O ct 2 01 4 Derived sections and categorical resolutions in homotopical context
نویسنده
چکیده
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 and proper category D(X̃). Another example of a categorical resolution comes from considering a finite CW-complex Y of homotopy type K(G, 1). The derived category of local systems over Y , which we denote as Loc(Y, k), is the derived category of complexes of locally constant sheaves over some field k. Denote by BG the fundamental groupoid of Y . We then see that Loc(Y, k) is exactly the derived category D(BG, k) of functors from BG to the category of complexes of vector spaces
منابع مشابه
Homotopical Resolutions Associated to Deformable Adjunctions
Given an adjunction F ⊣ G connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calcul...
متن کامل- qc / 0 41 01 04 v 1 2 0 O ct 2 00 4 CAUSAL SITES AS QUANTUM GEOMETRY
We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural " tangent 2-bundle, " analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relati...
متن کاملar X iv : h ep - p h / 01 06 32 4 v 3 1 5 O ct 2 00 1 Anderson Theorem for Color Superconductor
Ginzburg-Landau functional is derived for a system possessing both chiral and diquark condensates. Anderson theorem for such a system is formulated and proved.
متن کاملHomological Projective Duality for Grassmannians of Lines
We show that homologically projectively dual varieties for Grassmannians Gr(2, 6) and Gr(2, 7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that (1) the derived category of a Pfaffian cubic 4-fold admits...
متن کاملar X iv : m at h / 01 10 31 1 v 1 [ m at h . Q A ] 2 9 O ct 2 00 1 The categorical theory of relations and quantization March 19 , 2008
In this paper we develope a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of A−A bicomodules. Properties of relations are defi...
متن کامل