Feynman graphs, and nerve theorem for compact symmetric multicategories (extended 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.

💡 Research Summary

The paper introduces a new categorical framework built from Feynman graphs and shows that compact symmetric multicategories (also known as coloured modular operads) are precisely the Segal‑type presheaves on this graph category. The authors begin by defining a category G whose objects are finite Feynman graphs equipped with a set of external half‑edges (called “ports”) and internal vertices. Morphisms are isomorphisms preserving the port labeling; composition is performed by gluing ports of two graphs together, thereby allowing loops and multiple edges. This construction mirrors the way rooted trees encode the operations of ordinary multicategories, but it is richer because it admits internal cycles, reflecting the “compact” nature of the target structures.

Next, the paper recalls the notion of a compact symmetric multicategory (CSM). A CSM consists of a collection of colours (objects) together with multimorphisms that have an arbitrary finite list of input colours and a single output colour, together with a full symmetric group action on both inputs and outputs. The “compact” adjective indicates that the theory permits internal contractions (loops) and self‑edges, exactly the combinatorial features present in Feynman diagrams. In operadic language, a CSM is a coloured modular operad: the modularity encodes the ability to contract pairs of ports, while the symmetry encodes permutation invariance.

The central result is a nerve theorem for CSMs. The authors consider the presheaf category Psh(G) = Set^{G^{op}} and impose a Segal condition: for any graph Γ in G, the value of a presheaf F on Γ must be the limit of the diagram obtained by decomposing Γ into elementary graphs (graphs with a single internal vertex and exactly one input and one output port). In other words, F(Γ) is required to be the “gluing” of the values on the elementary pieces, exactly as the classical Segal condition forces a simplicial set to be determined by its 1‑simplices. The authors prove that a presheaf satisfies this condition if and only if it arises from a compact symmetric multicategory.

Two functors implement the equivalence. The nerve functor N : CSM → Psh(G) sends a CSM M to the presheaf that evaluates a graph Γ by taking the set of labellings of the ports of Γ by colours of M together with a choice of multimorphism at each internal vertex, respecting the symmetric actions. Conversely, the realisation functor R : Psh(G)_{Seg} → CSM reconstructs a CSM from a Segal presheaf by interpreting the value on the elementary graph as the set of multimorphisms with the given arity, and using the Segal condition to define composition and contraction. The authors verify that R∘N ≅ Id_{CSM} and N∘R ≅ Id_{Psh(G)_{Seg}}, establishing a categorical equivalence.

A substantial technical portion of the paper deals with the handling of loops and multiple edges. To ensure associativity of composition in G, the authors introduce a “compact closure” operation that formally adds formal inverses for each edge, guaranteeing that gluing along a loop behaves like a contraction in a modular operad. They also show that the symmetric group actions on ports extend naturally to actions on the whole graph, providing the full symmetry required for CSMs.

The authors position their result as a higher‑dimensional analogue of well‑known correspondences: linear orders ↔ categories (via the nerve) and rooted trees ↔ multicategories (via the dendroidal nerve). Here, Feynman graphs play the role of “higher‑dimensional trees” that encode not only branching but also feedback loops, and the Segal condition captures precisely the modular operadic composition laws.

Although the paper is an extended abstract, it outlines several directions for future work: (1) a detailed treatment of examples from quantum field theory, where the colours correspond to particle types and the Segal presheaves encode amplitudes; (2) an exploration of higher‑categorical extensions, replacing Set by ∞‑groupoids to obtain an ∞‑nerve theorem for ∞‑modular operads; (3) applications to computer science, such as graph‑based process calculi where the compact symmetric multicategory structure models concurrent communication with feedback.

In summary, the paper provides a clean categorical characterisation of compact symmetric multicategories as Segal‑type presheaves on a graph category built from Feynman diagrams. This bridges combinatorial physics, operad theory, and higher category theory, offering a unified language for structures that involve both branching and looping interactions.