SEMINARIO DI GEOMETRIA ALGEBRICA
Nell'ambito del seminario congiunto di geometria algebrica organizzato dai Dipartimenti di Matematica dell'Università e del Politecnico di Milano,
Giovedì 11 giugno alle 12 (durata 1 ora)
in Aula 305 (Via Celoria 20)
si terrà il seguente seminario:
Chiara Sava (Charles University)
The derivator of a pretriangulated dg-category
Abstract:
Derivators, introduced independently by Grothendieck, Heller, Franke and further developed by Groth, yield a model of higher categories based on the language of 2-categories. A prederivator is a 2-functor from the 2-category of small categories to the 2-category of large categories and a derivator is a prederivator with additional properties. Heuristically, a derivator can be viewed as a collection of homotopy categories of diagrams equipped with homotopy Kan extensions, and hence with homotopy limits and colimits. Stable derivators play for derivators the same role that stable $\infty$-categories play for higher categories: they provide enhancements of triangulated categories in which homotopy-theoretic constructions become functorial.
While derivators can be associated to $\infty$-categories, no direct construction of a derivator associated to a dg-category appears in the existing literature. In this work, we close this gap. We develop a theory of homotopy (co)limits, and homotopy Kan extensions in dg-categories via suitable weighted (co)limits, where the weights are replaced by appropriate resolutions to ensure invariance under quasi-isomorphisms. As a consequence, if $\mathscr{A}$ is a homotopy complete and cocomplete pretriangulated dg-category, then the 2-functor sending a small category $I$ to the dg-category of quasi-functors from (the free linear category generated by) $I$ to $\mathscr{A}$ defines a stable derivator. Indeed quasi functors are essentially “homotopically coherent functors” between dg-categories.
As an application, we study the derivator associated to a Frobenius exact category $\mathscr{F}$. Since stable Frobenius exact categories admit natural dg enhancements, our main result yields a description of their associated derivator. In particular, we prove that the derivator associated to the dg-category of acyclic complexes of projective objects of $\mathscr{F}$, evaluated at a small category $I$, is equivalent to the homotopy category of acyclic chain complexes of projective (equivalently injective) objects in the exact category of functors from $I$ to $\mathscr{F}$.
This talk is based on a joint work arXiv:2508.02612 with Francesco Genovese which also contain an appendix by Jan Šťovíček.