I am a postdoc at the MPI in Bonn. My research interest is focused on applications of Abstract Homotopy
Theory to other fields in Geometry and Algebra like Algebraic Topology, (Derived) Algebraic Geometry, and Representation Theory.
Besides being interested in the theory and applications of derivators, a notion introduced independently by Heller and Grothendieck,
the broad fields of Higher Category Theory and Higher Operad Theory also attract my attention.

Email: mgroth at mpim-bonn.mpg.de

Office: Room B14, Vivatsgasse 7, 53111 Bonn

Introduction to the formalism of derivators. This is built on work of Grothendieck, Heller and others and is a powerful and elegant approach to homological algebra, and homotopy theory, encompassing triangulated categories and derived functors. Details on the summer school *(Freiburg, August 18-22)* are here.

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Motivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be 'additive'. When the category is 'stable' in some sense, additivity along cofiber sequences is a question about the interaction of stability and the monoidal structure.

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.

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)

We show that stable derivators, like stable model categories, admit Mayer-Vietoris sequences arising from cocartesian squares. Along the way we characterize homotopy exact squares, and give a detection result for colimiting diagrams in derivators. As an application, we show that a derivator is stable if and only if its suspension functor is an equivalence.

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

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.

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.

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

5. Tilting theory for trees via stable homotopy theory (j/w Stovicek, February 2014, arXiv, submitted)

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.

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.

4. Tilting theory via stable homotopy theory (j/w Stovicek, January 2014, arXiv, submitted)

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.

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.

3. On autoequivalences of the (∞,1)-category of ∞-operads (j/w Ara and Gutierrez, December 2013, arXiv, submitted)

We study the (∞,1)-category of autoequivalences of ∞-operads. Using techniques introduced by Toen, Lurie, and Barwick and Schommer-Pries, we prove that this (∞,1)-category is a contractible ∞-groupoid. Our calculation is based on the model of complete dendroidal Segal spaces introduced by Cisinski and Moerdijk. Similarly, we prove that the (∞,1)-category
of autoequivalences of non-symmetric ∞-operads is the discrete monoidal category associated to Z/2Z. We also include a computation of the (∞,1)-category of autoequivalences of (∞,n)-categories based on Rezk's Θ_{n}-spaces.

2. Universality of multiplicative infinite loop space machines (j/w Gepner and Nikolaus, May 2013, arXiv, submitted)

We establish a canonical and unique tensor product for commutative monoids and groups in an ∞-category which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that E_n-(semi)ring objects give rise to E_n-ring spectrum objects. In the case that the ∞-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 ∞-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.

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.

1. A short course on ∞-categories (July 2010, arXiv, under revision, to be submitted)

These notes are meant to give a concise introduction to the theory of ∞-categories (aka. quasicategories) as developed in particular by Jacob Lurie and Andre Joyal. Some central ideas one finds in Lurie's *Higher Topos Theory* and *Derived Algebraic Geometry I-III* are covered (now subsumed in Lurie's book *Higher Algebra*).

'Abstract representation theory for acyclic quivers' (j/w Jan Stovicek).

'Tilting theory for posets via stable homotopy theory' (j/w Jan Stovicek).

'Abstract iterated tilting theory for Dynkin quivers of type A' (j/w Jan Stovicek).

'Gorenstein homological algebra via derivators' (j/w Greg Stevenson).

'Equivariant derivators' (j/w Ivo Dell'Ambrogio and Irakli Patchkoria).

'A short course on derivators'.

This preprint develops in a rather detailed way some foundational aspects of the theory of monoidal (pre)derivators and their modules. We give more details than typically given in a research article and hope consequently this account to be easy to digest. A more condensed treatment (and one which goes beyond what is covered here!) will be given in 'Additivity of traces in monoidal derivators' (j/w Kate Ponto and Mike Shulman).

1. Monoidal derivators and additive derivators (March 2012, arXiv)

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.

