Monoidal categories graded by crossed modules and 3-dimensional HQFTs (Q6110343)

Homotopy Quantum Field Theory (HQFT) is a type of quantum field theory closely related to Topological Quantum Field Theory (TQFT), but in which the objects of study are no longer simply manifolds and the cobordisms between them, but manifolds and cobordisms endowed with structural maps to a fixed target space, \(X\). The aim is to define and study homotopy invariants of such maps using ideas adapted from the study of TQFTs. The initial development of HQFTs was by \textit{V. Turaev} around 1999, and further developed and clarified in his monograph [Homotopy quantum field theory. With appendices by Michael Müger and Alexis Virelizier. Zürich: European Mathematical Society (EMS) (2010; Zbl 1243.81016)]. The case of 2D HQFTs is well understood and, for \(X=BG\), the classifying space of a group, \(G\), these are classified by a form of crossed \(G\)-algebra, which can also be thought of as a \(G\)-graded algebra. In a series of papers, [Int. J. Math. 23, No. 9, 1250094, 28 p. (2012; Zbl 1254.57012); Int. J. Math. 25, No. 4, Article ID 1450027, 66 p. (2014; Zbl 1296.57016); Int. J. Math. 31, No. 10, Article ID 2050076, 57 p. (2020; Zbl 1473.57032)], \textit{V. Turaev} and \textit{A. Virelizier} classified 3D HQFTs having \(X=BG\) in terms of \(G\)-graded spherical fusion categories, using both a state sum and a surgery approach. In this paper, the authors extend the state sum approach from those earlier papers to the case where the target space, \(X\), is the classifying spaces of a crossed modules, \(\chi\). To this end, they introduce a notion of \(\chi\)-graded modular category, and develop their theory including that of spherical \(\chi\)-fusion categories of a certain type. These are the analogues of spherical \(G\)-fusion categories as used in the earlier papers. They show how any such category leads to a 3D HQFT with target space, \(X= B\chi\). The exposition includes numerous examples and calculations of the invariants concerned.