@user40276 So, here we need to be a bit careful. If we consider $D_{\mathbb{A}^1}$, it is fairly tautological that it is equivalent to the category of $H\mathbb{Z}⊗\mathbb{S}$, where $H\mathbb{Z}$ is the Eilenberg-MacLane spectrum in spectra and I used the standard tensoring of spectra by motivic spectra. But one should remember that this spectrum is very different from the motivic Eilenberg-MacLane spectrum (as you say, no transfers). (cont.)

I'll admit I'm not up to speed with the various topologies you can consider, but I think the statement with transfers should follow if we knew that every scheme is cdh-locally smooth, which is again a form of resolution of singularities. I don't know if the qfh or h topology help

@DenisNardin Ah, ok. I guess I got confused on my own notation by using HZ to denote different things. I was considering the qfh topology because every \Lambda (X) has transfers whenever the characteristic of the residue fields of S are invertible. This defines an adjunction between qfh-sheaves and sheaves with transfers. But qfh-sheaves seems to be a bit bigger anyway and the equivalence would not be the correct one as you mentioned. ...

But regarding this cdh-locally smooth are you trying to apply the analogous argument using de Jong's alteration theorem? If so apparently you would need cdh-locally smooth projective...

Uhm... i was trying to run the argument of Hoyois-Kelly-Østvær but I haven't thought too much about it, to be fair

Sorry to barge in but what is a cd structure and what's the best place to read about it for an overview of the basics?

All I know is that its something that gives you a grothndieck topology from a collection of squares.

Ah, yes

This should be at least an overview: sciencedirect.com/science/article/pii/S0022404909002631

(it's actually the original paper referring to this)

Looks very readable, thanks!

While i'm here I have another question, given a sequence of spaces $\cdots \to X_n \to \cdots \to X_0$ what are some ways of determining whether or not the corresponding pro-space is equivalent to a constant one?

Hopefully methods that will work when its pro-constant for a non-stupid reason

Is it more correct to say "closed irreducible subset" or "irreducible closed subset"?

@SaalHardali Probably the main useful result in cd structures (which is the main result in the paper that Denis cited) is that if it's complete and regular, excision <--> Cech descent . Furthermore, when its also bounded (which is more or less equivalent to finite cohomological dimension), excision <--> hyperdescent. A proof in a more intelligible notation can be found in arxiv.org/pdf/1506.07093.pdf (3.2.5) or arxiv.org/pdf/1610.06871.pdf (2.2.7)

Does Cech descent mean the usual thing that cech nerve of covers sent to colimit diagrams after sheafification?