Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)

TL;DR

This paper introduces a new Hopf algebra structure on tracelets within decomposition spaces, providing a novel algebraic framework for analyzing causal information in rewriting systems with applications to stochastic models.

Contribution

It constructs a symmetric monoidal decomposition space of tracelets, leading to a cocommutative Hopf algebra that advances the algebraic understanding of rewriting systems.

Findings

Hopf algebra of tracelets captures combinatorial aspects of rewriting

Decomposition space structure enables algebraic analysis of causal information

Application potential in stochastic rewriting systems like chemical networks

Abstract

Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.