arXiv subjects

Tobias Lenz

Publications and source records attributed to Tobias Lenz.

At least 19 recordsLinked to original sources

arXiv2026

Global $\infty$-categories and global Thom spectra

We introduce a framework of (Lie-)global $\infty$-categories, which formalizes various families of $\infty$-categories indexed by compact Lie groups and equipped with suitable restriction functors along continuous group homomorphisms that occur naturally in equivariant homotopy theory and representation theory. As our main results, we show that in this framework unstable and stable equivariant and global homotopy theory admit universal properties, refining and generalizing the results for finite groups from arXiv:2301.08240 and arXiv:2307.11001. In particular, we characterize the passage from unstable to stable equivariant and global homotopy theory at the level of global $\infty$-categories as universally inverting the action of representation spheres in an appropriate sense. Building on this, we define parametrized equivariant and global Thom spectrum functors and show that they recover classical Thom spectrum constructions defined in terms of pointset models.

math.AT↗

arXiv2026

Universality of span 2-categories and the construction of 6-functor formalisms

Given an $\infty$-category $C$ equipped with suitable wide subcategories $I, P \subset E\subset C$, we show that the $(\infty,2)$-category $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I}$ of higher (or iterated) spans defined by Haugseng has the universal property that 2-functors $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I} \to \mathbb D$ correspond precisely to $(I, P)$-biadjointable functors $C^\text{op} \to \mathbb D$, i.e. functors $F$ where $F(i)$ for $i \in I$ admits a left adjoint and $F(p)$ for $p \in P$ admits a right adjoint satisfying various Beck-Chevalley conditions. We also extend this universality to the symmetric monoidal and lax symmetric monoidal settings. This provides a conceptual explanation for - and an independent proof of - the Mann-Liu-Zheng construction of 6-functor formalisms from suitable functors $C^\text{op}\to\text{CAlg}(\text{Cat})$.

math.CT↗

arXiv2026

Norms in equivariant homotopy theory

We show that the $\infty$-category of normed algebras in genuine $G$-spectra, as introduced by Bachmann-Hoyois, is modelled by strictly commutative algebras in $G$-symmetric spectra for any finite group $G$. We moreover provide an analogous description of Schwede's ultra-commutative global ring spectra in higher categorical terms. Using these new descriptions, we exhibit the $\infty$-category of ultra-commutative global ring spectra as a partially lax limit of the $\infty$-categories of genuine $G$-spectra for varying $G$, in analogy with the non-multiplicative comparison of Nardin, Pol, and the second author. Along the way, we establish various new results in parametrized higher algebra, which we hope to be of independent interest.

math.AT↗

arXiv2025

Revisiting $(\infty,2)$-naturality of the Yoneda embedding

We show that the Yoneda embedding 'is' $(\infty,2)$-natural with respect to the functoriality of presheaves via left Kan extension, refining the $(\infty,1)$-categorical result proven independently by Haugseng-Hebestreit-Linskens-Nuiten and Ramzi, and answering a question of Ben-Moshe. As the key technical ingredient, we show that the identity functor of the $(\infty,1)$-category of $(\infty,1)$-categories admits only one enhancement to an $(\infty,2)$-functor (namely, the identity functor).

math.CT↗

arXiv2025

Parametrized stability and the universal property of global spectra

We develop a framework of parametrized semiadditivity and stability with respect to so-called atomic orbital subcategories of an indexing $\infty$-category $T$, extending work of Nardin. Specializing this framework, we introduce global $\infty$-categories and the notions of equivariant semiadditivity and stability, yielding a higher categorical version of the notion of a Mackey 2-functor studied by Balmer-Dell'Ambrogio. As our main result, we identify the free presentable equivariantly stable global $\infty$-category with a natural global $\infty$-category of global spectra for finite groups, in the sense of Schwede and Hausmann.

math.AT↗

arXiv2025

Global homotopy theory via spectral Mackey functors

We show that Hausmann's model of global stable homotopy theory in terms of symmetric spectra is equivalent to the $\infty$-category of spectral Mackey functors in the sense of Barwick on a certain global effective Burnside category. We moreover provide an analogous description of Schwede's ultra-commutative monoids as space-valued global Mackey functors.

