Let X be a compact real analytic manifold, and let T ∗X be its cotangent bundle. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X. In this paper, we develop a Fukaya A∞-category Fuk(T ∗X) whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write TwFuk(T ∗X) for the A∞-triangulated envelope of Fuk(T ∗X) consi...

Let T be a tilting object in a triangulated category which is equivalent to the bounded derived category of a finite-dimensional hereditary algebra. The text investigages the strong global dimension, in the sense of Ringel, of the opposite algebra A of the endomorphism algebra of T . This invariant is expressed in terms of the lengths of the sequences T0, . . . , Tl of tilting objects such that...

We discuss some basic properties of the graded center of a triangulated category and compute examples arising in representation theory of finite dimensional algebras.

In the work of Hoshino, Kato and Miyachi, [8], the authors look at t-structures induced by a compact object, C, of a triangulated category, T , which is rigid in the sense of Iyama and Yoshino, [9]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on T whose heart is equivalent to Mod(End(C)). Rigid objects in a triangulated category can the thought of as b...

We study the vanishing of cohomology in a triangulated category, in particular vanishing gaps and symmetry.

On objects of a triangulated category with a stability condition, we construct a topology.

On objects of a triangulated category with a stability condition, we construct a topology.

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories ...

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any tstructure is abel...

We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As a first application, we show that the stable category of maximal Cohen-Macaulay modules over a certain isolated singularity of ...