Moritz Groth

Derivators, pointed derivators and stable derivators

We develop some aspects of the theory of derivators, pointed derivators and stable derivators. Stable derivators are shown to canonically take values in triangulated categories. Similarly, the functors belonging to a stable derivator are canonically exact so that stable derivators are an enhancement of triangulated categories. We also establish a similar result for additive derivators in the context of pretriangulated categories. Along the way, we simplify the notion of a pointed derivator, reformulate the base change axiom and give a new proof that a combinatorial model category has an underlying derivator. 55U35, 55U40, 55PXX

Universality of multiplicative infinite loop space machines

We establish a canonical and unique tensor product for commutative monoids and groups in an1‐category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En ‐(semi)ring objects in C give rise to En ‐ring spectrum objects in C . In the case that C is the1‐category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K‐theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable 1‐categories. In particular, we identify preadditive and additive 1‐categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring.D C/’ Ring.D/ C . Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in1‐categories. 55P48; 55P43, 19D23

Tilting theory via stable homotopy theory

We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillies strongly (co)cartesian n-cubes. As applications we construct abstract Auslander-Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi-Yau property. This is used to offer a new perspective on Mays axioms for monoidal, triangulated categories.

Abstract representation theory of Dynkin quivers of type A

Abstract We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields. These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result. Finally, using abstract representation theory of linearly oriented A n -quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable ∞-categories.

Tilting theory for trees via stable homotopy theory

Abstract We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory as well as in the equivariant, motivic, and parametrized variant thereof. As an application of these equivalences we obtain abstract tilting results for trees valid in all these situations, hence generalizing a result of Happel. The main tools introduced for the construction of these reflection functors are homotopical epimorphisms of small categories and one-point extensions of small categories, both of which are inspired by similar concepts in homological algebra.

Abstract tilting theory for quivers and related categories

We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However, even over a field our main result leads to new derived equivalences for example for not necessarily finite or acyclic quivers. The results obtained here rely on a careful analysis of the compatibility of gluing constructions for small categories with homotopy Kan extensions and homotopical epimorphisms, as well as on a study of the combinatorics of amalgamations of categories.