Categorical generalisations of quantum double models (Q6634613)

This paper shows that any involutive Hopf monoid \(H\)\ in a complete and finitely cocomplete symmetric monoidal category \(\mathcal{C}\)\ yields an invariant of oriented surfaces, computing this invariant for examples, such as simplicial groups as Hopf monoids in \textrm{SSet}\ and crossed modules as Hopf monoids in \textrm{Cat}.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] introduces the algebraic background, such as Hopf monoids in symmetric monoidal categories as well as their (co)modules and the construction of their (co)invariants via (co)equalizers and images in complete and finitely cocomplete symmetric monoidal categories.\N\N\item[\S 3] gives the background on ribbon graphs and surfaces.\N\N\item[\S 4] formulates the categorical counterpart of \textit{A. Yu. Kitaev}'s [Ann. Phys. 303, No. 1, 2--30 (2003; Zbl 1012.81006)] quantum double model for an involutive Hopf monoid \(H\)\ in a complete and finitely cocomplete symmetric monoidal category, which is a simple generalization of \textit{O. Buerschaper} et al. [J. Math. Phys. 54, No. 1, 012201, 46 p. (2013; Zbl 1305.81125)] for finite-dimensional semisimple complex Hopf algebras, and an almost identical construction of which can be seen in [\textit{C. Meusburger} and \textit{T. Voß}, Quantum Topol. 12, No. 3, 507--591 (2021; Zbl 1482.57014)].\N\N\item[\S 5] shows that the protected object defined by a ribbon graph depends only on the homeomorphism class of the associated oriented surface \(\Sigma\), first demonstrating that moving the markings for the (co)module structures and edge reversals yield isomorphic protected objects, then considering a number of graph transformations that are sufficient to reduce every connected ribbon graph to a standard graph, and establishing that these induce isomorphisms of the protected object.\N\N\item[\S 6] treats the example of a simplicial group \(H=\left( H_{n}\right) _{n\in\mathbb{N}_{0}}\)\ as a Hopf monoid in \textrm{SSet}.\N\N\item[\S 7] is concerned with the construction of the protected object for group objects in \textrm{Cat}.\N\N\item[\S 8] describes the mapping class group action on the protected object, establishing:\N\NTheorem. The protected object for a Hopf monoid \(H\)\ and a surface \(\Sigma\)\ of genus \(g\geq1\)\ is equipped with an action of the mapping class group \textrm{Map} \(\left( \Sigma\right) \)\ by automorphisms.\N\N\end{itemize}