Aaron Mazel-Gee
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View article: Perverse schobers and 3d mirror symmetry
Perverse schobers and 3d mirror symmetry Open
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 t…
View article: Symmetries of the cyclic nerve
Symmetries of the cyclic nerve Open
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 ho…
View article: A braided monoidal $(\infty,2)$-category of Soergel bimodules
A braided monoidal $(\infty,2)$-category of Soergel bimodules Open
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 …
View article: Perverse schobers and 3d mirror symmetry
Perverse schobers and 3d mirror symmetry Open
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 t…
View article: Derived Mackey functors and $C_{p^n}$-equivariant cohomology
Derived Mackey functors and $C_{p^n}$-equivariant cohomology Open
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 fo…
View article: A universal characterization of noncommutative motives and secondary algebraic K-theory
A universal characterization of noncommutative motives and secondary algebraic K-theory Open
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…
View article: Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements
Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements Open
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. Pred…
View article: The universality of the Rezk nerve
The universality of the Rezk nerve Open
We functorially associate to each relative [math] –category [math] a simplicial space [math] , called its Rezk nerve (a straightforward generalization of Rezk’s “classification diagram” construction for relative categories). We prove the f…
View article: Stratified noncommutative geometry
Stratified noncommutative geometry Open
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 theore…
View article: $\mathbb{E}_\infty$ automorphisms of motivic Morava $E$-theories
$\mathbb{E}_\infty$ automorphisms of motivic Morava $E$-theories Open
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 no…
View article: Goerss--Hopkins obstruction theory for $\infty$-categories
Goerss--Hopkins obstruction theory for $\infty$-categories Open
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 presentab…
View article: Factorization homology of enriched $\infty$-categories
Factorization homology of enriched $\infty$-categories Open
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 t…
View article: The geometry of the cyclotomic trace
The geometry of the cyclotomic trace Open
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…
View article: A naive approach to genuine $G$-spectra and cyclotomic spectra
A naive approach to genuine $G$-spectra and cyclotomic spectra Open
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 na…
View article: Goerss--Hopkins obstruction theory via model ∞-categories
Goerss--Hopkins obstruction theory via model ∞-categories Open
We develop a theory of model ∞-categories -- that is, of model structures on ∞-categories -- which provides a robust theory of resolutions entirely native to the ∞-categorical context. Using model ∞-categories, we generalize Goerss--Hopkin…