[Paper Review] Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract)

André Joyal, Joachim Kock|arXiv (Cornell University)|Aug 19, 2009

Homotopy and Cohomology in Algebraic Topology7 references4 citations

TL;DR

This paper establishes a nerve theorem for compact symmetric multicategories by introducing a category of Feynman graphs as a combinatorial framework. It shows that compact symmetric multicategories are precisely the presheaves on this category satisfying a Segal condition, generalizing the nerve theorem for categories and multicategories to the modular, symmetric setting via a monadic structure on graphical species.

ABSTRACT

We describe a category of Feynman graphs and show how it relates to compact symmetric multicategories (coloured modular operads) just as linear orders relate to categories and rooted trees relate to multicategories. More specifically we obtain the following nerve theorem: compact symmetric multicategories can be characterised as presheaves on the category of Feynman graphs subject to a Segal condition. This text is a write-up of the second-named author's QPL6 talk; a more detailed account of this material will appear elsewhere.

Motivation & Objective

• To extend the nerve theorem from categories and multicategories to compact symmetric multicategories (colored modular operads).
• To define a category of Feynman graphs as the combinatorial foundation for encoding operations with multiple inputs and outputs.
• To establish that compact symmetric multicategories are equivalent to presheaves on the category of Feynman graphs satisfying a Segal condition.
• To generalize the monadic approach to graphical species, using a free-forgetful adjunction and a monad structure on decorated graphs.

Proposed method

• Introduces a category of Feynman graphs as the indexing category for the nerve construction, generalizing linear orders (for categories) and rooted trees (for multicategories).
• Defines a monad on the category of graphical species via a composition operation on decorated graphs, where vertices are replaced by subgraphs with matching interfaces.
• Constructs the Kleisli category of this monad, denoted Gr, which captures morphisms from graphs to decorated subgraphs.
• Establishes a generic/free factorization system in Gr, where generic maps correspond to refinements and free maps to etale morphisms.
• Uses this factorization to prove that the nerve functor from compact symmetric multicategories to presheaves on Gr is fully faithful.
• Applies techniques from category theory, including left Kan extensions and sheaf conditions, to verify that the essential image of the nerve consists of presheaves satisfying the Segal condition.

Experimental results

Research questions

• RQ1 How can the nerve theorem for categories and multicategories be generalized to the setting of compact symmetric multicategories?
• RQ2 What combinatorial structure—specifically, what category of graphs—can serve as the indexing category for such a nerve theorem?
• RQ3 How does the monad structure on graphical species encode the composition and symmetry of operations in compact symmetric multicategories?
• RQ4 What role does the Segal condition play in characterizing presheaves that arise as nerves of compact symmetric multicategories?
• RQ5 Can the generic/free factorization in the category of Feynman graphs be used to prove the nerve theorem, analogous to the factorization in the simplex category?

Key findings

• The nerve functor from compact symmetric multicategories to presheaves on the category of Feynman graphs is fully faithful.
• A presheaf on the category of Feynman graphs is in the essential image of the nerve functor if and only if its restriction to the full subcategory of 0-vertex graphs satisfies the Segal condition.
• The category of Feynman graphs admits a generic/free factorization system, analogous to the one in the simplex category Δ.
• The monad structure on graphical species arises from a composition operation that replaces vertices with subgraphs, and this structure underlies the definition of compact symmetric multicategories.
• The construction generalizes the Getzler–Kapranov approach to modular operads, with the category of Feynman graphs playing the role of the category of graphs used in their work.
• The nerve theorem for compact symmetric multicategories is proven via a preservation result for left Kan extensions, relying on the generic/free factorization.