Natural transformations of A∞A_\infty-functors (between dg-categories) are “directed homotopies” (reference?)

Let A and B be dg-categories over a field, viewed as A∞-categories. The A∞-category (actually, dg-category) of strictly unital A∞-functors A→B will be denoted by Fun∞(A,B). It is described explicitly (in the non-unital case) for example in P. Seidel’s book (“Fukaya category and Picard-Lefschetz theory”). Let Δ1 be the 1-simplex category, namely, the linear category … Read more

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 DbCoh(X) is equivalent to the derived category of complexes of sheaves of OX-modules whose cohomology is coherent. The only reference I could find are these notes, which are written in some exotic language I can’t parse. Does anyone know … Read more

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 $\mathcal{D}_2$ along a bimodule $\phi \in \mathcal{D}_2^{op} \otimes \mathcal{D}_1$ is the following: they say that an element of such a category is a triple $(M_1, M_2, … Read more

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’s fix a universe and a base ring $k$, and work only with small things…), and is there an explicit description of the homotopy category … Read more

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 profunctor. My question is: is there any “universal property” of this gluing? Something like “a dg-functor defined on the … Read more

DG model of A-infinity category

Given a k-linear dg category Cdg, I can produce an (A∞-quasi-equivalent) k-linear A∞ category CA∞ by taking the homotopy category H0(Cdg) with zero differential and the given composition, and then defining all the higher A∞ structure morphisms to be the Massey products. Possibly up to some adjectives which I’ve omitted, this is an equivalence between … Read more

Definition of EnE_{n}-operad in dgCat

In “Derived Algebraic Geometry and Deformation Quantization” Toën defines in 5.1.2 an En-monoidal A-linear dg-category as an En-monoid in the symmetric monoidal ∞-category dgCatA of compactly generated (A-linear) dg-categories. Concretely, unwrapping this definition Toën says this is equivalent of having a dg-category T∈dgCatA and morphisms En(k)⊗T⊗k→T satisfying the usual conditions of an algebra over an … Read more

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 commutative ring $k$) schemes is quasi-equivalent to the dg-category of continuous dg functors $\mathbb{R}\underline{\mathrm{Hom}}_c(\mathrm{L}_{qcoh}(X),\mathrm{L}_{qcoh}(Y))$, thus providing a kernel for every triangulated functor between the derived categories … Read more

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” and in particular I would like to understand proposition 11 on page 52. The reference he cites is around 300 pages and seems to develop a lot more than maybe necessary for this particular statement. I’m happy … Read more

Formality of A∞A_\infty-category vs formality of its total algebra

Let C be an A∞-category and A its total algebra (elements in A are formal linear combinations of arbitrary morphisms in C and multiplications of arrows which can’t be concatenated are defined to be zero). If C is formal, it is obvious that A is formal, too. Does the converse hold? The reason I think … Read more