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It makes sense to consider larger versions of the (unstable and stable) motivic homotopy categories built out of the site $Sch_S$ of all schemes over $S$ (say of finite type to avoid dealing with size …

One could say that the story begins with Beilinson's conjectures on the existence of a theory of motivic cohomology. In accordance with the insights of the Grothendieck school that cohomology theorie …

This answer is an elaboration on Dylan's comments. 1) Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences. (Let' …

Your proof works in the derived case as well. That is, assume smoothness so that $D^bCoh$ is identified with the full subcategory of compact objects (in general the argument will apply to the subcate …

The argument sketched in Example 3.20 of [J. P. Pridham, Tannaka duality for enhanced triangulated categories, arXiv:1309.0637] demonstrates the comparison assuming the existence of the motivic t-stru …

There are plenty of interesting dg-categories one can associate to a scheme. From the point of view of six functor yoga, these should be viewed as "categories of coefficients" for cohomology theories …

This is not really an answer to your question, just an attempt to address your question from the comments. There are various flavours of homotopical or higher algebraic geometry that are commonly con …

Regarding the first question: If $X$ is quasi-compact quasi-separated and $U \to X$ is a quasi-compact open immersion, then Thomason-Trobaugh showed that there is a "proto-localization sequence", i.e …

I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of application …

The exact sequence of triangulated categories $$ Perf(X)\to D^b_{coh}(X)\to D_{sg}(X) $$ may be lifted to an exact sequence of stable $\infty$-categories or dg-categories in the sense of BGT: choose …

Let $DGCat_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categorie …

It is hard to know where to begin! A general principle is that as long as you are only concerned with the derived category of a single variety, it is generally sufficient to consider it as a triangul …

It may be helpful to have a look at these notes by Anatoly Preygel (see also MO/15910/2503). In particular, Proposition 3.3.6 says that an algebraic stack is an fpqc sheaf if the diagonal is quasi-af …

I have just discovered a chart comparing topologies on Sch/S, made by Pieter Belmans. It includes all the topologies discussed above, and some more I haven't even heard of. It's even interactive and …

Here is a guess about the remark of Orlov. Suppose that one wants to define a good notion of noncommutative scheme, given that an affine noncommutative scheme is an associative algebra. Trying to de …