ProPs of graphs and generalised traces

TL;DR

This paper introduces TraPs, a new algebraic structure related to ProPs, to assign generalized convolutions and traces to graphs with decorated edges, expanding the algebraic framework for analyzing such graph-based operators.

Contribution

It defines TraPs, relates them to wheeled ProPs, constructs their free objects, and establishes their universal properties, advancing the algebraic understanding of graph-based operators.

Findings

Defined TraPs and related them to wheeled ProPs.

Constructed free objects for TraPs.

Proved universal properties of ProPs and TraPs.

Abstract

We assign generalised convolutions (resp. traces) to graphs whose edges are decorated by smooth kernels (resp. smoothing operators) on a closed manifold. To do so, we introduce the concept of TraPs (Traces and Permutations), which roughly correspond to ProPs (Products and Permutations) without vertical concatenation and equipped with families of generalised partial traces. They can be equipped with a ProP structure in deriving vertical concatenation from the partial traces and we relate TraPs to wheeled ProPs first introduced by Merkulov. We further build their free object and give precise proofs of universal properties of ProPs and TraPs.