# Moritz Rahn (former surname: Groth)

Home · Research

### Preprints

14. Abstract cubical homotopy theory (j/w Beckert, March 2018, 71 pages, arXiv, submitted)The main players are the cardinality filtration of n-cubes, the induced interpolation between cocartesian and strongly cocartesian n-cubes, and the yoga of iterated cone constructions. In the stable case, the representation theories of chunks of n-cubes are related by compatible strong stable equivalences and admit a global form of Serre duality. As sample applications, we use these Serre equivalences to express colimits in terms of limits and to relate the abstract representation theories of chunks by infinite chains of adjunctions.

On a more abstract side, along the way we establish a general decomposition result for colimits, which specializes to the classical Bousfield-Kan formulas. We also include a short discussion of abstract formulas and their compatibility with morphisms, leading to the idea of universal formulas in monoidal homotopy theories.

13. Generalized stability for abstract homotopy theories (j/w Shulman, April 2017, arXiv, submitted)

12. Book project on derivators, Volume I (January 2016, under construction, draft version)

11. A short course on ∞-categories (new version: January 2015, arXiv, submitted)

_{∞}-ring spectra, E

_{∞}-ring spectra, and Derived Algebraic Geometry.

### Publications and accepted papers

10. Revisiting the canonicity of canonical triangulations (February 2016, to appear in Theory and Applications of Categories)These results rely on a more careful study of morphisms of derivators and this study is of independent interest. We analyze the interaction of morphisms of derivators with limits, colimits, and Kan extensions, including a discussion of invariance and closure properties of the class of Kan extensions preserved by a fixed morphism.

9. Abstract tilting theory for quivers and related categories (j/w Stovicek, to appear in Annals of K-Theory)

Specializing to representations over a field and to specific shapes, this re- covers 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.

8. Abstract representation theory of Dynkin quivers of type A (j/w Stovicek, Advances in Mathematics, Volume 293 (2016), pp. 856-941)

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.

7. Tilting theory for trees via stable homotopy theory (j/w Stovicek, J. Pure Appl. Algebra 220 (2016), no. 6, pp. 2324-2363)

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.

6. Tilting theory via stable homotopy theory (j/w Stovicek, January 2014, arXiv, to appear in Crelle's Journal, 62 pages, doi:10.1515/crelle-2015-0092)

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.

5. On autoequivalences of the (∞,1)-category of ∞-operads (j/w Ara and Gutierrez, Math.Z., Volume 281, Issue 3(2015), pp. 807-848, article available here)

_{n}-spaces.

4. Universality of multiplicative infinite loop space machines (j/w Gepner and Nikolaus, Algebraic & Geometric Topology 15 (2015), 3107-3153)

The main tool we use to establish these results is the theory of smashing localizations of presentable ∞-categories. In particular, we identify preadditive and additive ∞-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 \otimes C) = Ring(D) \otimes C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞-categories.

3. The additivity of traces in monoidal derivators (j/w Ponto and Shulman, Journal of K-Theory 14 (2014), issue 03, pp. 422-494)

May proved such an additivity theorem when the stable structure is a triangulation, based on new axioms for monoidal triangulated categories. In this paper we use stable derivators instead, which are a different model for 'stable homotopy theories'. We define and study monoidal structures on derivators, providing a context to describe the interplay between stability and monoidal structure using only ordinary category theory and universal properties. We can then perform May's proof of the additivity of traces in a closed monoidal stable derivator without needing extra axioms, as all the needed compatibility is automatic.

2. Mayer-Vietoris sequences in stable derivators (j/w Ponto and Shulman, HHA 16 (2014), 265-294, article available here)

1. Derivators, pointed derivators, and stable derivators (Algebraic & Geometric Topology 13 (2013), 313-374)

### In preparation

'Abstract stabilization: the universal absolute' (j/w Mike Shulman).'Spectral Picard groups and exceptional weights' (j/w Jan Stovicek).

'Global Serre dualities' (j/w Falk Beckert).

'Abstract cubical homotopy theory and the parasimplicial S.-construction' (j/w Falk Beckert).

'Derivator topoi and classical homotopy theory' (j/w Lyne Moser and George Raptis).

'The theory of derivators. Volume I.' (research monograph)

'The theory of derivators. Volume II' (research monograph)

### Older notes

3. Characterizations of abstract stable homotopy theories (February 2016, largely subsumed by Generalized stability for abstract homotopy theories which is j/w Shulman)2. Monoidal derivators (July 2012, will be subsumed by research monographs.)

1. Monoidal derivators and additive derivators (March 2012, will be subsumed by research monographs.)

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