Unitary transformations of fibre functors (Q2078404)

By Tannaka duality, a compact group \(G\) is equivalent to its category of finite-dimensional representations \(\mathrm{Rep}(G)\), with canonical unitary monoidal \textit{fiber functor} \[ F:\mathrm{Rep}(G)\rightarrow\mathrm{Hilb} \] where \(\mathrm{Hilb}\) is the category of finite-dimensional Hilbert spaces and linear maps. One can generalize this notion of representation to \(C^{\ast}\)\textit{-tensor categories with conjugates}, which, as well as their fiber functors, are to be described by the representation theory of compact quantum groups and their associated Hopf-Galois objects. The author [``Unitary pseudonatural transformations'', Preprint, \url{arXiv:2004.12760}] introduced a notion of \textit{unitary pseudonatural transformation} relating two monoidal functors, or more generally two pseudofunctors, demonstrating that the \(2\)-category \(\mathrm{Fun}(\mathcal{C},\mathrm{Hilb})\) of unitary fiber functors, unitary pseudonatural transformations and modifications is of certain nice properties. The principal objective in this paper is to study unitary pseudonatural transformations of \(C^{\ast}\)-tensor categories with conjugates, or equivalently, provided that a fiber functor exists, representation categories of compact quantum groups. A synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] provides necessary background material for this paper. \item[\S 3] addresses the relationship between unitary pseudonatural transformations and Hopf-Galois theory. Let \(\mathcal{C}\) be a \(C^{\ast}\)-tensor category with conjugates. Whenever a fiber functor \(F:\mathcal{C} \rightarrow\mathrm{Hilb}\) exists, one can construct a monoidal equivalence \[ \mathcal{C}\simeq\mathrm{Rep}(G) \] for a compactquantum group \(G\), which means that the category \(\mathcal{C}\) is to be understood in terms of the compact quantum group \(G\), or rather its dual Hopf \(\ast\)-algebra \(A_{G}\). Let \(F_{1},F_{2}:\mathcal{C} \rightarrow\mathrm{Hilb}\) be fiber functors corresponding to compact quantum groups \(G_{1},G_{2}\). Then one can construct an \(A_{G_{2}}\)-\(A_{G_{1}}\)-bi-Hopf-Galois object \(Z\) linking the two fiber functors. As a generalization of the known fact [\textit{T. Banica}, ``Symmetries of a generic coaction'', Preprint, \url{ arXiv:math/9811060}, Theorem 4.4.1] claiming that the \(1\)-dimensional \(\ast\)-representations of an \(A_{G_{2}}\)-\(A_{G_{1}}\)-bi-Hopf-Galois object correspond to unitary monoidal natural transformations \(F_{1}\rightarrow F_{2}\), it is shown (Theorem 3.14) that there is an isomorphism of categories between \begin{itemize} \item The category \(\mathrm{Rep}(Z)\) of finite-dimensional \(\ast\)-representations of \ and intertwining linear maps. \item The category \(\mathrm{Hom}(F_{1},F_{2})\) of unitary pseudonatural transformations \(F_{1}\rightarrow F_{2}\) and modifications. \end{itemize} \item[\S 4] discusses the Morita classification/construction of accessible unitary pseudonatural transformations and fiber functors. Given a fiber functor \(F:\mathcal{C}\rightarrow\mathrm{Hilb}\), the author exploits Morita theory to classify \begin{itemize} \item unitary fiber functors accessible from \(F\) by a unitary pseudonatural transformation in terms of Morita equivalence classes of simple Frobenius monoids in \(\mathrm{Rep}(A_{G})\), and \item unitary pseudonatural transformations from \(F\) in terms of \(\ast\)-isomorphism classes\ of simple Frobenius monoids in \(\mathrm{Rep}(A_{G})\). \end{itemize} \item[\S 5] shows that finite-dimensional quantum graph isomorphisms are unitary pseudonatural transformations. The author establishes an equivalence between the following \(2\)-categories (Theorem 5.20): \begin{itemize} \item The \(2\)-category \(\mathrm{QGraph}_{X}\) of quantum graphs quantum isomorphic to \(X\) as objects, quantum isomorphisms as \(1\)-morphisms, and intertwiners as \(2\)-morphisms. \item The \(2\)-category \(\mathrm{Fun}(\mathrm{Rep}(G_{X}),\mathrm{Hilb})_{X}\) of fiber functors accessible by a unitary pseudonatural transformation from the canonical fiber functor on \(\mathrm{Rep}(G_{X})\) as objects, unitary pseudonatural transformations as \(1\)-morphisms, and modifications as \(2\)-morphisms. \end{itemize} \end{itemize}