[Paper Review] Factorization homology of enriched $\infty$-categories

David Ayala, Francis, John+2 more|arXiv (Cornell University)|Oct 17, 2017

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper introduces factorization homology for enriched $(∞,1)$-categories over 1-manifolds, generalizing Hochschild homology and topological Hochschild homology. It establishes a framework using symmetric monoidal $∞$-categories as enrichments, showing that factorization homology over a circle computes topological Hochschild homology in the spectral case, thereby realizing a categorified integration formalism for higher field theories.

ABSTRACT

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 the case of spectral enrichment, we show that the value of factorization homology on a circle is topological Hochschild homology.

Motivation & Objective

• To generalize factorization homology beyond En-algebras to enriched $(∞,1)$-categories using symmetric monoidal $∞$-categories as enrichments.
• To establish a functorial framework for factorization homology that captures symmetries and structures inherent in Hochschild-type invariants.
• To demonstrate that factorization homology over the circle recovers topological Hochschild homology in the spectral enrichment case.
• To provide a blueprint for a full theory of factorization homology in higher dimensions, particularly for enriched $(∞,n)$-categories with adjoints.

Proposed method

• Defining factorization homology via right-lax functors on flagged enriched $(∞,1)$-categories, leveraging a factorization system on the category of manifolds.
• Using the theory of cocartesian fibrations and factorization systems to construct a left adjoint that models the factorization homology functor.
• Introducing cartesian enriched factorization homology for symmetric monoidal $∞$-categories with finite limits and colimits.
• Applying the theory of Segal spaces and enriched $∞$-categories to relate flagged enriched structures to factorization homology.
• Constructing the factorization homology of an $E_1$-algebra in a symmetric monoidal $∞$-category $\mathcal{V}$ as a colimit over disk-stratified manifolds.
• Proving that the value of factorization homology on the circle is equivalent to topological Hochschild homology when $\mathcal{V}$ is the $∞$-category of spectra.

Experimental results

Research questions

• RQ1 How can factorization homology be extended from En-algebras to enriched $(∞,1)$-categories?
• RQ2 What is the role of symmetric monoidal $∞$-categories as enrichments in defining factorization homology?
• RQ3 How does factorization homology over the circle recover topological Hochschild homology in the spectral case?
• RQ4 What is the functorial structure of factorization homology under right-lax functors between enriched $(∞,1)$-categories?
• RQ5 Can the framework be generalized to higher-dimensional field theories with defects?

Key findings

• Factorization homology of a spectral enrichment over the circle is equivalent to topological Hochschild homology, providing a geometric interpretation of THH.
• The construction of factorization homology is functorial in a right-lax sense, extending to enriched $(∞,1)$-categories with appropriate structure.
• Cartesian enriched factorization homology is well-defined for symmetric monoidal $∞$-categories with finite limits and colimits, enabling a stable homotopy-theoretic framework.
• The theory realizes a categorified version of integration, where the input is an enriched $(∞,1)$-category and the output is a $∞$-category encoding global invariants.
• The factorization homology construction is compatible with the natural $T$-action on Hochschild-type invariants, manifesting symmetries geometrically.
• The framework generalizes classical Hochschild homology and provides a direct link to algebraic K-theory via the cyclotomic trace, with a natural extension to the topological setting.