I've known about this ambiguity (w/r/t Goodwillie calculus) for a while, but what is the deal with these things? ncatlab.org/nlab/show/polynomial+functor

Like, why do people care about "polynomial functors," the kind not associated to Goodwillie calculus?

I guess they're related to type theory?

can someone help me translate an infinity-categorical statement into a model categorical one? Does Proposition 7.1.4.11 of Higher Algebra say that there is a Quillen equivalence between HQ-commutative algebras and Q-cdgas with their standard model structures?

if so, what is it exactly (Lurie says it's "canonical") and is it strong monoidal...?

@JonBeardsley yes, one use of them is as presentations for certain types called W-types ncatlab.org/nlab/show/W-type

@BrunoStonek as a Quillen equivalence this is due to Shipley 02, see here: ncatlab.org/nlab/show/…

@UrsSchreiber but that's for associative algebras. In that paper, Shipley says: "In general then, the Quillen equivalence in Theorem 1.1 cannot be extended to categories of commutative algebras. Rationally though, it should hold. We do not consider this extension here though because it would require different techniques."

@BrunoStonek oh, sure, sorry, you were asking about the commutative case.

@JonBeardsley They are also related to operads. For example, there's a paper by Joachim Kock that gives a very nice description of the dendroidal category using polynomial functors. (If you upgrade to infinity-categories this works even better: infinity-operads with space of objects X can be identified as precisely being polynomial monads on spaces over X; this is in a joint paper with Joachim and David Gepner that will appear "soon".)

In case anyone cares (@RuneHaugseng maybe? thanks for your earlier reply!): Possibly I made progress on the question I mentioned a few days back, regarding identifying an algebraic model of rational parameterized stable homotopy theory that unifies the Sullivan/Quillen model of classical rational homotopy theory with the Schwede-Shipley model of rational stable homotopy theory:

Namely Schwede97, theorem 3.2.3 (here: ncatlab.org/nlab/show/parametrized+spectrum#Schwede97 ) states that spectra in rational simplicial commutative algebras augmented over some A are equivalent to spectra in simplicial A-modules. The latter in turn, via Schwede-Shiply, should be equivalent to chain complexes of A-modules. But the theorem essentially only uses that rational simplicial commutative algebras form a right proper simplicial model category.

The same however is true for rational simplicial Lie algebras (here: ncatlab.org/nlab/show/…), and these of course form a model for classical rational homotopy theory.

@BrunoStonek This proposition is not quite giving what you want. You would need to also prove that the model category of commutative HQ-module is a model for the \infty category of commutative algebras over Q (this is true and I think is proved for symmetric spectra in Pavlov and Scholbach's papers on operads in symmetric spectra).

Then since both the model categories of CDGAs over Q and of HQ-commutative algebras are combinatorial there must exist a zig-zag of Quillen equivalences (although there is no method for producing it concretely).

@GeoffroyHorel thanks! I don't understand what "is a model for" means in this context, though. also, you meant commutative algebras over HQ, right?

What I mean by "M is a model for X" where M is a model category and X is an infinity category is that when you do the infinty-categorical localization of M at its weak equivalences, you get an \infinity-category that is equivalent to X

yes I mean commutative algebras over HQ

@UrsSchreiber If I understand correctly, an abstract version of this is that for A a commutative ring spectrum, A-modules is the stabilization of the infinity-category of augmented A-algebras, and if A is an HQ-algebra then the latter is equivalent to rational spaces over and under the rational space corresponding to A.

Rational spaces over A might well be equivalent to functors from A to rational spaces, in which case this stabilization would give functors from A to the stabilization of rational spaces. But is that stabilization the same thing as HQ-modules?

@RuneHaugseng, your first sentence would be the kind of statement that I am after (we'll need to throw in some connectivity assumptions). In your second sentence I am guessing that just as Q-module structure on an abelian group is unique if it exists, so an HQ-module structure on a spectrum is unique up to equivalence, if it exists.

@UrsSchreiber The uniqueness follows from the fact that the multiplication map HQ∧HQ→HQ is an equivalence of spectra (and it also implies that there is only one E_∞-algebra structure on HQ)

But this does not imply that the stabilization of rational spaces is HQ-modules (or if it does I do not see how)

@DenisNardin thanks for the uniqueness. Regarding existence: rationalization is smashing, no?

For spectra, yes. I'm not sure what a smashing localization is in an unstable setting though

I mean a spectrum in rational spaces will be a rational spectrum (no?) and hence will be equivalent to smash product with HQ, and hence will be an HQ-module. Let me know if I am mixed up.

I think that the question is precisely why is a spectrum in rational spaces a rational spectrum. I'm probably missing something though

I am thinking: by the formula for stable homotopy groups of a spectrum, if all the homotopy groups of the component spaces are rational, then the colimit over them is rational, too. Maybe I am making a mistake here?

There are some subtle finiteness issues about exactly what a rational space is. For finite spaces this is the same as all homotopy groups are rational, but I am not sure that it is true in general. To be clear, I am quite sure that there is a functor from spectra in rational spaces to rational spectra, it is the other way round that's confusing me

Augh sorry I should not have said "finiteness", I meant "nilpotency"

What I meant is that if π_1X is not abelian, understanding what "rational" means is a bit confusing

But I'm very far from an expert of this so, I might be missing something obvious

Yes, true, in the statement that I am after, we'd be looking for nilpotent rational spaces, the objects of classical homotopy theory (I guess Bousfield-Gugenheim generalized a bit further even, but I keep forgetting how). So what I am after is a algebraic model of these spaces (any one of the many that exists) and then the algebraic nature of spectrum objects in a slice category of these.

@JonBeardsley Isn't the monad that describes Tambara functors supposed to be a polynomial functor in this sense? (I think that's basically what Mike Hill explained to me once.)

In equivariant Anderson duality, we need to compute levelwise Hom(-,Z) and levelwise Ext(-,Z) and flip restriction/transfer upside down, to get new Mackey functors fitting into the short exact sequence of Anderson dual. Is this procedure can be described as internal hom and internal Ext of Mackey functors in some way?

I mean, "Can this procedure be...", I guess my grammar is quite bad