The Grothendieck Construction in Categorical Network Theory

TL;DR

This thesis introduces a flexible categorical framework for network construction using operads derived from network models, leveraging a monoidal Grothendieck construction to represent concepts like priority and dependency.

Contribution

It develops a monoidal lift of the Grothendieck construction for operad specification and demonstrates its application to network concepts such as priority and dependency.

Findings

Generalizes Green's graph products to universal algebra

Models priority and dependency in network frameworks

Connects monoidal fibrations with catalysts in Petri nets

Abstract

In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a \emph{network model}, a monoidal variant of Joyal's combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green's graph products of groups to the context of universal algebra. For the latter, we examine the emergence of monoidal fibrations from the presence of catalysts in Petri nets.