Questions tagged [dg-categories]
A differential graded category is a category enriched over complexes of modules for some commutative ring.
86 questions
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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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Stable infinity categories vs dg-categories
What is the relation between dg-categories and stable $\infty$-categories?
Given a dg-category one can form its dg-nerve and get a $\infty$-category
(which will be stable if the dg-category is?).
...
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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 ...
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Does the Riemann-Hilbert Correspondence work at the DG level?
let $X$ denote a smooth complex algebraic variety. Let $D_{rh}(X)$ denote the category of regular holonomic $D$-modules on $X$ and $D_{rh}^b(D(X))$ denote the bounded derived category of $D$-modules ...
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What is the Hochschild cohomology of the dg category of perfect complexes on a variety?
Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of ...
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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 ...
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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\...
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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-...
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(Co-)Limits and fibrations of DG-Categories?
First of all, let me see if I got the 1-categorical version right:
Let $\mathcal F:C\to Cat $ be a
(pseudo-) functor. The 2-colimit
$\mathrm{colim}_C\mathcal F$ is then
given by the Grothendieck
...
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Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence
Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a ...
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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 $D$...
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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 ...
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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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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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DG vs. abelian quotients
The following, if true, should probably be "standard," but I don't know where to look. I'd rather be slightly imprecise about hypotheses in the hope that there's a good general answer. Feel free to ...
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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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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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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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Why Faonte called "small" and "big" dg-nerves?
I read G.Faonte "Simplicial nerve of an $A_\infty$-category" (https://arxiv.org/abs/1312.2127). In his paper, he calls two dg-nerve construction "small" and "big".
"...
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[Reference Request] The Definition of Adjoint Functors between dg-categories
Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...
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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 ...
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dg objects: Z-graded vs. Z/2Z-graded
I am wondering: Are there any general theorems or principles relating the theory of Z-graded dg objects and the theory of Z/2Z-graded dg objects? I am mainly interested in dg algebras, dg Lie algebras,...
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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 ...
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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
$$ HH(U\mathfrak g) \cong HH(\wedge^* \mathfrak g^\vee,...
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Is the bar resolution of complexes dg-functorial?
Let $k$ be a commutative ring, and let $V$ be a complex of $k$-modules (more in general, we can take an $\mathcal A$-dg-module, where $\mathcal A$ is a dg-category. We can construct the bar resolution ...
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A Question about a theorem in Toën's notes "Lectures on dg-categories"
So I am trying to learn a bit about dg categories from Toën's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand ...
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A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
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1
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Derived categories of smooth proper varieties?
We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical ...
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Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?
Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,...
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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 (...
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1
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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 ...
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Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)
Let $\mathcal A, \mathcal B$ be dg-categories over a field $k$. I denote by $\mathcal{RHom}(\mathcal A,\mathcal B)$ the dg-category (defined up to quasi-equivalence) which gives the internal hom in $\...
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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 ...
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Formality of $A_\infty$-category vs formality of its total algebra
Let $\cal C$ be an $A_\infty$-category and $A$ its total algebra (elements in $A$ are formal linear combinations of arbitrary morphisms in $\cal C$ and multiplications of arrows which can't be ...
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2
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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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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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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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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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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 ...
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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{...
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Lifting commutative diagrams of functors from the homotopy level to the "higher" level
Let $\mathcal A$ and $\mathcal B$ be differential graded categories over a field. Let $F, G, K : \mathcal A \to \mathcal B$ be quasi-functors (see here for definitions). Assume you have morphisms of ...
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2
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Universal property of gluing [collage, cograph] of dg-categories
In some recent works, such as this one (3.2, page 15), a definition of "gluing of dg-categories along a dg-bimodule" is given. It is obviously the analogue of the notion of collage (or cograph) of a ...
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homotopy limits of dg categories
The question is related to the following MO question
(Co-)Limits and fibrations of DG-Categories?
My question is,
how to define the homotopy limit (and colimit) of a system of dg-categories (let'...
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votes
3
answers
476
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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 ...
- 295
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2
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737
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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 ...
- 1,325
4
votes
1
answer
490
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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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4
votes
1
answer
299
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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 $...
- 1,325
4
votes
1
answer
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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 $\mathcal{O}_X$-...
- 7,952
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1
answer
824
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(Homotopy) limits and colimits in a dg-category
It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of $(\infty,1)$-...
- 2,079
4
votes
0
answers
123
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Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
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