### QPL 09

Day One of QPL 09 went relatively smoothly and there were a few interesting talks, but now I will just mention the beautiful talk by Joachim Kock on his recent work with A. Joyal.

They prove a (nerve) theorem characterising compact symmetric multicategories (modular operads) in terms of Feynman graphs (which is an extension to the work in this paper). Recall that whereas categories are about edges (objects) and vertices (morphisms) which compose, a multicategory allows (planar) rooted tree reductions. Unrooted trees take us to cyclic operads, and finally with loops we have (undecorated) Feynman graphs, with external legs.

The classical nerve theorem for ordinary categories looks at an adjunction between Cat and the category of directed graphs. A graph $G$ is sent to a category with objects the vertices and morphisms the paths in $G$. There is a certain factorization property for this adjunction that gives a functor $\Delta_{0} \rightarrow \Delta$, where $\Delta_{0}$ is the class of distance preserving maps, that is to say graphs with matching path lengths between distinct vertex sets. $\Delta$ includes certain extra diagrams.

In the case of the modular operads, the theorem takes a similar form. The analogue of directed graphs here is the category of presheaves on elementary graphs (the basic building blocks). A generic map is a refinement of a star graph (with one vertex) where the vertex is replaced by another graph so that the outputs match up. But there is also a class of etale maps, or covers, such that the cover of a graph is a pullback square. Anyway, there is a monad $T$ which expands a presheaf on elementary graphs to one on all Feynman graphs. The modular operads are the $T$ algebras for this monad.

They prove a (nerve) theorem characterising compact symmetric multicategories (modular operads) in terms of Feynman graphs (which is an extension to the work in this paper). Recall that whereas categories are about edges (objects) and vertices (morphisms) which compose, a multicategory allows (planar) rooted tree reductions. Unrooted trees take us to cyclic operads, and finally with loops we have (undecorated) Feynman graphs, with external legs.

The classical nerve theorem for ordinary categories looks at an adjunction between Cat and the category of directed graphs. A graph $G$ is sent to a category with objects the vertices and morphisms the paths in $G$. There is a certain factorization property for this adjunction that gives a functor $\Delta_{0} \rightarrow \Delta$, where $\Delta_{0}$ is the class of distance preserving maps, that is to say graphs with matching path lengths between distinct vertex sets. $\Delta$ includes certain extra diagrams.

In the case of the modular operads, the theorem takes a similar form. The analogue of directed graphs here is the category of presheaves on elementary graphs (the basic building blocks). A generic map is a refinement of a star graph (with one vertex) where the vertex is replaced by another graph so that the outputs match up. But there is also a class of etale maps, or covers, such that the cover of a graph is a pullback square. Anyway, there is a monad $T$ which expands a presheaf on elementary graphs to one on all Feynman graphs. The modular operads are the $T$ algebras for this monad.

## 2 Comments:

To one who has spent years around the Pacific, Oxford seems like the centre of the far side of the world. I am glad you are enjoying the conferences. The Category Theory event at Princeton in '007 was fascinating.

In some ways it doesn't feel so alien. We all went out to dinner last night, to a nice Indian place that could have been at home.

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