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Questions tagged [dg-categories]

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

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81 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 ...
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107 views

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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105 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 ...
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149 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 $A^{en}$-...
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81 views

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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70 views

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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3answers
332 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 ...
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0answers
168 views

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 ...
7
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218 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,...
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0answers
192 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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221 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-...
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105 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
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1answer
120 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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162 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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237 views

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
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1answer
314 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 ...
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199 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 ...
24
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3answers
2k views

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
146 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 ...
2
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1answer
113 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
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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
1answer
207 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
261 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 ...
5
votes
2answers
258 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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2answers
403 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 ...
4
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239 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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2answers
439 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 ...
4
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115 views

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
404 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\...
2
votes
1answer
120 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
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1answer
317 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
190 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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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 ...
8
votes
2answers
337 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 $D$...
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0answers
127 views

Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$

Let $\mathcal T = D^b(\mathbb P^1)$, the bounded derived category of $\mathbb P^1$ over a field $k$. It is well known that $\mathcal T$ admits a strong and full exceptional sequence $\{\mathcal O, \...
1
vote
1answer
132 views

Is the extension of a quasi-functor again a quasi-functor?

Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to \...
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221 views

Perf($\mathscr{A}$) and perfect chain complexes

Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ \...
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0answers
161 views

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 ...
5
votes
2answers
567 views

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 $\mathcal V =\mathbf C(k)$ ...
3
votes
1answer
584 views

(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
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1answer
152 views

Taking zeroth cohomology of a dg-module over a dg-category is “good” on representables, but probably “bad” on cones

This question is possibly related to this other one. Let $\mathcal A$ be a dg-category over a commutative ring $k$. I denote by $\text{dgm-}\mathcal A$ the dg-category of right dg-$\mathcal A$-...
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67 views

Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones

This question is strictly related to this other one. Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let http://presheaf.com/cache/d86dhk341k582e6j252q2b3qn3v32.png ...
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1answer
156 views

Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”

I set this problem in the framework of (pretriangulated) dg-categories; everything can probably be translated in the world of stable $(\infty,1)$-categories. Let $\mathcal A$ be a pretriangulated dg-...
5
votes
1answer
168 views

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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votes
2answers
183 views

Does semi-free behave well under totalization

Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this so?...
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2answers
300 views

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 ...
14
votes
2answers
2k views

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?). ...
7
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1answer
454 views

[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 $F\dashv G$ ...
5
votes
0answers
274 views

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 ...