[Paper Review] Operads, modules and topological field theories

Geoffroy Horel|arXiv (Cornell University)|May 21, 2014

Homotopy and Cohomology in Algebraic Topology26 references3 citations

TL;DR

This paper develops a general theory of modules over algebras in operads, showing that such modules correspond to associative algebras in right modules over the operad. It constructs a functorial structure on categories of modules indexed by manifolds and bordisms, leading to a map from an $∞$-operad of cobordisms to the $∞$-operad of $∞$-categories, thereby realizing a topological field theory via factorization homology for $ \mathscr{E}_d$-algebras.

ABSTRACT

In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold gives rise to a theory of modules over E_d-algebras and each bordism gives rise to a functor from the category defined by its incoming boundary to the category defined by its outgoing boundary. We describe how to assemble these categories into a map from a certain operad to the operad of (infinity)-categories.

Motivation & Objective

• To classify all sensible notions of modules over an algebra over a given operad $ \mathscr{O}$, showing they correspond to associative algebras in the category of right $ \mathscr{O}$-modules.
• To construct a homotopically coherent framework for operations on categories of modules using model categories and simplicial operads.
• To establish a correspondence between $(d-1)$-manifolds and categories of modules over $ \mathscr{E}_d$-algebras, and between bordisms and functors between such categories.
• To define a map from a cobordism-related $∞$-operad to the $∞$-operad of $∞$-categories, realizing a topological field theory via factorization homology.
• To generalize Lurie's results on $ \mathscr{E}_d$-monoidal structures on module categories using model category techniques instead of $∞$-categories.

Proposed method

• Introduces the Morita operad $ \mathscr{M}or(\mathscr{O})$ whose objects are associative algebras in right $ \mathscr{O}$-modules, and morphisms are bimodules and weak equivalences.
• Constructs a simplicial operad of model categories, allowing the definition of $ \mathscr{O}$-algebras in model categories via maps from $ \mathscr{O}$ to this operad.
• Uses Rezk's nerve construction to associate complete Segal spaces to model categories, ensuring homotopical coherence of the theory.
• Equips categories of $P$-shaped modules over an $ \mathscr{O}$-algebra with a model structure, enabling homotopical algebra.
• Defines a functor from the $∞$-operad $f\widehat{\mathscr{C}ob}_d$ (related to cobordism categories) to the $∞$-operad of $∞$-categories, encoding the field theory.
• Applies the theory to $ \mathscr{E}_d$-algebras, constructing module categories for $(d-1)$-manifolds and functors for bordisms, using factorization homology.

Experimental results

Research questions

• RQ1 What is the complete classification of module theories over an algebra over a given operad $ \mathscr{O}$?
• RQ2 How can one construct a homotopy-coherent monoidal or operadic structure on categories of modules over $ \mathscr{O}$-algebras?
• RQ3 Can factorization homology be used to assign categories of modules to manifolds and functors to bordisms in a way that forms a topological field theory?
• RQ4 How does the theory of $ \mathscr{O}$-algebras in model categories relate to $∞$-categorical constructions, particularly in the context of $ \mathscr{E}_d$-algebras?
• RQ5 What is the role of the Morita operad in organizing module shapes and their associated functors?

Key findings

• Modules over an $ \mathscr{O}$-algebra are classified by associative algebras in the category of right $ \mathscr{O}$-modules, with each such algebra defining a distinct 'shape' of module.
• The category of $P$-shaped modules over an $ \mathscr{O}$-algebra $A$ admits a well-defined model structure, making it suitable for homotopical analysis.
• A functorial assignment of model categories to $(d-1)$-manifolds and to bordisms is constructed, compatible with composition.
• The resulting structure is a map of $∞$-operads from $f\widehat{\mathscr{C}ob}_d$ to the $∞$-operad of $∞$-categories, realizing a topological field theory.
• For $ \mathscr{E}_d$-algebras, the theory recovers and generalizes Lurie's result on $ \mathscr{E}_d$-monoidal structures on left module categories.
• The construction is homotopically coherent: weak equivalences of operads induce Quillen equivalences on module categories, ensuring invariance under homotopy.