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Constanze Roitzheim

Publications and source records attributed to Constanze Roitzheim.

At least 19 recordsLinked to original sources
arXiv2026

Bousfield Localizations on the Nonmodular Lattice $N_5$

We provide a complete description of the model category structures on the nonmodular lattice $N_5$. Furthermore we explain how these model category structures are related to each other via Bousfield localization. This work heavily relies on the use of combinatorical objects from equivariant homotopy theory known as \emph{transfer systems}, and it results in a wealth of interesting interactions between combinatorial and homotopical methods.

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arXiv2025

Left and right Bousfield localization on lattices

The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. We describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating system of morphisms which are responsible for the change in model structure. This leads to new concrete insights into the behavior of model categories on posets in general.

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arXiv2025

Characterizing model structures on finite posets

Transfer systems on finite posets have recently been gaining traction as a key ingredient in equivariant homotopy theory. Additionally, they also naturally occur in the data of a model structure. We give a complete characterization of all model category structures on a finite lattice, using transfer systems as our main tool, resulting in new connections between abstract homotopy theory and equivariant methods.

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arXiv2024

Uniquely compatible transfer systems for cyclic groups of order $p^rq^s$

Bi-incomplete Tambara functors over a group $G$ can be understood in terms of compatible pairs of $G$-transfer systems. In the case of $G = C_{p^n}$ , Hill, Meng and Li gave a necessary and sufficient condition for compatibility and computed the exact number of compatible pairs. In this article, we study compatible pairs of $G$-transfer systems for the case $G = C_{p^rq^s}$ and identify conditions when such transfer systems are uniquely compatible in the sense that they only form trivially compatible pairs. This gives us new insight into collections of norm maps that are relevant in equivariant homotopy theory.

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arXiv2023

The Derived Deligne Conjecture

Derived $A_\infty$-algebras have a wealth of theoretical advantages over regular $A_\infty$-algebras. However, due to their bigraded nature, in practice they are often unwieldy to work with. We develop a framework involving brace algebras on operads which allows us to study derived $A_\infty$ algebras in a new conceptual context. One particular advantage is that this construction allows us to generalize the Lie algebra structure on the Hochschild complex of an $A_\infty$-algebra, obtaining new and rigorous versions of the Deligne conjecture.

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arXiv2023

Model structures on finite total orders

We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro's Catalan triangle. This is an application of previous work of the authors on the theory of $N_\infty$-operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of $[n]$.

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arXiv2023

Levels of algebraicity in stable homotopy theories

We study several different notions of algebraicity in use in stable homotopy theory and prove implications between them. The relationships between the different meanings of algebraic are unexpectedly subtle, and we illustrate this with several interesting examples arising from chromatic homotopy theory.

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arXiv2023

Tensor-triangular rigidity in chromatic homotopy theory

We study the uniqueness of enhancements of tensor-triangulated categories. To do so, we provide conditions under which these enhancements interact well with categorical decompositions. As an application we obtain new results about the uniqueness of enhancements in chromatic homotopy theory.

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arXiv2023

Monoidal Properties of Franke's Exotic Equivalence

Franke's reconstruction functor R is known to provide examples of triangulated equivalences between homotopy categories of stable model categories, which are exotic in the sense that the underlying model categories are not Quillen equivalent. We show that, while not being a tensor-triangulated functor in general, R is compatible with monoidal products.

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arXiv2021

$N_\infty$-operads and associahedra

We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the collection of N-infinity-operads stands in bijection with the poset structure of the (n+1)-associahedron. We further provide a lower bound for the number of possible N-infinity-operads for any finite cyclic group G.

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arXiv2020

Equivariant homotopy commutativity for $G=C_{pqr}$

We investigate the combinatorial data arising from the classification of equivariant homotopy commutativity for cyclic groups of order $G=C_{p_1 \cdots p_n}$ for $p_i$ distinct primes. In particular, we will prove a structural result which allows us to enumerate the number of $N_\infty$-operads for $C_{pqr}$, verifying a computational result.

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arXiv2018

Homotopy Theory of Bicomplexes

We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure's weak equivalences are detected by the $E^2$-term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes.

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arXiv2017

Endomorphisms of Exotic Models

We calculate the endomorphism dga of Franke's exotic algebraic model for the $K$-local stable homotopy category at odd primes. We unravel its original abstract structure to give explicit generators, differentials and products.

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arXiv2017

Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory

We prove that the $v_1$-local $G$-equivariant stable homotopy category for $G$ a finite group has a unique $G$-equivariant model at $p=2$. This means that at the prime $2$ the homotopy theory of $G$-spectra up to fixed point equivalences on $K$-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for $K$-local spectra of the second author with the equivariant rigidity result for $G$-spectra of the first author. Further, when the prime $p$ is at least $5$ and does not divide the order of $G$, we provide an algebraic exotic model as well as a $G$-equivariant exotic model for the $v_1$-local $G$-equivariant stable homotopy category, showing that for primes $p \ge 5$ equivariant rigidity fails in general.

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arXiv2017

Bousfield localisations along Quillen bifunctors

Consider a Quillen adjunction of two variables between combinatorial model categories from $\mathcal{C}\times\mathcal{D}$ to $\mathcal{E}$, and a set $\mathcal{S}$ of morphisms in $\mathcal{C}$. We prove that there is a localised model structure $L_{\mathcal{S}}\mathcal{E}$ on $\mathcal{E}$, where the local objects are the $\mathcal{S}$-local objects in $\mathcal{E}$ described via the right adjoint. These localised model structures generalise Bousfield localisations of simplicial model categories, Barnes and Roitzheim's familiar model structures, and Barwick's enriched Bousfield localisations. In particular, we can use these model structures to define Postnikov sections in more general left proper combinatorial model categories.

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arXiv2016

Towers and fibered products of model structures

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.

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arXiv2012

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E-localisation of this model category. We study the properties of this new construction and relate it to some well-known categories.

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