Introduction to derivators (three talks)

August 2014, Universität Freiburg, Germany:

Summer school on derivators (four talks)

November 2013, University of Copenhagen, Denmark:

Master course on ∞-categories (six talks)

September 2013, Université catholique de Louvain, Belgium:

Workshop on Homotopy Theory of ∞-categories (four talks)

July 2013, EPFL, Lausanne, Switzerland:

Short course on ∞-categories (five talks)

July 2013, EPFL, Lausanne, Switzerland:

Mini-course on derivators (three talks)

October-December 2011, Radboud University, Nijmegen, Netherlands:

The canonical triangulations in stable derivators (six talks)

July 2011, Universität Osnabrück, Germany:

Short course on ∞-categories (three talks)

June 2011, Universität Regensburg, Germany:

Short course on ∞-categories (four 2-hour talks)

September 2010, University of Notre Dame, Indiana:

Short course ∞-categories (four talks)

January 2010, University of Warsaw, Poland:

Short course on ∞-categories (three talks)

Talk on 'Abstract representation theory of Dynkin quivers of type A'

May 2014, Universität Freiburg, Germany:

Talk on Tilting theory via stable homotopy theory and Tilting theory of trees via stable homotopy theory

May 2014, 21st NRW Topology Meeting, Universität Wuppertal, Germany:

Talk on Tilting theory via stable homotopy theory and Tilting theory of trees via stable homotopy theory

April 2014, Universität Augsburg, Germany:

Talk on Tilting theory via stable homotopy theory and Tilting theory of trees via stable homotopy theory

April 2014, Universität Bielefeld, Germany:

Talk on Tilting theory via stable homotopy theory and Tilting theory of trees via stable homotopy theory

March 2014, University of Sheffield, UK:

Talk on Derivators, pointed derivators, and stable derivators and Tilting theory via stable homotopy theory

February 2014, Universität Regensburg, Germany:

Talk on Tilting theory via stable homotopy theory

January 2014, Charles University in Prague, Czech Republic:

Talk on Tilting theory via stable homotopy theory

November 2013, University of Copenhagen, Denmark:

Talk on Tilting theory via stable homotopy theory

July 2013, Young Topologists Meeting, EPFL, Switzerland:

Talk on Universality of multiplicative infinite loop space machines

June 2013, CMS Summer Meeting, Halifax, Nova Scotia:

Talk on Universality of multiplicative infinite loop space machines

May 2013, Universität Osnabrück, Germany:

Talk on Derivators, pointed derivators, and stable derivators

May 2013, GQT meeting, Utrecht, Netherlands:

Talk on Derivators, pointed derivators, and stable derivators

May 2013, Charles University in Prague, Czech Republic:

Talk on Derivators, pointed derivators, and stable derivators

May 2013, Max Planck Institute for Mathematics, Bonn, Germany:

Talk on Derivators, pointed derivators, and stable derivators

April 2013, University of Illinois at Urbana-Champaign, Illinois:

Talk on Derivators, pointed derivators, and stable derivators

April 2013, University of Notre Dame, Indiana:

Two talks on Derivators, pointed derivators, and stable derivators and The additivity of traces in monoidal derivators

April 2013, UC Riverside, Calilfornia:

Talk on Derivators, pointed derivators, and stable derivators

April 2013, Stanford University, California:

Talk on The additivity of traces in monoidal derivators

April 2013, UC Berkeley, California:

Talk on Derivators, pointed derivators, and stable derivators

March 2013, University of Lille, France:

Talk on The additivity of traces in monoidal derivators

February 2013, University of Copenhagen, Denmark:

Talk on The additivity of traces in monoidal derivators

December 2012, Radboud University, Nijmegen, Netherlands:

Talk on The additivity of traces in monoidal derivators

November 2012, EPFL, Lausanne, Switzerland:

Talk on Derivators, pointed derivators, and stable derivators

May 2012, Universität Regensburg, Germany:

Two talks on Derivators, pointed derivators, and stable derivators

April 2012, Ruhr-Universität Bochum, Germany:

Two talks on Derivators, pointed derivators, and stable derivators

January 2012, UQAM, Montreal, Canada:

Two talks on Derivators, pointed derivators, and stable derivators

November 2011, Universität Bielefeld, Germany:

Talk on Derivators, pointed derivators, and stable derivators

September 2011, Universität Bonn, Germany:

Thesis defense, 30-minutes talk on Derivators, pointed derivators, and stable derivators and Monoidal and enriched derivators, the slides are here

August 2011, Universität Hamburg, Germany, Conference on Structured Ring Spectra:

10-minutes bell-show talk on Derivators, pointed derivators, and stable derivators and Monoidal and enriched derivators

September 2010, Harvard University, Massachusetts:

Two talks on Derivators, pointed derivators, and stable derivators