Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra

TL;DR

This paper presents a categorical framework linking combinatorial graph structures to surface topology, geometry, and algebra, with applications in string topology and topological field theory.

Contribution

It introduces a Feynman categorical approach that unifies surface geometry and combinatorics, providing new formalism for cyclic and modular operads and their generalizations.

Findings

Formalism for cyclic and modular operads derived from graph dissection.

Use of left Kan extensions for efficient computation of Feynman operations.

Categories of structured graphs identified as potentially significant for future research.

Abstract

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical viewpoint allows us to use left Kan extensions of Feynman operations as an efficient computational tool. The computations involve the study of certain categories of structured graphs which are expected to be of independent interest.