Questions tagged [dg-categories]

A differential graded category is a category enriched over complexes of modules for some commutative ring.

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Derived categories of smooth proper varieties?

We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical ...
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dg-Künneth formula for qcqs schemes

Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)...
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(Commutative) Algebras in $\mathsf{dgCat}_k$

Suppose $k$ is a fixed commutative ring, and let $\mathsf{dgCat}_k$ denote the category of $k$-linear dg-categories. We will equip $\mathsf{dgCat}_k$ with the Morita model structure (see theorem 2.27 ...
Stahl's user avatar
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Colimits of DG-categories and functors between them

Suppose I have two diagrams $\{\mathcal{C}_i\}_{i\in \mathcal{I}}$ and $\{\mathcal{D}_i\}_{i\in \mathcal{I}}$ in the $\infty$-category of DG-categories over a field $k$ with continuous functors (i.e. ...
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Find a Morita equivalent finite cell DG category

I am trying to understand the following statement: Suppose that $\mathcal{E}$ is a pre-triangulated proper DG category with a full exceptional collection. Then $\mathcal{E}$ is Morita equivalent to a ...
Harold Finch's user avatar
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Signs in dg Yoneda embedding: proof of existence of Dwyer-Kan model structure on $\mathit{dgcat}$

I'm studying a proof of the fact that the category of dg-categories admits a (Dwyer-Kan) model structure. As references, I'm using Pieter Belmans' master thesis and Goncalo Tabuada's paper Une ...
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Integral lattice in noncommutative Hodge theory

Associated to a $DG_{\mathbb{C}}$-category, $\mathcal{C}$, we have some Hodge theoretic data - $HH_{*}(\mathcal{C})$ plays the role of Hodge cohomology and $HP$ plays the role of de Rham cohomology. ...
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Definition of $E_{n}$-operad in dgCat

In "Derived Algebraic Geometry and Deformation Quantization" Toën defines in 5.1.2 an $E_{n}$-monoidal A-linear dg-category as an $E_{n}$-monoid in the symmetric monoidal $\infty$-category $...
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Intuition for points of the moduli of objects for a dg-category

Problem summary: I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in this paper by Brav and Dyckerhoff) actually looks like/how to give an alternative ...
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Symmetric monoidal structure(s) on the $\infty$-category of dg-categories

Let $k$ be a commutative ring with $1$, and let $\mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in [1, Section 2]. We may equip $\mathsf{dgCat}_k$ with the Morita model ...
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Relationship between different definitions of the Hochschild homology

Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to ...
Stahl's user avatar
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Are dg-modules over a cofibrant dg-category cofibrant?

Fix a commutative ring $k;$ all dg-categories will be dg-categories over $k.$ Throughout the question, I will be following the notation and conventions of Toën's "The homotopy theory of dg-...
Stahl's user avatar
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Is there a notion of projective dg category?

Since the paper Smooth and proper noncommutative schemes and gluing of DG categories by Orlov, dg categories are considered the non-commutative counterpart of algebraic geometry. More specifically, we ...
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Kan liftings and projective varieties

Regard the following two bicategories: $\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition ...
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Calabi-Yau structures on dg-categories

A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here) $$ A^! = A[-n]$$ Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In ...
Markus Zetto's user avatar
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1 answer
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Morphisms on fibre products

Let $X$ and $Y$ be two quasi compact, separated schemes over $k$, and consider the fibre product $X \times Y$. If we call $p_1$ and $p_2$ the two projections, and we take perfect complexes $F_1, F_2 \...
Federico Barbacovi's user avatar
7 votes
2 answers
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$k$-linear $\infty$ stable categories and dg categories

This question is related to this question, where I asked about the relation between the derived category of a fiber product $Y \times_Z W$ and the push out of the diagram of derived categories one ...
Federico Barbacovi's user avatar
4 votes
2 answers
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Perfect DG modules

I was wondering whether there is a characterization of perfect DG modules over a DG algebra as there is one for modules over a ring. Namely, an object in $D(R)$, where $R$ is a ring, is perfect if and ...
Federico Barbacovi's user avatar
1 vote
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Application of the cube lemma

In the paper Spherical DG-functors, the authors introduce the notion of twisted cubes and prove a lemma that they call "The cube lemma". One of the applications they prove is the following, given a ...
Federico Barbacovi's user avatar
4 votes
1 answer
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Definition of gluing of dg categories

I am reading the paper by Kuznetsov and Lunts, Categorical resolutions of irrational singularities, and I’m struggling with a few things. The definition of gluing of DG-categories $\mathcal{D}_1$ and $...
Federico Barbacovi's user avatar
5 votes
2 answers
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Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...
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Functorial cones

This is probably more a reference request than a real question. I was studying dg-categories in order to understand how one can derive a functorial cone construction when a triangulated category (...
Federico Barbacovi's user avatar
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Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?

In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\...
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"DG categories" over other DG categories

The standard definition of a DG-category is "a category enriched in chain complexes over a commutative ring $k$". Suppose I have a category enriched over some symmetric monoidal DG-category, for ...
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How to show mapping cones are homotopy cofibers

In a dg-category $\mathcal{C}$, the $n$-translation of an object $C$ is an object $C[n]$ representing the functor $$ {\rm Hom}(-,C)[n]. $$ The cone of a closed morphism $f\colon C \to D$ of degree ...
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DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...
Dasha Poliakova's user avatar
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Is there any survey of dg-categories from the $\infty$-category point of view?

