arXiv subjects

Aaron Mazel-Gee

Publications and source records attributed to Aaron Mazel-Gee.

At least 19 recordsLinked to original sources

arXiv2024

Symmetries of the cyclic nerve

We undertake a systematic study of the Hochschild homology, i.e. (the geometric realization of) the cyclic nerve, of $(\infty,1)$-categories (and more generally of category-objects in an $\infty$-category), as a version of factorization homology. In order to do this, we codify $(\infty,1)$-categories in terms of quiver representations in them. By examining a universal instance of such Hochschild homology, we explicitly identify its natural symmetries, and construct a non-stable version of the cyclotomic trace map. Along the way we give a unified account of the cyclic, paracyclic, and epicyclic categories. We also prove that this gives a combinatorial description of the $n=1$ case of factorization homology as presented in [AFR18], which parametrizes $(\infty,1)$-categories by solidly 1-framed stratified spaces.

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arXiv2024

A braided monoidal $(\infty,2)$-category of Soergel bimodules

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and 4. Here we provide the first categorification of this Hecke braided monoidal category, which takes the form of an $\mathbb{E}_2$-monoidal $(\infty,2)$-category whose hom-$(\infty,1)$-categories are $k$-linear, stable, idempotent-complete, and equipped with $\mathbb{Z}$-actions. This categorification is designed to control homotopy-coherent link homology theories and to-be-constructed topological quantum field theories in dimension 4 and 5. Our construction is based on chain complexes of Soergel bimodules, with monoidal structure given by parabolic induction and braiding implemented by Rouquier complexes, all modelled homotopy-coherently. This is part of a framework which allows to transfer the toolkit of the categorification literature into the realm of $\infty$-categories and higher algebra. Along the way, we develop families of factorization systems for $(\infty,n)$-categories, enriched $\infty$-categories, and $\infty$-operads, which may be of independent interest. As a service aimed at readers less familiar with homotopy-coherent mathematics, we include a brief introduction to the necessary $\infty$-categorical technology in the form of an appendix.

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arXiv2023

Stratified noncommutative geometry

We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable $\infty$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as $E_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$ is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory.

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arXiv2023

Perverse schobers and 3d mirror symmetry

The proposed physical duality known as 3d mirror symmetry relates the geometries of dual pairs of holomorphic symplectic stacks. It has served in recent years as a guiding principle for developments in representation theory. However, due to the lack of definitions, thus far only small pieces of the subject have been mathematically accessible. In this paper, we formulate abelian 3d mirror symmetry as an equivalence between a pair of 2-categories constructed from the algebraic and symplectic geometry, respectively, of Gale dual toric cotangent stacks. In the simplest case, our theorem provides a spectral description of the 2-category of spherical functors - i.e., perverse schobers on the affine line with singularities at the origin. We expect that our results can be extended from toric cotangent stacks to hypertoric varieties, which would provide a categorification of previous results on Koszul duality for hypertoric categories $\mathcal{O}$. Our methods also suggest approaches to 2-categorical 3d mirror symmetry for more general classes of spaces of interest in geometric representation theory. Along the way, we establish two results that may be of independent interest: (1) a version of the theory of Smith ideals in the setting of stable $\infty$-categories; and (2) an ambidexterity result for co/limits of presentable enriched $\infty$-categories over $\infty$-groupoids.

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arXiv2023

Derived Mackey functors and $C_{p^n}$-equivariant cohomology

We establish a novel approach to computing $G$-equivariant cohomology for a finite group $G$, and demonstrate it in the case that $G = C_{p^n}$. For any commutative ring spectrum $R$, we prove a symmetric monoidal reconstruction theorem for genuine $G$-$R$-modules, which records them in terms of their geometric fixedpoints as well as gluing maps involving their Tate cohomologies. This reconstruction theorem follows from a symmetric monoidal stratification (in the sense of \cite{AMR-strat}); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from genuine $G$-spectra to genuine $G$-$\mathbb{Z}$-modules (a.k.a. derived Mackey functors) provides a convenient intermediate category for calculating equivariant cohomology. Indeed, as $\mathbb{Z}$-linear Tate cohomology is far simpler than $\mathbb{S}$-linear Tate cohomology, the above reconstruction theorem gives a particularly simple algebraic description of genuine $G$-$\mathbb{Z}$-modules. We apply this in the case that $G = C_{p^n}$ for an odd prime $p$, computing the Picard group of genuine $G$-$\mathbb{Z}$-modules (and therefore that of genuine $G$-spectra) as well as the $RO(G)$-graded and Picard-graded $G$-equivariant cohomology of a point.

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arXiv2021

A universal characterization of noncommutative motives and secondary algebraic K-theory

We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to Blumberg--Gepner--Tabuada. As a consequence, we obtain a corepresentability theorem for secondary K-theory. We envision this as a fundamental tool for the construction of trace maps from secondary K-theory. Towards these main goals, we introduce a preliminary formalism of "stable $(\infty, 2)$-categories"; notable examples of these include (quasicoherent or constructible) sheaves of stable $\infty$-categories. We also develop the rudiments of a theory of presentable enriched $\infty$-categories -- and in particular, a theory of presentable $(\infty, n)$-categories -- which may be of intependent interest.

