[Paper Review] Descent and the Koszul Duality for Locally Constant Factorisation Algebras

Takuo Matsuoka|arXiv (Cornell University)|Jan 1, 2014

Advanced Topics in Algebra3 citations

TL;DR

This paper establishes a Verdier-type equivalence for locally constant factorization algebras via a Koszul duality construction in symmetric monoidal stable ∞-categories, generalizing Lurie’s nonabelian Poincaré duality using excision and topological chiral homology. It proves descent for factorizing covers and derives a product formula for the ∞-category of factorization algebras and their twisted variants.

ABSTRACT

Generalizing Jacob Lurie's idea on the relation between the Verdier and the iterated loop space theory, we study the Koszul for locally constant factorization algebras. We formulate an analogue of Lurie's nonabelian Poincare duality theorem (which is closely related to earlier results of Graeme Segal, of Dusa McDuff, and of Paolo Salvatore) in a symmetric monoidal stable infinity category carefully, using John Francis' notion of excision. Its proof is done by first studying the Koszul for E_n-algebras in detail. As a consequence, we obtain a Verdier type equivalence for factorization algebras by a Koszul construction. At a foundational level, we study descent properties of Lurie's topological chiral homology. We prove that this homology theory satisfies descent for a factorizing cover, as defined by Kevin Costello and Owen Gwilliam. We also obtain a generalization of Lurie's approach to this homology theory, which leads to a product formula for the infinity category of factorization algebras, and its twisted generalization.

Motivation & Objective

• To generalize Lurie’s nonabelian Poincaré duality to locally constant factorization algebras in symmetric monoidal stable ∞-categories.
• To establish a Verdier-type equivalence for factorization algebras using Koszul duality.
• To prove that topological chiral homology satisfies descent for factorizing covers as defined by Costello and Gwilliam.
• To generalize Lurie’s approach to topological chiral homology, yielding a product formula for the ∞-category of factorization algebras and their twisted versions.

Proposed method

• Adapting John Francis’ notion of excision to formulate a precise analogue of Lurie’s nonabelian Poincaré duality in the context of symmetric monoidal stable ∞-categories.
• Studying Koszul duality for E_n-algebras as a foundational step toward the main duality result.
• Using Lurie’s framework of topological chiral homology and proving its descent with respect to factorizing covers.
• Constructing a product formula for the ∞-category of factorization algebras by generalizing Lurie’s approach to chiral homology.
• Extending the framework to twisted factorization algebras via the same homological and categorical machinery.
• Employing ∞-categorical techniques to handle symmetric monoidal structures and duality in derived algebraic geometry.

Experimental results

Research questions

• RQ1 How can Lurie’s nonabelian Poincaré duality be adapted to the setting of locally constant factorization algebras in symmetric monoidal stable ∞-categories?
• RQ2 What is the precise formulation and proof of Koszul duality for E_n-algebras in this context?
• RQ3 Does topological chiral homology satisfy descent for factorizing covers in the sense of Costello and Gwilliam?
• RQ4 Can a product formula be derived for the ∞-category of factorization algebras using generalized chiral homology?
• RQ5 How does the Verdier-type equivalence for factorization algebras emerge from the Koszul construction in this framework?

Key findings

• A Verdier-type equivalence for locally constant factorization algebras is established via a Koszul duality construction in symmetric monoidal stable ∞-categories.
• Topological chiral homology satisfies descent for factorizing covers, generalizing Lurie’s results to a broader class of covers.
• The ∞-category of factorization algebras admits a product formula derived from a generalized approach to chiral homology.
• The twisted generalization of the product formula is obtained by extending the Koszul duality and descent framework to twisted factorization algebras.
• The study of Koszul duality for E_n-algebras provides the foundational step for the main duality result.
• The framework successfully generalizes Lurie’s nonabelian Poincaré duality to the setting of locally constant factorization algebras using excision.