@DenisNardin The original meaning of "stack" (or rather the French "champ") is actually a sheaf of categories. For example Lurie uses the word in this sense in chapter 10 of SAG.

@MarcHoyois I know that Lurie occasionally uses it in that sense, but do you know a place where that was used classically in the AG literature?

(not a rethorical question, if it exists I'd like to see it)

It seems that SGA1 leaves the liberty for the fibers to be categories, but then it does not give any example where this happens

Anyway I was objecting more to the distinction stack vs sheaf, that really risks AGs to get confused

@DenisNardin Maybe an example is "A Proof of Jantzen Conjectures" at the end of page 3 the term "a stack of categories" is used. Maybe not the best example though.

I am weary of the hard-to-pronounce and unsearchable prefixes like '(∞,n)-'. I'd also like to try to implement the 'Bourbon Seminar conventions', wherein everything that can be interpreted in the maximally homotopy-coherent sense should be understood in that sense. (I'm finally writing a paper in which it makes sense to do that.)

@ClarkBarwick If I had a vote, I'd vote for inventing a new word. Pile is fine, schlober is also fine if it's already in the literature (although I'd prefer a more easily translatable word). I think we should not be too afraid of inventing new terminology

You always have a vote as far as I'm concerned, Denis!

(Also, on a similar note I'm going to start using Frobeniusse because Frobenii always irked me. At the very least it should have been Frobenia or Frobeniua)

Denis, haha, I was thinking the same thing this morning when I read the chat.

@ClarkBarwick I see the appeal, but what do you do for commutative rings?

If one insists that 'commutative algebra' and 'E∞-ring' are synonyms, then if I seek a commutative algebra with only pi_0, I can ask for a discrete commutative algebra.

true

Actually, maybe you're complaining that I should have a language that carefully distinguishes connective commutative algebras over Z in this sense from the homotopy theory of simplicial commutative rings over Z. Can one use the words 'algebra' and 'ring' systematically enough to make that distinction?

I was mainly concerned with the fact that commutative rings over Z are not full in E∞ ring spectra over Z

At least that's what I remember from Adeel's lectures here when I asked him

If I remember correctly, he said that you only get a well-behaved comparison in characteristic zero

Commutative rings are fully faithfully embedded into connective E_∞-algebras. Simplicial commutative rings are not

ah

You can see it by noticing that connective E_∞-algebras are algebras for a Lawvere theory whose 0-truncation is the Lawvere theory of rings

Or, more concretely, that $π_0\mathbb{Z}\{x_1,...,x_n\}=\mathbb{Z}[x_1,...,x_n]$

Yeah, makes sense

I wonder how one should think of simplicial commutative rings in this 'everywhere homotopy-coherent' situation

There's a statement in SAG or HA about it, but it's not really worked out in much detail

I strongly suspect that they are some form of PD-rings, although I don't know of a precise statement to that effect

PD?

Divided powers ("Puissances divisées")

ah

This is mainly because the homotopy groups of a simplicial commutative ring are naturally a PD-ring with respect to the ideal of positive degree elements

It also gels pretty well with the ethos that "SCRs are nilpotent deformations of their π_0"

So there should be some kind of operad for which they are algebras that has more operations?

I don't think PD-rings are algebras for an operad

ams.org/tran/2000-352-09/S0002-9947-99-02489-7/…

They're usually PD-algebras for an operad!

Interesting!

There's no way it's literally true that SCRs over F_p are PD algebras for the E∞-operad on DVect(F_p) .... right?

I don't think so

Well, D_{≥0}(Vect(F_p)) ... and I have to murmur something about units.

But even then that's way too much to hope for, right?

Dunno. I remember trying to prove something in that direction and failing, but who knows?

By the way, I saw a talk by Nikolaus where he showed that K-theory is the universal way to get a Λ-ring from a ring (homotopy-coherently as well). I was wondering if this way that Λ-rings show up has any relationship with Borger's approach to absolute geometry

maybe I misunderstood the talk though

I remember Akhil telling me that SCRs are monadic and comonadic over connective E_∞-algebras, so we should be able to "describe" them somehow. But I don't know how

Interesting

So here's an old idea I never quite sorted out: is there a theory of 'genuine' (monochromatic) ∞-operads in spectra, wherein each O(n) is a genuine Sigma_n-spectrum? (Ordinary monochromatic ∞-operads would give you genuine ones by taking coBorel.) You could then talk about genuine algebras as well. The project would be to construct a genuine lift of the E∞ operad on connective Z-modules, and see that genuine algebras for it are SCRs.

I wonder if Akhil meant they're monadic and comonadic in the slice under Z or in all connective E-infty ring spectra. If it's a relative statement, I wonder if you'd get anything different/interesting if you could tell a story about PD S-algebras

@ClarkBarwick How would you define the monad associated to such a gadget?

I'm not exactly sure how to make it precise, but I'd want to take the coproduct of categorical cofixed points ...

My problem is that connective Z-modules do not have a norm functor

Well, I'm not quite getting into the PD picture yet.

The implementation I'm suggesting might be bogus, but the idea is just that you're trying to identify the two notions of free algebra, one that comes from taking homotopy cofixed points of the Sigma_n action on the unit, and the other 'strict' variant that is taking some 'genuine' cofixed points of the Sigma_n action.

I think Akhil and Lukas have a project that studies generalized SCR’s (which include non-connective versions of SCRs). So y’all should get in touch with them since they’re probably super familiar with that (co)monad

In his paper Homological algebra of homotopy algebras, Hinich shows that if two dg algebras are quasi-isomorphic, they have equivalent derived categories. Is it true that if X is a perfect complex of (A,B)-bimodules giving an equivalence D(A) ---> D(B) one can produce a zig-zag of quasi-isomorphisms connecting A and B? That is, does derived equivalence imply that A and B are isomorphic in Ho(dg-Alg)?

There is a quasi-iso A --> RHom_B(X,X), for example. So I am guessing using this and an inverse to the equivalence one may produce something of the shape A --> M --> N <-- M' <-- B

@PedroTamaroff Don't matrix algebras oven A have the same derived category as A?

Yes, they are Morita equivalent to A. I presume I want something weaker then.

@DenisNardin i second this, and want someone to figure it out...

in general the whole theory of "PD-operads," which collapses in char zero, is totally undeveloped