A symplectic manifold gives rise to a triangulated A∞-category, the derived Fukaya category, which encodes information on Lagrangian submanifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into what the Fukaya category is, where it comes from, and what symplectic topologists want to do with it. ...everything you wanted to say required a context. If you gav...

The slice filtration on Voevodsky’s triangulated category of motives is defined by effectivity conditions. It is constructed and studied in [HK]. An analogous filtration on the homotopy category was introduced by Voevodsky. In this note we try to get a conjectural picture of the properties of the slice filtration by systematic use of the realization functor to the derived category of Hodge stru...

For each recollement of triangulated categories, there is an epivalence between the middle category and comma associated with a triangle functor from on right to left. morphic enhancement $\mathcal {T}$ , are three explicit ideals enhancing category, whose corresponding factor categories all equivalent module over . Examples related inflation weighted projective lines discussed.

After a brief review of derived categories, triangulated categories and t-structures, we shall consider the bounded derived category D(X,R) of the category of étale sheaves of R-modules on X (a k-scheme of finite type), where R is an l-torsion ring with l a prime number distinct from the characteristic of the field k. We shall construct a t-structure on the (triangulated) full subcategory D c(X...

We study the homotopy category K(InjA) of all injective A-modules InjA and derived category D(ModA) of the category ModA of all A-modules, where A is finite dimensional algebra over an algebraically closed field. We are interested in the algebra with discrete derived category (derived discrete algebra. For a derived discrete algebra A, we get more concrete properties of K(InjA) and D(ModA). The...

In this paper we give necessary and sufficient conditions for a functor to be representable in strongly generated triangulated category which has linear action by graded ring, discuss some applications examples.

In (non-commutative) geometry, a categorical resolution of a (potentially singular) variety X is a full and faithful embedding of its derived category D(X) into a smooth and proper triangulated category T [11, 12]. The notion generalises the situation of rational singularities, where the geometric resolution functor F : X̃ → X induces the full and faithful functor F ∗ : D(X) → D(X̃) to the smooth...

Triangulated categories were introduced in the mid 1960’s by J.L. Verdier in his thesis, reprinted in [15]. Axioms similar to Verdier’s were independently also suggested in [2]. Having their origins in algebraic geometry and algebraic topology, triangulated categories have by now become indispensable in many different areas of mathematics. Although the axioms might seem a bit opaque at first si...

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal oneorthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.