Universal factorization spaces and algebras

TL;DR

This paper develops a categorical framework for factorization spaces and algebras, establishing their equivalence to classical notions, and introduces universal objects and pullback constructions, extending vertex algebra concepts.

Contribution

It introduces categories of weak factorization algebras and spaces, proves their equivalence to classical categories, and defines universal factorization objects via étale morphisms.

Findings

Categories of weak factorization algebras/spaces are equivalent to classical ones.

Defined pullback of factorization objects via étale morphisms.

Extended the notion of vertex algebra to higher dimensions and non-linear settings.

Abstract

We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an \'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.