Dg Categories Research Articles

Constructible sheaves and the Fukaya category

Let X X be a compact real analytic manifold, and let T ∗ X T^*X be its cotangent bundle. Let S h ( X ) Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X X . In this paper, we develop a Fukaya A ∞ A_\infty -category F u k ( T ∗ X ) Fuk(T^*X) whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write T w F u k ( T ∗ X ) Tw Fuk(T^*X) for the A ∞ A_\infty -triangulated envelope of F u k ( T ∗ X ) Fuk(T^*X) consisting of twisted complexes of Lagrangian branes. Our main result is that S h ( X ) Sh(X) quasi-embeds into T w F u k ( T ∗ X ) Tw Fuk(T^*X) as an A ∞ A_\infty -category. Taking cohomology gives an embedding of the corresponding derived categories.

Homotopy theory of well-generated algebraic triangulated categories

AbstractFor every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially dg categories which are stable under suspensions, cosuspensions, cones and α-small sums.Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

On the structure of Calabi-Yau categories with a cluster tilting subcategory

We prove that for d\geq 2 , an algebraic d -Calabi-Yau triangulated category endowed with a d -cluster tilting subcategory is the stable category of a DG category which is perfectly (d+1) -Calabi-Yau and carries a non degenerate t -structure whose heart has enough projectives.

Grothendieck ring of pretriangulated categories

We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal decompositions of corresponding triangulated categories. We introduce an operation of “multiplication” • on the collection of DG categories, which makes this abelian group into a commutative ring. A few applications are considered: representability of “standard” functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

DG quotients of DG categories

Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.

99/02299 The development of video-based sensors for off-and onstream analysis: Cruz, E. B. and Adel, G. T. Proc. Annu. ISA Anal. Div. Symp., 1998, 31, 259–268

The periodic cyclic homology of any proper dg category comes equipped with a canonical pairing. We show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with Kyoji Saito's higher residue pairing on the twisted de Rham cohomology of the singularity.

Deriving DG categories

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a triangulated analogue (4.3) of a of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a Morita theorem (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories.

Homotopy theory of $dg$ categories via localizing pairs and Drinfeld’s $dg$ quotient

Using localizing pairs and Drinfeld's dg quotient we construct a new Quillen model for the homotopy theory of dg categories. We prove that, in contrast with the original model, this new Quillen model carries a natural closed symmetric monoidal structure. As an application, we obtain a simple construction of the internal Hom-objects and a conceptual characterization of Toen's previous work. Making use of this new Quillen model, Lowen has recently developed a derived deformation theory.

The character field theory and homology of character varieties

We construct an extended oriented $(2+\epsilon)$-dimensional topological field theory, the character field theory $X_G$ attached to a affine algebraic group in characteristic zero, which calculates the homology of character varieties of surfaces. It is a model for a dimensional reduction of Kapustin-Witten theory ($N=4$ $d=4$ super-Yang-Mills in the GL twist), and a universal version of the unipotent character field theory introduced in arXiv:0904.1247. Boundary conditions in $X_G$ are given by quantum Hamiltonian $G$-spaces, as captured by de Rham (or strong) $G$-categories, i.e., module categories for the monoidal dg category $D(G)$ of $D$-modules on $G$. We show that the circle integral $X_G(S^1)$ (the center and trace of $D(G)$) is identified with the category $D(G/G)$ of class $D$-modules, while for an oriented surface $S$ (with arbitrary decorations at punctures) we show that $X_G(S)\simeq{\rm H}_*^{BM}(Loc_G(S))$ is the Borel-Moore homology of the corresponding character stack. We also describe the on the character theory, a one parameter degeneration to a TFT whose boundary conditions are given by classical Hamiltonian $G$-spaces, and which encodes a variant of the Hodge filtration on character varieties.

Compact generation of the category of $\mathrm{D}$-modules on the stack of $G$-bundles on a curve

Let G be a reductive group. Let BunG denote the stack of G-bundles on a smooth complete curve over a field of characteristic 0, and let D-mod(BunG) denote the DG category of D-modules on BunG. The main goal of the paper is to show that D-mod(BunG) is compactly generated (this is not automatic because BunG is not quasi-compact). The proof is based on the following observation: BunG can be written as a union of quasi-compact open substacks j : U ↪→ BunG, which are ”co-truncative”, i.e., the functor j! is defined on the entire category D-mod(U).