Is there anything known about the heart of the natural t-structure on rationalized non-commutative motives $\mathcal{M}_{loc} \otimes Sp_{\mathbb{Q}}$ in the sense of Blumberg-Gepner-Tabuada? Are there any conjectures about what it should be related to?

@SaalHardali I don't know, but as a first guess, is it the case that the rationalization of non-commutative motives (over the sphere, I suppose) is non-commutative motives over Q? The latter fully faithfully contains ordinary rational motives over Q "modulo the action of the tate motive".

@TomBachmann Does non-commutative motives over $\mathbb{Q}$ mean we do the exact same construction but we start with the category of rational (idempotent complete) stable infinity categories?

Yes.

Oh, I have no idea but I would think this should be a lot smaller

As I said, just a guess.

Thanks for the input! What did you mean by "modulo tate motive"?

See e.g. Theorem 4.6 here: math.mit.edu/~tabuada/BuenosAires.pdf

I guess your question is very close to the difference between the category of rational small stable infinity categories and the rationalization of the category of all small stable infinity category (both with exact functors).

That's why I said I think it should be smaller. but i have no idea...

@TomBachmann Thanks!

@SaalHardali What do you mean with "rationalize" a category? Just rationalizing all mapping spaces?

Yes.

But in this case it's a semi-additive category so it's not that terrible of an operation,

It's unclear to me why this would produce anything reasonable, but ok

What do you mean by reasonable?

Yeah I guess even Q-linearly you have functors (such as the identity) which do not become divisible. So my guess is maybe not so plausible.

Do we agree that if you have a presentable stable infinity category and you rationalize the mapping spaces then this is the same as tensoring with $Sp_{\mathbb{Q}}$?

@SaalHardali No

Tensoring with $\operatorname{Sp}_{\mathbb{Q}}$ is the same as considering $\mathbb{Q}$-modules in there, which will contain more stuff in general (e.g. because you're adding idempotents)

How do I see this, btw?

@TomBachmann It's true in general that Mod_E(C)=Mod_E(Sp)⊗C. I guess the lazy version is to use the standard argument to identify module categories by considering the adjunction Mod_E(Sp)⊗C→C

Nice. Thanks!

Oh wait the argument I gave works only for C symmetric monoidal

Hrmm... then dunno, Schwede-Shipley Morita theory is probably enough

hmm

Ugh wait

@DenisNardin What idempotents am I adding?

@SaalHardali For example if you have an element e:X→X such that e^2=2e, you are adding (e-1)/2

Ok, the argument I wanted to give doesn't work out of the box

And why doesn't it become an idempotent upon rationalization?

@SaalHardali It does. And this gives you a summand in the tensor that doesn't exist in the category where you only rationalize mapping spaces without adding new objects

Also I would expect "rationalize the mapping spaces" would only be correct if the source is compact.

@TomBachmann Yes, that too

What do we mean by source here?

@DenisNardin I see

@SaalHardali I think he's saying that $\operatorname{Map}_{\mathbb{Q}}(\mathbb{Q}⊗x,\mathbb{Q}⊗y)!=\mathbb{Q}⊗\operatorname{Map}(x,y)$ unless x is compact

I'm confused now though. Is the statement I said true for presheaf categories at least?

I thought the general case reduces to presheaf categories and i'm trying to detect my mistake

@SaalHardali I don't think so. The problem is that the operation "rationalize mapping spaces" doesn't even preserve presentability of the category

I see.

I guess even for the category of spaces it's not true...

I lived a lie

Wait, actually my argument for why it fails in spaces doesn't work

so this might still be accidently true for spaces.

In general all operations that get a new category from the old by "doing stuff to mapping spaces" are highly suspicious in my book

I do agree with the sentiment but somehow I thought rationalization is so good that it's an exception

Mod_E⊗C=Fun^R(Mod_E^{op},C)=Fun^L(Mod_E,C^{op})^{op}=RMod_E(C^{op})^{op}=Mod_E(C)

Whew, found an argument

Anyway then, I do understand and fully agree now with the term "unreasonable" thanks @DenisNardin

I have different question: Are limits of null maps in spectra null?

For colimits it's false of course because of nontrivial phantom maps.

(hit enter before I finished writing)

A stable $\infty$-category $C$ has a canonical Waldhausen structure where all maps are cofibrations. There is also a canonical Waldhausen structure on $C^{op}$, again where all maps are cofibrations; call these maps "fibrations". These structures are compatible in the sense that pushouts of fibrations along cofibrations are fibrations, and the resulting square is exact, and pullbacks of cofibrations along fibrations are cofibrations, and the resulting square is exact.

Of course, this just amounts to saying that all pushouts are pullbacks and vice-versa.

Is there some more general theory of these "two commuting Waldhausen and co-Waldhausen structures"?

I'm thinking about the example of the category $St^{idem}$ of small, idempotent complete stable $\infty$-categories, where cofibrations are fully faithful exact functors and fibrations are fully faithful in $St^{idem, op}$.

Maybe "commuting" wasn't the right word, but "interacting"

@ReubenStern They're called "exact ∞-categories"

(you actually need to throw in a few more conditions for the theory to be well-behaved, but I think this is the best answer to your question)

@DenisNardin wonderful, that's exactly the notion I was looking for!