I was reading this question on dg-categories and a comment by David Ben-Zvi says "An excellent pre-$\infty$-categorical overview is Keller's ICM address https://arxiv.org/abs/math/0601185". I was ...
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Wrong way Poincare duality for Calabi-Yau dg-algebras?

Let $A$ be a smooth compact Calabi-Yau dg $k$-algebra of dimension $w$. It is widely known (e.g. Atsusi Takahashi proposition 2.4) that in such situation we have non canonical isomorphism of $A^{en}$-...
Mykola Pochekai's user avatar
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Explicit description of periodic map $S : HC_{i} \to HC_{i-2}$ for dg and $A_\infty$ algebras

Let $A$ some associative unital $k$-algebra, let $HC_*(A)$ is cyclic homology of $A$ and $HH_*(A)$ is hochschild homology of $A$. Then we have Connes exact sequence: $$ ... \xrightarrow[]{} HH_n(A) \...
Mykola Pochekai's user avatar
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Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for ...
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4 votes
3 answers
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Deriving the functor $ \int_{\Gamma} F(-,-)$

Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and ...
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On homotopically inverting maps in DG categories

In a DG category $\mathcal{C}$, call a (closed degree 0) morphism $f: X \to Y$ a homotopy equivalence, if there exists $g: Y \to X$ such that $gf-1_X$ and $fg-1_Y$ are exact. For a full DG ...
Dasha Poliakova's user avatar
7 votes
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707 views

Hochschild cohomology of a universal enveloping algebra of a Lie algebra

I was told that the following equation is true: Given a finitely generated Lie algebra $\mathfrak g$, there is a Gerstenhaber algebra isomorphism $$ HH(U\mathfrak g) \cong HH(\wedge^* \mathfrak g^\vee,...
sock's user avatar
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Tensor product of dg-modules in the framework of dg categories

Let $\mathcal{A}$ and $\mathcal{B}$ be two dg categories, and let $M \in \text{Mod}_{\mathcal{B}}^{\mathcal{A}}$ be a $\mathcal{A}$-$\mathcal{B}$ bimodule. In many references in the framework of dg-...
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Simple question about DG-algebras

Considering the following conditions for two DG-algebras $A$ and $B$: 1) There exists quasi-isomorphic DG-algebra morphism $A \to B$. 2) There exists a DG-algebra $C$ and two quasi-isomorphic DG-...
Sasha's user avatar
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6 votes
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Drinfeld quotient of 'finite' dg-categories

I have been reading Gonçalo Tabuada's paper Higher K-theory via universal invariants in a seminar and the following question arose. At one point in his construction (specifically $\S$10) he looks at ...
Ian Coley's user avatar
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1 answer
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Could $RHom(A,-)$ distinguish non-quasi-equivalent dg-categories?

One of the remarkable results in Toen's paper is the existence of internal homs of dg-categories. Actually for two dg-categories $A$ and $B$, there exists a dg-category $RHom(A,B)$ such that for any ...
Zhaoting Wei's user avatar
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2 votes
0 answers
217 views

What is the motivation behind the definition for a smooth differential graded category?

Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\...
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DG model of A-infinity category

Given a $k$-linear dg category $\mathcal{C}_{dg},$ I can produce an ($A_\infty$-quasi-equivalent) $k$-linear $A_\infty$ category $\mathcal{C}_{A_\infty}$ by taking the homotopy category $H^0(\mathcal{...
Ben G's user avatar
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1 answer
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Twisted derived Morita theory of schemes

It has been proved by Toën and Lunts-Schnürer that the dg category $\mathrm{L}_{qcoh}(X\times Y)$ of quasi-coherent sheaves over the product of two quasi-compact, quasi separated (and flat over a ...
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0 answers
237 views

On the category of $D$-modules

Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$. 1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck ...
Sasha's user avatar
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30 votes
3 answers
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DG categories in algebraic geometry - guide to the literature?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric ...
Saal Hardali's user avatar
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7 votes
0 answers
264 views

Is the bar resolution of complexes dg-functorial?

Let $k$ be a commutative ring, and let $V$ be a complex of $k$-modules (more in general, we can take an $\mathcal A$-dg-module, where $\mathcal A$ is a dg-category. We can construct the bar resolution ...
Francesco Genovese's user avatar
2 votes
1 answer
149 views

Why do some literatures prefer right module to left module when dealing with DG modules?

I've been trying to read some papers on differential graded modules (for example, Keller, Deriving DG categories) In most of literature I found about dg-modules, they define them as right modules (Of ...
Anonymous's user avatar
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21 votes
2 answers
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How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
3 votes
1 answer
221 views

comparison of truncations

I am trying to understand the proof of Lemma 3.0.15 of this paper (Ben-Zvi, Nadler, Preygel - Integral transforms for coherent sheaves). The context is of two triangulated categories $C,D$ with t-...
Matthias Volkov's user avatar
2 votes
1 answer
449 views

DG natural transformation Serre functors

This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer. Let $X$ be a smooth projective variety ...
Libli's user avatar
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6 votes
2 answers
330 views

Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined $$ RHom(C,...
Zhaoting Wei's user avatar
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9 votes
1 answer
529 views

derived categories as presentable DG-categories

Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and ...
Sasha's user avatar
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4 votes
1 answer
325 views

Reference request: category of sheaves of O-modules with coherent cohomology

Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of $\mathcal{O}_X$-...
Dmitry Vaintrob's user avatar