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.

@skd out of curiosity, what other examples are you going to discuss?

@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)

@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