# Moritz Groth

Home · Research

### Preprints

14. Abstract cubical homotopy theory (j/w Beckert, March 2018, 71 pages, arXiv, submitted) Triangulations and higher triangulations axiomatize the calculus of derived cokernels when applied to strings of composable morphisms. While
there are no cubical versions of (higher) triangulations, in this paper we use coherent diagrams to develop some aspects of a rich cubical calculus. Applied
to the models in the background, this enhances the typical examples of triangulated and tensor-triangulated categories.

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.

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)

We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if
homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations
generalize to an abstract notion of "stability relative to a class of functors", which includes in particular pointedness, semiadditivity, and ordinary
stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.

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

Assuming basic acquaintance with homological algebra only, in the first volume we give a careful motivation for the notion of a derivator. We discuss
the existence of canonical triangulations in stable derivators, indicating that stable derivators provide an enhancement of triangulated categories.
This result is taken as a pretext to study various basic constructions in derivators, like iterated (co)fiber constructions, Barratt-Puppe sequences,
and refined octahedron diagrams. To illustrate the added flexibility of derivators we also include a short discussion of total (co)fibers of squares.
On a more technical side, we establish various tools related to (parametrized) Kan extensions and (co)limiting (co)cones, which are of constant use also
in the sequels.

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

In this short survey we give a non-technical introduction to some
main ideas of the theory of ∞-categories, hopefully facilitating the digestion
of the foundational work of Joyal and Lurie. Besides the basic ∞-categorical
notions leading to presentable ∞-categories, we mention the Joyal and Bergner
model structures organizing two approaches to a theory of (∞,1)-categories.
We also discuss monoidal ∞-categories and algebra objects, as well as stable
∞-categories. These notions come together in Lurie's treatment of the smash
product on spectra, yielding a convenient framework for the study of A

_{∞}-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) Stable derivators provide an enhancement of triangulated categories as is indicated by the existence of canonical triangulations.
In this paper we show that exact morphisms of stable derivators induce exact functors of canonical triangulations, and similarly for arbitrary natural
transformations. This 2-categorical refinement also provides a uniqueness statement concerning canonical triangulations.

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.

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)

We generalize the construction of reflection functors from classi-
cal 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 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.

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)

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

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)

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.

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)

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.

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)

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

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.

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

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.

### 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) Stable derivators provide an enhancement of triangulated categories as is indicated by the existence of canonical triangulations.
In this paper we establish new characterizations of stable derivators, thereby obtaining additional interpretations of the passage from (pointed)
topological spaces to spectra and, more generally, of the stabilization. We show that a derivator is stable if and only if homotopy finite limits and homotopy
finite colimits commute, and there are variants for sufficiently finite Kan extensions. As an additional reformulation, a derivator is stable if and only if it
admits a zero object and if partial cone and partial fiber morphisms commute on squares.

2. Monoidal derivators (July 2012, will be subsumed by research monographs.)

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, will be subsumed by research monographs.)

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.

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