?

Well, either way, that seems to work.

I'm still somewhat unclear on the symmetric monoidal structure on $Exc(S_{\ast}^{fin},S)$. Is it in fact the Day convolution? Similar question for $Spectra=Exc_{\ast}(S_{\ast}^{fin},S)$. Also, is the inclusion $Exc_{\ast}(S_{\ast}^{fin},S)\to Exc(S_{\ast}^{fin},S)$ symmetric monoidal? I guess this is really just asking if pointed functors are closed under whatever the tensor product is.

@CWcx this has a pretty good bibliography for all computational things that sound like “loop spaces of spheres”: pdfs.semanticscholar.org/9747/…

the fact i mentioned is there early on, on page 3

@JonathanBeardsley i'd bet a lot you can just directly construct a functor between cartesian $\infty$-operads, no fanciness about laxness-via-adjoints required

@AaronMazel-Gee hm, I'm very anxious about directly constructing just about anything in ∞-categories.

though also clearly the right adjoint S --> S_{/X} is (strictly) monoidal

The map S→S_{/X} here being "cross with X"?

@Dedalus it depends what you mean by "$\infty$-category", and it depends what you mean by "construct". as jon says, it can also be characterized. so can the free abelian group $\mathbb{Z}$ be both constructed by hand and characterized abstractly.

@JonathanBeardsley yes, the right adjoint to the "forget the map to X" functor

I'm a little confused by that actually. Isn't it the case that U(A×_X B)=A×_X B, which is not equivalent to A×B?

I guess if it's right adjoint it should preserve Cartesian monoidal structure, so obviously I've confused myself.

@JonathanBeardsley the symmetric monoidal structure on Spectra = {reduced excisive functors} is the day convolution. as for the parametrized thing, is the projection to Spaces symmetric monoidal as well? (so this is something like the "external" tensor product on parametrized spectra?)

@AaronMazel-Gee well actually I was trying to figure out why the projection to spaces was symmetric monoidal, which was a statement Denis made earlier.

@JonathanBeardsley yes, the projection $A \times_X B \to A \times B$ is the lax structure map. the fact that the left adjoint is oplaxly (a.k.a. left-laxly) symmetric monoidal follows formally from the fact that the right adjoint is strictly symmetric monoidal.

Er. Okay I'm confused. I thought you said the right adjoint, which I'll write as $U:S_{/X}\to S$, is strictly monoidal. So this should mean that $U(A\times_X B)\simeq A\times B$, since the monoidal structure in $S_{/X} $ is $-\times_X -$ and that of $S$ is $-\times -$.

Oh, okay. You're not saying the right adjoint OF $S\to S_{/X}$, you're referring to this map AS the right adjoint.

Nevermind.

I just misunderstood the English, haha.

okay, i'm glad we're on the same page!

But yeah, perhaps I was overcomplicating things by thinking about $S_{/X}$ as $Comod_X$.

well, the beauty of the game is that we all have our own points of view

so back to the parametrized spectra, i would guess that this boils down to two assertions: (i) day convolution is monoidal on the value of $S^0$, and (ii) excisivification doesn't change the value on $S^0$

But alright... yeah the reason I'm interested in the monoidal structure on $Exc(S_\ast^{fin},S)$ is that I want the inclusion of the fiber over $X$, $Fun(X,Sp)\to Exc(S_\ast^{fin},S)$ to be oplax monoidal. Denis gave an argument for this above, but it doesn't identify the monoidal structure on $Fun(X,Sp)$, which I want to be the pointwise one.

(these may be false, even if the assertion is true)

What do you mean by "day convolution is monoidal on the value of $S^0$?"

yeah i guess in terms of the tangent fibration, the pointwise monoidal structure on $Fun(X,Sp)$ is really some sort of convolution between the cocommutative coalgebra structure on X and the commutative algebra structure on Sp

@JonathanBeardsley like, if F and G are functors, then $(F \otimes G)(S^0) \simeq F(S^0) \wedge G(S^0)$

Oh I see.

if true, this would just be a finality statement, which seems easy enough to verify

it's about the slice over $S^0$ with respect to the functor $S^f_\ast \times S^f_\ast \xrightarrow{- \wedge -} S^f_\ast$

Whoa typesetting

how'd i do? at some point the online-tex-compiler thing stopped working for me, and i never got around to fixing it

Looks good!

booya! i've spent too many hours of my life typing tex. it's like playing the piano for me, if i hit a wrong note i can feel it happening even before i hear it

I don't think I even knew the command \xrightarrow

oh yeah, at some point i started preferring over-labeled arrows instead of like $f : X \to Y$

it's nice because $\xrightarrow{\textup{it's infinitely extensible}}$

For sure. I sometimes prefer them, depending on how much it pushes the line spacing around, but yeah, I do like how you can extend it.

uhoh

But I've always used the awful \overset command.

oh bummer, i think that might require some additional package

@DenisNardin Tom gave an argument for the existence of a right adjoint to the pushforward, I believe. I don't immediately see why this right adjoint should be oplax monoidal, unless the pushforward is symmetric monoidal? Also, does this identify the monoidal structure of $Fun(X,Sp)$?

okay so i think here's how you find $Fun(X,Sp)$ with the pointwise monoidal structure inside of the tangent fibration: the object $X \in S$ is (canonically and uniquely) a cocommutative comonoid, and this is selected by a functor $(Fin_*)^{op} \to S$. then, you ask for sections over this that are inert-cartesian... or something like that, i may have the handedness off.

sections over this?

sections of the tangent fibration

So you mean maps $Fin_\ast^{op}\to T_S$? I'm using $T_S$ here to denote $Exc(S_\ast^{fin},S)$.

so over X you select an X-parametrized spectrum E, over XxX you select an (XxX)-parametrized spectrum, and the projections from XxX to X (being selected by inert morphisms in $Fin_\ast$) should be forced to be co/cartesian, so you identify the XxX-parametrized spectrum as the external product $E \boxtimes E$

then, the active morphism $2_+ \to 1_+$ in $Fin_\ast$ should go to the diagonal map $X \xrightarrow{\Delta} X \times X$, and the section over this should pick up a multiplication map $\Delta^*(E \boxtimes E) \simeq (E \otimes E) \to E$

again, i haven't worked out the handedness, this just seems to be a quick way to find the pointwise s.m. structure on $Fun(X,Sp)$ inside of the tangent fibration somehow

Hm.

This is assuming the map $T_S\to S$ is symmetric monoidal?

i don't think that's logically necessary for what i've described, you just have to know that $T_S \to S$ is cocartesian & cartesian with cartesian monodromy given by pullback of parametrized spectra (i.e. precomposition of functors to Sp)

So when you say that the projection maps X×X→X are co/cartesian, what is this with respect to? The symmetric monoidal structure on S?

sorry, i meant that $T_S \to S$ is a cocartesian fibration and a cartesian fibration. as for those projection maps, they are part of the cocommutative comonoid structure (which references the cartesian symmetric monoidal structure on $S$)

ah, no it's something slightly more complicated than what i was describing (because the symmetric monoidal structure on $T_S$ is not cocartesian) -- to do this "for real" i think you'd have to work with the $\infty$-operads that record the symmetric monoidal structures on $T_S$ and $S$. in any case, i am confident that the pointwise symmetric monoidal structure on $Fun(X,Sp)$ can be encoded "inside of $T_S$" in this way

speaking roughly: you name an object $E_1$ in the fiber over $X$, you name an object $E_2$ in the fiber over $X \times X$, you make co/cartesianness demands ensuring that $E_2 \simeq E_1 \boxtimes E_1$, over the diagonal map $X \xrightarrow{\Delta} X \times X$ you associate the multiplication map $\Delta^*(E_2) \to E_1$, and so on