actually, i just talked with marc hoyois and apparently it is unknown how to recover topological K-theory in terms of algebraic K-theory, though certainly it is hoped that this is somehow possible. but so maybe my previous suggestion was premature.

@AdrianClough right. among other things $P(V)$ is presentably monoidal: that is, presentable and monoidal and the monoidal structure structure commutes with colimits separately in each variable. the presentability guarantees that it can be presented by a model category, and the presentably-monoidalness tells you that you can even hope for a monoidal model category. [cont.]

(note that these by definition require the monoidal structure to be a left quillen bifunctor, and these present bi-cocontinuous bifunctors of $\infty$-categories, so this is necessary a priori to have any hope of presenting your monoidal $\infty$-category by a monoidal model category.)

of course, the yoneda embedding preserves limits but destroys colimits, but in practice i think this shouldn't pose so many issues. that is, i'm pretty sure that all (enriched) universal properties only refer to limits in the enriching category.

@AdrianClough in re this proposal, you're probably right about rectification. honestly i'm still amazed that $\infty$-categories (which i think of as being fully homotopy-coherent, no strictness about them) can always be presented by Top-enriched categories. but my impression is that rectification for (possibly many-object) associative algebras is generally easy, while for commutative algebras it's way more delicate.

anyways, what exactly would you be hoping to use such a strictification for?

I've been having a lot of trouble with the following. Suppose that $C$ is a $\infty$-category with finite coproducts (maybe even additive) and that $X: C^{op} \rightarrow Sp_{\geq 0}$ is a sheaf of connective spectra. Suppose moreover that $X$ is hypercomplete and that for each $k$, the homotopy group $\pi_{k}X$ sheaves send coproducts in $C$ to products of sets. Does $X$ itself necessarily send coproducts in $C$ to products of spaces?

I can imagine I could prove this by induction on $n$ when $X$ is $n$-truncated. This should then also prove the statement in the case when Postnikov towers in hypercomplete sheaves on $C$ converge. Would hypercompleteness in itself be enough?

@AaronMazel-Gee I thought I remember reading that one can obtain ku as the algebraic K-theory of the topological category of complex vector spaces (and then KU by inverting the Bott element). Given what you've just said, that can't be true. Am I remembering the statement incorrectly, or do you mean something else by "recovering topological K-theory from algebraic K-theory"?

@AaronMazel-Gee Thank you for the clarification! I'm was reading about $\infty$-cosmoi out of idle curiosity. As the slogan is that an $\infty$-cosmos is a place to ($\infty$-)category theory, it seemed natural for me to wonder whether an $\infty$-cosmos encodes the formal properties of the $(\infty,2)$-category of categories enriched over some given $\infty$-category.

@AaronMazel-Gee We already know that the $(\infty,2)$-category of $(\infty,n)$-categories can be modelled by an $\infty$-cosmos. These are $\infty$-categories enriched in $(\infty,n-1)$-categories. As the monoidal $\infty$-category of $(\infty,n-1)$-categories is Cartesian, this is what led me to suggest "Cartesian" as a possible adjective one has to put in front of the base monoidal $\infty$-category for things to work.

@ArunDebray It depends on what you mean by "taking the algebraic K-theory" and "topological category of complex vector spaces". If you take the topological category of complex vector spaces and isomorphisms and you group complete the resulting ∞-groupoid, you get ku. However this is not really "taking algebraic K-theory" for various reasons: 1) The topological category of complex vector spaces and isomorphisms is not the interior of any stable/Waldhausen (∞-)category (...)

2) You are not splitting any exact sequences. You are just taking a E_∞-space and group-completing. I tried to think for a while if you could use Fredholm complexes to construct a stable ∞-category whose algebraic K-theory is ku but I cannot see how to do it

it's been claimed to me that 1-excisive functors from (pointed, finite) spaces to spectra are just spectra

i have a vague argument for why this is true: 1-excisive functors take pushouts to pullbacks, so if F is such a 1-excisive functor, {F(S^k)}_{k >= 0} is an Omega-spectrum

is that the right idea?

@skd Yes, that is right. The construction going the other way sends a spectrum E to the functor Ω^∞(-∧E)

Actually it's easier, $Exc_(S^{fin}_, Sp)=Fun^{rex}(S^{fin}_,Sp)=Sp$, the argument above is why 1-excisive functors from pointed finite spaces to *spaces are spectra

great, thanks @DenisNardin!

i'm preparing a juvitop talk for examples of goodwillie calculus, and this is (a part of) an example that i'm planning on talking about

@DenisNardin pardon for the stupid question, what's the interior of a Waldhausen category?

@TylerLawson Well, the interior of a category is the maximal subgroupoid. Or is it used with a different meaning in general? I guess above I should really have just said that ku is neither the direct sum K-theory nor the Waldhausen K-theory of anything

@DenisNardin thanks! I'm glad to have that clarified.

@ArunDebray you can get the snaith splitting from considering the goodwillie tower of S^oo Map_*(K, -): Top_* -> Sp, where K is a finite CW-complex

i'll probably spend a fair amount of time on this, and then briefly talk about the goodwillie tower of the identity on pointed topological spaces

there's also something about the tate spectrum of (X^n)^{tS_n} being a desuspension of D_n F(X), where F(X) = S^oo Omega^oo S^oo X

i may talk about this, if there's time (when n=2 and X = S^{-1} this seems like it has something to do with lin's theorem but idk)

i'm writing up notes for the talk, so if you're interested, you should check out the juvitop website in a few days!

@TylerLawson @DenisNardin I like n-forum's usage of "core" for what you are calling "interior".

@skd thanks! I will definitely check the notes out!

@DenisNardin Ah, thanks. So the objection is that the "equivalences" aren't a union of path components? I guess it depends on your point of view -- some people object to saying the sphere is the K-theory of the category of sets too.

I guess so, but at least the sphere is the algebraic K-theory of a Waldhausen category (pointed finite sets with injections as cofibrations), so I feel ok about that :)

I'm going to continue calling ku algebraic K-theory anyway, if that's OK

We can't stop you, Tyler