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I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are si …

Suppose that a triangulated category $C$ contains a full additive subcategory $B$ of (strong) generators (i.e. there does not exist a proper strict triangulated subcategory $C'\subset C$ that contains …

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 …

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 …

Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular fac …

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; whi …

Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any exampl …

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 …

Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we …

For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them …

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\t …

Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of …

Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$. So here is my list …

Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $ …