Questions tagged [dg-categories]
A differential graded category is a category enriched over complexes of modules for some commutative ring.
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questions
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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 ...
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1answer
172 views
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-...
3
votes
1answer
199 views
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 ...
3
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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 ...
4
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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 ...
1
vote
1answer
129 views
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 $F_1, F_2 \...
6
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2answers
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$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 ...
3
votes
2answers
276 views
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 ...
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89 views
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 ...
3
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1answer
203 views
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 $...
5
votes
2answers
364 views
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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2answers
396 views
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 (...
8
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1answer
294 views
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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0answers
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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 ...
3
votes
1answer
174 views
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 ...
7
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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 ...
7
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157 views
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 ...
3
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191 views
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 $A^{en}$-...
3
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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:
$$ ... \xrightarrow[]{} HH_n(A) \...
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Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?
Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for ...
4
votes
3answers
373 views
Deriving the functor $ \int_{\Gamma} F(-,-)$
Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and ...
3
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0answers
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On homotopically inverting maps in DG categories
In a DG category $\mathcal{C}$, call a (closed degree 0) morphism $f: X \to Y$ a homotopy equivalence, if there exists $g: Y \to X$ such that $gf-1_X$ and $fg-1_Y$ are exact.
For a full DG ...
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Hochschild cohomology of a universal enveloping algebra of a Lie algebra
I was told that the following equation is true:
Given a finitely generated Lie algebra $\mathfrak g$, there is a Gerstenhaber algebra isomorphism
$$ HH(U\mathfrak g) \cong HH(\wedge^* \mathfrak g^\vee,...
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273 views
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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267 views
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-...
6
votes
0answers
146 views
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 ...
3
votes
1answer
129 views
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 ...
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votes
0answers
187 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{...
5
votes
1answer
357 views
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 ...
2
votes
0answers
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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 ...
27
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3answers
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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 ...
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0answers
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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 ...
2
votes
1answer
123 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 ...
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2answers
1k views
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
1answer
213 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-...
2
votes
1answer
357 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 ...
6
votes
2answers
291 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,...
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votes
1answer
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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 ...
4
votes
0answers
259 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 $\mathcal{O}_X$-...
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votes
2answers
517 views
Is dgCat a category or a 2-category?
Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...
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Natural transformations of $A_\infty$-functors (between dg-categories) are “directed homotopies” (reference?)
Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to \...
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Is the dg-nerve functor a Quillen equivalence?
Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq \text{Hom}_{\mathcal{...
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1answer
501 views
Why do the model structures on dg-algebras and on dg-categories are not compatible?
First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write $CdgAlg\...
3
votes
1answer
130 views
Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective
Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}_{\mathrm{dg}}(\mathbf A))$...
6
votes
1answer
398 views
Matrix factorizations as a dg-category?
Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in ...
3
votes
1answer
228 views
A question about the morphisms in the homotopy category of dg-Cat
Let $dg-Cat$ denote the category of (small) dg-categories and $Ho(dg-Cat)$ denote the localization of $dg-Cat$ at quasi-equivalence. Using the model structure on $dg-Cat$ we can describe the morphisms ...
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0answers
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Lifting DG-categories to characteristic zero
The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...
9
votes
2answers
476 views
An example of two cofibrant dg categories whose tensor product is not cofibrant
I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and $D$...
