Merlin Christ 2/4 - A gentle introduction to sheaves of stable infinity-categories

Описание: The derived category of a gentle algebra describes the partially wrapped Fukaya category of a marked surface with boundary. Further, this category can be described as the stable ∞-category of global sections of a constructible sheaf of stable ∞-categories. This constructible sheaf is defined on any choice of graph homotopy equivalent to the surface (the graph is sometimes called the skeleton of the surface). We will see how the abstract description as global sections, meaning the limit of a diagram of stable ∞-categories, can be used for concrete computations. To turn the universal property into what is essentially combinatorics, we will recall and employ the description of the limit via so-called coCartesian sections of the Grothendieck construction.

The sheaf theory perspective will naturally give rise to well known statements from the representation theory of gentle algebras. For instance, we will construct global sections (=objects in the derived category) from curves in the surface via the gluing of local sections corresponding to curve segments. Furthermore, some of the proofs will directly generalize from the base field to any base ring spectrum.
The setting of gentle algebras gives a particular accessible introduction to a circle of ideas. Other classes of examples were similar methods apply include many triangulated categories arising from surfaces and Fukaya-type categories. Finally, we remark that the cosheaf properties of topological Fukaya categories/derived categories of graded gentle algebras are originally due to Dyckerhoff-Kapranov ('15,'18) and Haiden-Katzarkov-Kontsevich ('17).