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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 …

As far as I remember, there is an important example of a functor that transforms mono- and epimorphisms into mono- and epimorphisms, respectively, but is not half-exact; this is the functor of interm …

Under which conditions localizing an additive category by some class S of morphisms yields and additive category? It seems easy to define certain addition on morphisms if we fix their representatives …

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 think that the Grothendieck group DOES depend on A. Indeed, any additive category C could be embedded (by the Yoneda embedding) into the abelian category of contravariant additive functors from C to …

The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those statem …

A silly remark is that "trivial" $t$-structures always exist. You should probably say that you want a bounded or a non-degenerate $t$-structure. As far as I remember, non-zero negative $K$-groups of $ …

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper? Since weak equivalen …

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 …

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist co …

I have two functors $F_1,F_2$ from a category $C$ into two distinct categories $D_1,D_2$. I would like to say that $F_1$ and $F_2$ are equivalent if there exists a commutative square $\require{AMScd}$ …

For functors $E,F,E',F':X\to Y$ I would like to say that transformations $\tau:E\to F$ and $\tau':E'\to F'$ are isomorphic if they are isomorpic in the arrow category of functors $X\to Y$, that is, t …

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ …

Let $T$ be a compactly generated triangulated category, that is, $T$ is closed with respect to small coproducts and equals its own smallest triangulated subcategory closed with respect to coproducts a …

1) I believe that the (Murre's) vanishing needed for Hanamura's argument is stronger than the BS conjecture. 2) There are certain standard conditions ensuring that a triangulated category is equivale …