arXiv subjects

David Kern

Publications and source records attributed to David Kern.

6 recordsLinked to original sources

arXiv2024

Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories

Lessard's $\mathbb{Z}$-categories are an analogue of $\omega$-categories possessing cells in all positive and negative dimensions. Categorical spectra, developed by Stefanich, are an analogue of spectra obtained by replacing the suspension of pointed $\infty$-groupoids by that of pointed $(\infty,\omega)$-categories. We give an $\infty$-categorical definition of weak $\mathbb{Z}$-categories (alias $(\infty,\mathbb{Z})$-categories), and show categorical spectra to be equivalent to pointed $(\infty,\mathbb{Z})$-categories. In particular, we show that the stable cells of categorical spectra coincide with the natural cells of $(\infty,\mathbb{Z})$-categories, and recover Lessard's description of spectra as pointed weak $\mathbb{Z}$-groupoids.

math.CT↗

arXiv2024

All Segal objects are generalised monads in spans

We extend Barwick's and Haugseng's construction of the double $\infty$-category of spans in a pullback-complete $\infty$-category $\mathfrak{C}$ to more general shapes: for a large class of algebraic patterns $\mathfrak{P}$, we define a $\mathfrak{P}$-monoidal $\infty$-category of $\mathfrak{P}$-shaped spans in $\mathfrak{C}$, and we identify $\mathfrak{P}$-monads in it with Segal $\mathfrak{P}$-objects in $\mathfrak{C}$. For the cell pattern $\Theta^{\mathrm{op}}$, this recovers a homotopical reformulation of Batanin's original definition of weak $\omega$-categories, and in general can be seen as a variant of the generalised multicategories of Burroni, Hermida, Leinster and Cruttwell-Shulman.

math.CT↗

arXiv2023

Constraint Vector Bundles and Reduction of Lie (Bi-)Algebroids

We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is proven, identifying constraint vector bundles with certain finitely generated projective modules, and a Cartan calculus for constraint differentiable forms and multivector fields is introduced. All of these constructions will be shown to be compatible with reduction. Finally, we apply this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac manifolds.

math.DG↗

arXiv2023

Monoidal envelopes and Grothendieck construction for dendroidal Segal objects

We propose a construction of the monoidal envelope of $\infty$-operads in the model of Segal dendroidal spaces, and use it to define cocartesian fibrations of such. We achieve this by viewing the dendroidal category as a "plus construction" of the category of pointed finite sets, and work in the more general language of algebraic patterns for Segal conditions. Finally, we rephrase Lurie's definition of cartesian structures as exhibiting the categorical fibrations coming from envelopes, and deduce a straightening/unstraightening equivalence for dendroidal spaces.

math.CT↗

arXiv2022

Derived moduli of sections and push-forwards

We introduce a derived enhancement of the moduli space of sections defined by Chang-Li, and we compute its tangent complex. Special cases of this moduli space include stable maps and stable quasi-maps. As an application, we prove that G-theoretic stable map and quasi-map invariants of projective spaces are equal.

math.AG↗

arXiv2020

A categorification of the quantum Lefschetz principle

The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus $0$, regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the $\infty$-categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to recover the classical Quantum Lefschetz formula when its hypotheses are satisfied.

math.AG↗