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Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid_M$ is invertible. We will say that $M'$ is ind-torsi …

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 $A^{op}$ …

I have an abelian category $A$ that is AB4, AB3* and has an injective cogenerator. Do these conditions "help" in checking whether a given family $a_i$ of (compact) objects of $A$ is generating in it? …

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 …

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 …

You can start with the derived category of graded polarizable Hodge structures or with the category of Hodge modules (over a complex variety $X$). These categories possess natural weight structures (a …

Let $\mathcal{A}$ be a small additive category. Consider the category $PreSh(\mathcal{A})$ of all additive functors from $\mathcal{A}^{op}$ into abelian groups; note that this category is abelian and …

Does there exist any intrinsic characterization of additive categories equivalent to $\operatorname{Ex}(A,Ab)$, that is, of exact functors from a small abelian category $A$ into abelian groups? Any hi …

I didn't check that thoroughfully, but it seems that in the ("abstract") setting you are interested in there exist (exact) "projections" $j^{i*}:D\to D^i$ and $j^{i!}:D\to D^i$ ($D$ is the "big" trian …

I would like to apologize for this rather stupid abstract nonsense question. Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it see …

How would you say that a small additive category $C$ embedds (contravariantly) into the category of exact functors from a 'large' abelian $C'$ into abelian groups (this is something like Yoneda's embe …

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 …

I am trying to understand (Saito's?) category of mixed Hodge modules as a category (i.e. I am not interested in its construction, just in properties of objects and morphisms). I would be grateful for …

Let $X$ be a manifold, $i: Z\to X$ is a closed embedding. For a sheaf $S$ (of abelian groups) on a manifold $X$ and each $\varepsilon>0$ we denote by $Z_\varepsilon$ the set of points of $X$ that l …