geometric realization of simplicial spaces does not commute with infinite products. but how about just for Segal spaces? it feels like this should be the case, but i don't know how to prove it. (i'm most interested in a proof that never uses models (e.g. Kan complexes) or facts about sets.)

I just realized that I know of no notion of a filtered $E_{\infty}$ ring spectrum which would naturally give rise to a multiplicative spectral sequence in homotopy. In all the multiplicative spectral sequences I know the multiplicative structure needs to be added by hand. I think i have an idea about how to define it but mainly i'm interested in whether this has been adressed and if so where?

@SaalHardali If you take a cosimplicial E_infty ring spectrum, do you then believe that there is a multiplicative structure on the spectral sequence?

hmm, i think so, though i think there should be a more general notion, for instance on which would apply to a filtered dg-algebra. Maybe you can apply this by using the equivalence in char 0 but it's still not really satisfactory...

@SaalHardali what about the filtrations associated to building E∞ rings by E∞ cells, like Mandell does?

These "multiplicative" Postnikov towers or whatever.

After a conversation with someone about this I think a reasonable first definition would be lax monoidal functors out of the symmetric monoidal poset Z. This seems like the bair minimum you want out of a Z-filtered algebra and I think its enough...

I feel like something like this is developed in Sean Tilson's thesis.

Though not in the ∞-categorical language.

Thanks, is it available online?

I think here: arxiv.org/pdf/1602.06736.pdf

Maybe it would be best to spell out explicitly that the end goal i'm after is to be able to speak about an infinity category of filtered modules over a filtered algebra.

The infinity part is not crucial for me. If its a not too terrible model category its also okay i guess.

@SaalHardali It's not that hard to define filtered objects. You just take $Fun(\mathbb{Z},C)$ equipped with Day convolution

That's fine though that doesn't give you filtered algebras.

Why not? Just take algebra objects there

A filtered algebra is not a filtered object in the category of algebras

Indeed it is not. It is an algebra object in the category of filtered spectra, say

This would be an algebra object in the category of filtered objects?

Ah wait, you're saying i just take first filtered then algebra

Yep

hmmm, I guess there's no reason why that shouldn't work

I'm pretty convinced that a filtered ring spectrum will give you a multiplicative spectral sequence, although the proof is likely combinatorially nontrivial and likely uses some of the old Cartan formalism

And actually I think its equivalent to my definition from before

This is the "standard" way of defining filtered rings in a homotopical context, as far as I can tell

Someone needs to explain to me the part at the end of John Rognes' Galois theory book where he describes taking an automorphism of Morava E-theory and producing a map S[BU]-->E_n, and how this is describing a relationship between the Morava stabilizer group S_n (as the Galois group of E_n over L_{K(n)}S) and BU as the Hopf-algebra of functions on the Galois group of MU over S...

That's cool, so now filtered modules over a filtered ring are just modules over it in this category of filtered objects?

Yeah

That actually makes a lot of sense

Funny that it wasn't obvious to me before you said it

I got carried away with finding an explicit description of the objects I needed.

You also get a nice associated graded functor by taking the adjoint of the functor sending a graded object to the corresponding filtered object (formally you get it via a push-pull kind of construction)

Thanks!

And then Jack Morava says that some kind of geometric realization of a suitable category of Tate motives is supposed to produce a category of complex oriented spectra.... @_@

My quib earlier about cosimplicial objects was that the equivalance between cosimplicial spectra and filtered spectra (in the correct direction) is not symmetric monoidal for the pointwise symmetric monoidal structure on cosimplicial spectra and the day convolution structure on filtered spectra

So while algebras under day convolution in filtered spectra may or may not be what you are interested in, they are not the objects studied in Sean Tilson's thesis.

(which I contend come from algebras in cosimplicial spectra)

hmmm, in what i have in mind algebras i. filtered spectra with day convolution makes more sense but it does trouble me that these approaches are not equivalent, at least not obviously so...

Oh sure. I was just connecting dots using words like filtered and multiplicative spectral sequence.

What happens is that if you take an algebra in filtered spectra then the higher multiplicative structure on the spectral sequence gives you algebraic steenrod operations acting on the spectral sequence. This also happens with an algebra in cosimplicial spectra, but you get fewer such operations.

If you look at the operations on the adams spectral sequence, they are of the latter kind, not the former.

@TomBachmann I see. That's actually a bit reassuring because the ASS comes to us naturally as a cosimplicial ring rather than a filtered ring so its reasonable that you lose some stuff when you forget this structure.

@TomBachmann So if i understand correctly the upshot is that there are multiplicative spectral sequences which come from cosimplicial algebras and those that come from filtered algebras. And the structure on the former is more rich than the structure on the latter. Do you know where i can read about these steenrod operations in the cosimplicial and filtered situations and the differences? Its not really what i was initially interested in though it interests me regardless...

Wait! I think i got it the other way around and now i'm utterly confused. Are you saying that the SS of a cosimplicial spectrum has less structure than that of an N-filtered spectrum?

(ring spectrum of course*)

Yes, the cosimplicial one has less structure.

You can read Sean Tilson's thesis for something about these operations. He refers to a chapter of Bob Bruner in the H-infinity book.

Apparently some work of Phil Hackney shows that the structure coming from a cosimplicial algebra is strictly less than the filtered one, but I do not totally understand that (Tyler Lawson told me so).

I do not know any good references for this. In fact I don't really know anything more than what I wrote!