math.AT↗

arXiv2025

Mackey functors and classical equivariant $K$-theory

We show that the spectral Mackey functors associated to the equivariant algebraic $K$-theory spectra of Guillou-May and Merling (originally constructed using pointset models) can be described purely $\infty$-categorically in terms of the monoidal Borel construction of Barwick-Glasman-Shah and Hilman. We moreover show how Pützstück's global version of the Borel construction provides an analogous description of the global spectral Mackey functors arising from Schwede's global algebraic $K$-theory spectra. Our arguments crucially rely on techniques from parametrized higher category theory as well as on structural results on global and equivariant $K$-theory to avoid any explicit computations.

math.AT↗

arXiv2025

Parametrized (higher) semiadditivity and the universality of spans

Semiadditivity of an $\infty$-category, i.e. the existence of biproducts, provides it with useful algebraic structure in the form of a canonical enrichment in commutative monoids. This ultimately comes from the fact that the $\infty$-category of commutative monoids is the universal semiadditive $\infty$-category equipped with a finite-product-preserving functor to spaces, or equivalently that the $(2,1)$-category of spans of finite sets is the universal semiadditive $\infty$-category. In this article, we prove a vast generalization of these facts in the context of parametrized semiadditivity, a notion we define using Hopkins-Lurie's framework of ambidexterity. This simultaneously generalizes a result of Harpaz for higher semiadditivity and a result of Nardin for equivariant semiadditivity. We deduce that every parametrized semiadditive $\infty$-category is canonically enriched in Mackey functors/sheaves with transfers. As an application, we reprove the Mackey functor description of global spectra first obtained by the second-named author and generalize it to $G$-global spectra. Moreover, we obtain universal characterizations of the $\infty$-categories of $\mathbb Z$-valued $G$-Mackey profunctors and of quasi-finitely genuine $G$-spectra as studied by Kaledin and Krause-McCandless-Nikolaus, respectively.

math.AT↗

arXiv2025

Homotopical commutative rings and bispans

We prove that commutative semirings in a cartesian closed presentable $\infty$-category, as defined by Groth, Gepner, and Nikolaus, are equivalent to product-preserving functors from the $(2,1)$-category of bispans of finite sets. In other words, we identify the latter as the Lawvere theory for commutative semirings in the $\infty$-categorical context. This implies that connective commutative ring spectra can be described as grouplike product-preserving functors from bispans of finite sets to spaces. A key part of the proof is a localization result for $\infty$-categories of spans, and more generally for $\infty$-categories with factorization systems, that may be of independent interest.

math.CT↗

arXiv2025

Partial parametrized presentability and the universal property of equivariant spectra

We introduce a notion of partial presentability in parametrized higher category theory and investigate its interaction with the concepts of parametrized semiadditivity and stability from arXiv:2301.08240. In particular, we construct the free partially presentable $T$-categories in the unstable, semiadditive, and stable contexts and explain how to exhibit them as full subcategories of their fully presentable analogues. Specializing our results to the setting of (global) equivariant homotopy theory, we obtain a notion of equivariant presentability for the global categories of arXiv:2301.08240, and we show that the global category of genuine equivariant spectra is the free global category that is both equivariantly presentable and equivariantly stable. As a consequence, we deduce the analogous result about the $G$-category of genuine $G$-spectra for any finite group $G$, previously formulated by Nardin.

math.AT↗

arXiv2025

Universality of Barwick's unfurling construction

Given an $\infty$-category $\mathcal{C}$ with pullbacks, its $(\infty,2)$-category $\mathbf{Span}(\mathcal{C})$ of spans has the universal property of freely adding right adjoints to morphisms in $\mathcal{C}$ satisfying a Beck--Chevalley condition. We show that this universal property is implemented by an $(\infty,2)$-categorical refinement of Barwick's \emph{unfurling construction}: For any right adjointable functor $\mathcal{C} \to \mathrm{Cat}_{\infty}$, the unstraightening of its unique extension to $\mathbf{Span}(\mathcal{C})$ can be explicitly written down as another span $(\infty,2)$-category, and on underlying $(\infty,1)$-categories this recovers Barwick's construction. As an application, we show that the constructions of cartesian normed structures by Nardin--Shah and Cnossen--Haugseng--Lenz--Linskens coincide.