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arXiv2021

Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements

Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them. Predicated on this, we then characterize dualizability in any monoidally stratified $\infty$-category in terms of stratumwise dualizability and a projection formula for the links. Using our criterion, we prove a 1-dimensional bordism hypothesis for symmetric monoidal recollements. Namely, we provide an algebraic enhancement of the 1-dimensional framed bordism $\infty$-category that corepresents dualizable objects in symmetric monoidal recollements. We also give a number of examples and applications of our criterion drawn from algebra and homotopy theory, including equivariant and cyclotomic spectra and a multiplicative form of the Thom isomorphism.

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arXiv2019

$\mathbb{E}_\infty$ automorphisms of motivic Morava $E$-theories

We apply Goerss--Hopkins obstruction theory for motivic spectra to study the motivic Morava $E$-theories. We find that they always admit $\mathbb{E}_\infty$ structures, but that these may admit "exotic" $\mathbb{E}_\infty$ automorphisms not coming from the usual Morava stabilizer group.

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arXiv2018

Goerss--Hopkins obstruction theory for $\infty$-categories

Goerss--Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model $\infty$-categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable $\infty$-category (such as that of equivariant spectra or of motivic spectra).

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arXiv2017

The geometry of the cyclotomic trace

We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\infty$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X) \to TC(X)$ from algebraic K-theory to topological cyclic homology for any scheme $X$. This construction rests on a new identification of the cyclotomic structure on $THH(C)$, which we find to be a consequence of (i) the geometry of 1-manifolds, and (ii) linearization (in the sense of Goodwillie calculus). Our construction of the cyclotomic trace likewise arises from the linearization of more primitive data.

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arXiv2017

Factorization homology of enriched $\infty$-categories

For an arbitrary symmetric monoidal $\infty$-category $\mathcal{V}$, we define the factorization homology of $\mathcal{V}$-enriched $(\infty,1)$-categories over (possibly stratified) 1-manifolds and study some of its basic properties. In the case of spectral enrichment, we show that the value of factorization homology on a circle is topological Hochschild homology.

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arXiv2017

A naive approach to genuine $G$-spectra and cyclotomic spectra

For any compact Lie group $G$, we give a description of genuine $G$-spectra in terms of the naive equivariant spectra underlying their geometric fixedpoints. We use this to give an analogous description of cyclotomic spectra in terms of naive $T$-spectra (where $T$ denotes the circle group), generalizing Nikolaus--Scholze's recent work in the eventually-connective case. We also give an explicit formula for the homotopy invariants of the cyclotomic structure on a cyclotomic spectrum in these terms.

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arXiv2015

Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces

Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of simplicial sets in this capacity; this is due to the existence of a suitable model structure thereon, which is particularly convenient to work with since it enjoys the technical properties of being *proper* and of being *cofibrantly generated*. This paper is devoted to showing that, if one is willing to work $\infty$-categorically, then one can manipulate simplicial spaces exactly as one manipulates simplicial sets. Precisely, this takes the form of a proper, cofibrantly generated model structure on the *$\infty$-category* of simplicial spaces, the definition of which we also introduce here.

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arXiv2015

Model $\infty$-categories III: the fundamental theorem

We prove that a model structure on a relative $\infty$-category $(M,W)$ gives an efficient and computable way of accessing the hom-spaces $hom_{M[[W^{-1}]]}(x,y)$ in the localization. More precisely, we show that when the source $x \in M$ is *cofibrant* and the target $y \in M$ is *fibrant*, then this hom-space is a "quotient" of the hom-space $hom_M(x,y)$ by either of a *left homotopy relation* or a *right homotopy relation*.

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arXiv2015

Model $\infty$-categories II: Quillen adjunctions

We prove that various structures on model $\infty$-categories descend to corresponding structures on their localizations: (i) Quillen adjunctions; (ii) two-variable Quillen adjunctions; (iii) monoidal and symmetric monoidal model structures; and (iv) enriched model structures.

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arXiv2015

Hammocks and fractions in relative $\infty$-categories

We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Kan, we define a *hammock localization* functor from relative $\infty$-categories to $sS$-enriched $\infty$-categories, thus providing a rich source of examples of $sS$-enriched $\infty$-categories. Simultaneously unpacking and generalizing one of their key results, we prove that given a relative $\infty$-category admitting a *homotopical three-arrow calculus*, one can explicitly describe the hom-spaces in the $\infty$-category presented by its hammock localization in a much more explicit and accessible way. As an application of this framework, we give sufficient conditions for the Rezk nerve of a relative $\infty$-category to be a (complete) Segal space, generalizing joint work with Low.

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arXiv2015

All about the Grothendieck construction

We provide, among other things: (i) a Bousfield--Kan formula for colimits in $\infty$-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) $\infty$-categorical generalizations of Barwick--Kan's Theorem B$_n$ and Dwyer--Kan--Smith's Theorem C$_n$ (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of diagram $\infty$-categories.

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arXiv2015

The universality of the Rezk nerve

We functorially associate to each relative $\infty$-category $(R,W)$ a simplicial space $N^R_\infty(R,W)$, called its Rezk nerve (a straightforward generalization of Rezk's "classification diagram" construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve $N^R_\infty(R,W)$ is precisely the one corresponding to the localization $R[[W^{-1}]]$; and (ii) that the Rezk nerve functor defines an equivalence $RelCat_\infty [[ W_{BK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of relative $\infty$-categories to the $\infty$-category of $\infty$-categories.

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