## 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

## 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 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