arXiv subjects

Emanuele Pavia

Publications and source records attributed to Emanuele Pavia.

6 recordsLinked to original sources

arXiv2026

Higher Koszul duality and $n$-affineness

In this paper we study $\mathbb{E}_n$-Koszul duality in the topological setting, and the closely related question of \emph{$n$-affineness} for Betti stacks. The $\mathbb{E}_n$-Koszul dual of the algebra of chains on the $n$-fold loop space of a space $X$ is the algebra of cochains on $X$. It was expected that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules, but even the precise formulation of such a statement was not known. We give a rigorous formulation, and a proof, of such an $\mathbb{E}_n$-Koszul duality in the topological setting as an equivalence of $(\infty,n)$-categories. Conceptually, our main innovation is highlighting the coaffine stack defined by the \emph{cospectrum} of $\mathrm{C}^{\bullet}(X;\Bbbk)$ as a key geometric object supporting Koszul duality. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $π_{n+1}(X)$ is an obstruction to $n$-affineness.

math.AG↗

arXiv2026

Derived hyperquot schemes

We define a derived enhancement of the hyperquot scheme (also known as nested Quot scheme), which classically parametrises flags of quotients of a perfect coherent sheaf on a projective scheme. We prove it is representable by a derived scheme, and we compute its global tangent complex. As an application, we provide a natural obstruction theory on the classical hyperquot scheme. The latter recovers the virtual fundamental class recently constructed by the first and third author in the context of the enumerative geometry of hyperquot schemes on smooth projective curves.

math.AG↗

arXiv2025

An axiomatic approach to analytic $1$-affineness

The notion of $1$-affineness was originally formulated by Gaitsgory in the context of derived algebraic geometry. Motivated by applications to rigid and analytic geometry, we introduce two very general and abstract frameworks where it makes sense to ask for objects to be $1$-affine with respect to some sheaf of categories. The first framework is suited for studying the problem of $1$-affineness when the sheaf of categories arises from an operation in a six-functor formalism over $\mathscr{C}$; we apply it to the setting of analytic stacks and condensed mathematics. The second one concerns $1$-affineness in the context of quasi-coherent sheaves of categorical modules over stable module categories: it simultaneously generalizes the algebro-geometric setting of Gaitsgory and makes it possible to formulate the problem also when dealing with rigid analytic varieties and categories of nuclear modules.

math.AG↗

arXiv2025

Higher local systems and the categorified monodromy equivalence

We study local systems of $(\infty,n)$-categories on spaces. We prove that categorical local systems are captured by (higher) monodromy data: in particular, if $X$ is $(n+1)$-connected, then local systems of $(\infty,n)$-categories over $X$ can be described as $\mathbb{E}_{n+1}$-modules over the iterated loop space $Ω_{n+1}X$. This generalizes the classical monodromy equivalence presenting ordinary local systems as modules over the based loop spaces. Along the way we revisit from the perspective of $\infty$-categories Teleman's influential theory of topological group actions on categories, and we extend it to topological actions on $(\infty,n)$-categories. Finally, we show that the group of invertible objects in the category of local systems of $(\infty,n)$-categories over an $n$-connected space $X$ is isomorphic to the group of characters of $π_n(X)$. This should be thought of as a topological analogue of the higher Brauer group of the space $X$. We conclude the paper with applications of the theory of categorical local systems to the fiberwise Fukaya category of symplectic fibrations.

math.AT↗

arXiv2023

Mixed graded structure on Chevalley-Eilenberg functors

In this paper, we shall provide a purely $\infty$-categorical construction of the mixed graded structure over Chevalley-Eilenberg complexes computing homology and cohomology of Lie algebras defined over a field $\Bbbk$ of characteristic $0$. While this additional piece of structure on Chevalley-Eilenberg complexes is expected, and already described in terms of explicit models given by chain complexes, there is not a completely formal and model independent description of the mixed graded Chevalley-Eilenberg $\infty$-functors in available literature. After constructing in all details the Chevalley-Eilenberg $\infty$-functors and studying their main formal properties, we present some further conjectures on their behavior.

math.AG↗

arXiv2022

A $t$-structure on the $\infty$-category of mixed graded modules

In this work, we shall study in a purely model-independent fashion the $\infty$-category of mixed graded modules over a ring of characteristic $0$, and collect some basic results about its main formal properties. Finally, we shall endow such $\infty$-category with a both left and right complete accessible $t$-structure, showing how this identifies the $\infty$-category of mixed graded modules with the left completion of the Beilinson $t$-structure on the \infinity-category of filtered modules. Most of the content of this paper is already available in literature, and it serves mainly as a reference for future work.

math.CT↗