The problem is that there are not a lot of actual colimits in the homotopy category of (connected) CW complexes, so knowing that $\pi_1$ preserves them (which is true) is pretty much useless. The ...

It's worth noting first that smooth schemes are essentially the smallest possible category from which one can define the motivic homotopy category: to make sense of $\mathbb A^1$-homotopy and of the ...

It is true in complete generality that $X$ is the homotopy colimit of $C_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version ...

Categories $J$ such that limits of shape $J$ commute with filtered colimits in sets are called L-finite. There are several known characterization of them: see the nLab page about it. The page refers ...

Here's a direct link to the book by Hovey–Palmieri–Strickland. The category of motivic spectra is known to satisfy the axioms of Definition 1.1.4 in the book when the base is a countable field of ...

The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player,...

There certainly is a notion of higher Tannakian category which would have meaningful higher homotopy groups. I'm not sure how much of the theory has been worked out already, but higher Tannakian ...

ETA The answer is yes in general. Replace 2 below with a reference to HTT, Prop. 7.1.5.8. Since this has been open for a while, let me give a partial answer which hopefully is already interesting: I ...

It doesn't. For instance, the $\infty$-category of spectra is the colimit of the tower $$ \mathcal{S}_* \stackrel\Sigma\to \mathcal{S}_* \stackrel\Sigma\to ... $$ in $Pr^L$, but its colimit in $Cat_\...

(1) is true if $char(k)=0$. This follows from a combination of results. First of all, it is true over any field that the spectrum $MGL$ is connective, which means that $$MGL^{p,q}(X)=0$$ if $p>q+...

In modern language, one would say that $D_{qcoh}(-)$ is a sheaf of $(\infty,1)$-categories on the scheme $X$ (so "homotopy stack" = "sheaf of $(\infty,1)$-categories"). If $X$ is affine, or more ...

The most general functor of this form was constructed by Ayoub in La réalisation étale et les opérations de Grothendieck. Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly ...

Every contractible finite CW complex $X$ satisfies these conditions. This follows from results in Section 7.3 of HTT and Appendix A of HA: we have $Shv(X) \otimes Shv(X)=Shv(X\times X)$ since $X$ is ...

Regarding 2, there is no difficulty in defining $G$-spectra in the setting of $\infty$-categories. The only complications I can think of are that (1) the orbit category $\mathrm{Orb}^G$ is now an $\...

I'm going to restrict the discussion to Grothendieck abelian categories, because I'm not sure what can be said more generally. The main reference for what follows is Appendix C in Lurie's book ...

There are conceptually simple definitions, but they require a more symmetric definition of Hochschild homology. The Hochschild homology of $X/k$ (with coefficients in $\mathcal O_X$) is the homology ...

This would be the "comparison lemma" from SGA 4-1, Éxposé III, Thm. 4.1: if $C$ is a full subcategory of a site $D$, equipped with the induced topology, and if every object of $D$ is covered by ...

Let $k$ be a field and $\operatorname{DM}_{gm}(k)_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor $$ R: \...

I think your idea to reduce the question to small slice topoi works perfectly. I will use it to show that every sheaf on $Man$ (either the continuous or the smooth version) is the limit of its ...

The only reference I know for Künneth theorems in this generality are SGA4 and SGA4.5. Specifically, SGA4 has theorems for étale cohomology with proper support (which hold in ridiculous generality), ...

The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful ...

This is not true even if $\mathcal X$ is an Artin stack. For example, let $G$ be a smooth group scheme over the base $T$, and let $\mathbf BG$ be its classifying stack (the category of $G$-torsors ...

This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then $$ H^1_{et}(X,F) = H^1(\Pi_1^{et}(X), \tilde F), $$ where $\Pi_1^{...

You have to distinguish between pullbacks of cycles, pullbacks on Chow groups, and pullbacks of relative cycles. You cannot always pull back cycles. If $f: Y\to X$ is a morphism of (arbitrary) ...

I do not know the answer for a general Quillen adjunction, but I will attempt to give a complete answer in the case you're interested in, when the adjunction $(F,G)$ is of the form $(f_!,f^\ast)$ for $...

I believe the answer is YES and, more generally, that $\tau_{\leq n}\mathcal{C}\subset\mathcal{C}$ preserves filtered colimits for any $\infty$-topos $\mathcal{C}$. For the $\infty$-topos of $\infty$-...

Every Grothendieck quasitopos is presentable and locally cartesian closed. These are categories of separated presheaves on a site. The simplest example of a site whose separated presheaves do not form ...

$MGL$ does not admit a structure of $H\mathbb Z$-module. There are many ways to prove this. As Sean said in the comments, if it were true over $\mathbb C$, topological realization would imply that $MU$...

The image of the inclusion $CHM_k\hookrightarrow DM_k$ is indeed not contained in the heart $M_k$ of the motivic t-structure. $CHM_k$ does map to the heart, but via a different functor, namely, the ...

The formula also holds for perfect complexes. This can be deduced from the case of vector bundles, although it requires a lot of structure in that case. Namely, we need to use the fact that the ...