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You might be interested in the paper: Vaknin A., Virtual Triangles// K-Theory, 22 (2001), no. 2, 161--197.

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 …

I would like to understand the following setting: for an additive $B$ localize $K^b(B)$ by a set of $B$-morphisms (i.e. by a thick triangulated subcategory generated by some set of two-term complexes) …

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 …

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 …

You might be interested in the paper: Vaknin A., Virtual Triangles// K-Theory, 22 (2001), no. 2, 161--197.

I would like to draw an octahedral diagram in my paper; I would prefer to present it as the 'upper hat' + the 'lower hat' (as it is common in the texts on triangulated categories). Could anyone tell m …

Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if …

Let us call a commutative square $$ \require{AMScd} \begin{CD} A @>{g'}>> B \\ @V{f'}VV @VV{f}V \\ C @>>{g}> D \end{CD} $$ in a triangulated category split homotopy cartesian if the ("s …

I have a rather abstract paper on triangulated categories; I would say that it is of average size and quality. I want to find an appropriate journal to publish it; I would like it to be accepted in tw …

Yes, sure. The left adjoint to the corresponding embedding is simultaneously a right adjoint. Just note that $A\cong B\bigoplus B^{\perp}$.

For $t$ being the homotopy $t$-structure for Voevodsky effective motivic complexes when $M\otimes N\in DM_{eff}^{t\le -1}$ ensures that either $M$ or $N$ belongs to $DM_{eff}^{t\le -1}$ (so, we use th …

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 …

If $h_i:A_i\to A_{i+1}$ is a countable chain of morphisms in an abelian category $A$ that is AB3 then one can consider the (Bökstedt-Neeman) homotopy colimit of $A_i$ in $D^b(A)$. This is a two-term c …

Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above complexe …