Categorical generalisations of quantum double models

TL;DR

This paper develops a categorical framework linking involutive Hopf monoids to surface invariants, generalizing quantum double models and representation varieties through ribbon graphs and mapping class group actions.

Contribution

It introduces a new categorical construction that generalizes quantum double models using involutive Hopf monoids and ribbon graphs, applicable to various algebraic structures.

Findings

Constructs surface invariants from involutive Hopf monoids.

Provides a categorical generalization of Kitaev's quantum double ground state.

Connects to representation varieties and simplicial groups.

Abstract

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the category, defined up to isomorphism, that depends only on the homeomorphism class of the associated surface. This object is constructed via (co)equalisers and images and equipped with a mapping class group action. It can be viewed as a categorical generalisation of the ground state of Kitaev's quantum double model or of a representation variety for a surface. We apply the construction to group objects in cartesian monoidal categories, in particular to simplicial groups as group objects in SSet and to crossed modules as group objects in Cat. The former yields a simplicial set consisting of representation varieties, the latter a groupoid whose sets of objects and morphisms are obtained from representation varieties.