math.AT↗

arXiv2024

The Adams isomorphism revisited

We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more generally in the settings of $G$-Mackey functors, $G$-global homotopy theory, and equivariant Kasparov categories.

math.AT↗

arXiv2024

Global model categories and topological André-Quillen cohomology

We introduce global model categories as a general framework to capture several phenomena in global equivariant homotopy theory. We then construct genuine stabilizations of these, generalizing the usual passage from unstable to stable global homotopy theory. Finally, we define the global topological André-Quillen cohomology of an ultra-commutative ring spectrum and express it in terms of a genuine stabilization in our framework in analogy with the classical non-equivariant description obtained by Basterra and Mandell.

math.AT↗

arXiv2024

Normed equivariant ring spectra and higher Tambara functors

In this paper we extend equivariant infinite loop space theory to take into account multiplicative norms: For every finite group $G$, we construct a multiplicative refinement of the comparison between the $\infty$-categories of connective genuine $G$-spectra and space-valued Mackey functors, first proven by Guillou-May, and use this to give a description of connective normed equivariant ring spectra as space-valued Tambara functors. In more detail, we first introduce and study a general notion of homotopy-coherent normed (semi)rings, and identify these with product-preserving functors out of a corresponding $\infty$-category of bispans. In the equivariant setting, this identifies space-valued Tambara functors with normed algebras with respect to a certain normed monoidal structure on grouplike $G$-commutative monoids in spaces. We then show that the latter is canonically equivalent to the normed monoidal structure on connective $G$-spectra given by the Hill-Hopkins-Ravenel norms. Combining our comparison with results of Elmanto-Haugseng and Barwick-Glasman-Mathew-Nikolaus, we produce normed ring structures on equivariant algebraic K-theory spectra.

math.AT↗

arXiv2024

Simplicial $*$-modules and mild actions

We develop an analogue of the theory of $*$-modules in the world of simplicial sets, based on actions of a certain simplicial monoid $E\mathcal M$ originally appearing in the construction of global algebraic $K$-theory. As our main results, we show that strictly commutative monoids with respect to a certain box product on these simplicial $*$-modules yield models of equivariantly and globally coherently commutative monoids, and we give a characterization of simplicial $*$-modules in terms of a certain mildness condition on the $E\mathcal M$-action, relaxing the notion of tameness previously investigated by Sagave-Schwede and the first author.

math.AT↗

arXiv2023

Genuine vs. naïve symmetric monoidal G-categories

We prove that through the eyes of equivariant weak equivalences the genuine symmetric monoidal $G$-categories of Guillou and May [Algebr. Geom. Topol. 17 (2017), no. 6, 3259-3339; arXiv:1809.03017] are equivalent to just ordinary symmetric monoidal categories with $G$-action. Along the way, we give an operadic model of global infinite loop spaces and provide an equivalence between the equivariant category theory of genuine symmetric monoidal $G$-categories and the $G$-parsummable categories studied by Schwede [J. Topol. 15 (2022), no. 3, 1325-1454; arXiv:1912.08872] and the author [New York J. Math. 29 (2023), 635-686; arXiv:2009.07004].

math.AT↗

arXiv2023

Categorical models of unstable G-global homotopy theory

We prove that the category $\textbf{G-Cat}$ of small categories with $G$-action forms a model of unstable $G$-global homotopy theory for every discrete group $G$, generalizing Schwede's global model structure on $\textbf{Cat}$. As a consequence, we prove that $\textbf{G-Cat}$ models proper $G$-equivariant homotopy theory not only when we test weak equivalences on fixed points, but also when we test them on categorical homotopy fixed points.

math.AT↗

arXiv2023

On the global homotopy theory of symmetric monoidal categories

Parsummable categories were introduced by Schwede as input for his global algebraic $K$-theory construction. We prove that their whole homotopy theory with respect to the so-called global equivalences can already be modelled by the more mundane symmetric monoidal categories. In another direction, we show that the resulting homotopy theory is also equivalent to the homotopy theory of a certain simplicial analogue of parsummable categories, that we call parsummable simplicial sets. These form a bridge to several concepts of 'globally coherently commutative monoids' like ultra-commutative monoids and global $Γ$-spaces, that we explore in arXiv:2012.12676.

math.AT↗