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People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask …

I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations …

I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably …

Which applications of intersection (co)homology and of the (Topological) Decomposition Theorem have most chances to be understood by students?

Amusingly, I have considered the category of complexes where maps of the form du+vd are killed in: Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for mo …

This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly. For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $ …

As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex? shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex (and al …

Let us call a triangulated category algebraic if it admits a differential graded enhancement (i.e., an enrichment in complexes of abelian groups). Certainly, there is a notion of a tensor product on d …

I think that the simplest example is $K(B)$ (the homotopy category of complexes; you can also consider $K^b(B)$) where $B$ is any tensor additive category. Certainly, this example is not independent f …

For a commutative square of spaces (of manifolds, or of simplicial sets): $$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & D\end{array}\right)$$ I am tr …

People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in …

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (s …

First we assume that your equations have rational coefficients; if this is not so then you can probably 'approximate' your variety by a variety defined over rationals without changing its topology (th …

Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories? I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) …

For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules …