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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 …

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 …

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this questi …

The question already has good answers but I think there is still more to be said. References As already mentioned, algebraic K-theory satisfies Zariski descent. For regular noetherian schemes this …

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' …

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 …

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 …

I have also wondered about this question and recently came across some papers that seem to answer it. First of all, the paper Manuel L. Reyes, Obstructing extensions of the functor Spec to noncommu …

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 …

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 …

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 …

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 …

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 …

One way to understand your question is in the framework of $\mathbf{A}^1$-homotopy theory. This is because your nerve functor is better understood when defined on a cocomplete category like the categ …

Regarding 3), Andrew Kresch just told me that they gave up on the project.