298 research outputs found
Monoidal derivators and additive derivators
One aim of this paper is to develop some aspects of the theory of monoidal
derivators. The passages from categories and model categories to derivators
both respect monoidal objects and hence give rise to natural examples. We also
introduce additive derivators and show that the values of strong, additive
derivators are canonically pretriangulated categories. Moreover, the center of
additive derivators allows for a convenient formalization of linear structures
and graded variants thereof in the stable situation. As an illustration of
these concepts, we discuss some derivators related to chain complexes and
symmetric spectra
Abstract representation theory of Dynkin quivers of type A
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
-quivers, we construct canonical higher triangulations in stable
derivators and hence, a posteriori, in stable model categories and stable
-categories
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 Goodwillie's 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
May's axioms for monoidal, triangulated categories.Comment: minor improvements in the presentation (the definition of a strong
stable equivalence made more precise, references updated and added
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