Gauge transformations for categorical bundles
A categorical bundle is a structure, in terms of category theory, that encodes a classical bundle equipped with connections and some additional entities. The authors have developed the theory of categorical principal bundles in [\textit{S. Chatterjee} et al., Theory Appl. Categ. 31, 388--417 (2016; Zbl 1343.18009); J. Geom. Phys. 98, 128--149 (2015; Zbl 1339.18005); J. Geom. Phys. 112, 147--174 (2017; Zbl 1355.53018); J. Geom. Phys. 75, 129--161 (2014; Zbl 1280.81087); Theory Appl. Categ. 29, 215--255 (2014; Zbl 1305.18014)]. The principal objective in this paper is to develop a counterpart of the classical gauge transformation in the setting of categorical bundles. There is a considerable literature on category-theoretic approaches to gauge theories [\textit{J. C. Baez} and \textit{D. K. Wise}, Commun. Math. Phys. 333, No. 1, 153--186 (2015; Zbl 1308.83017)], [\textit{J. C. Baez} and \textit{J. Huerta}, Gen. Relativ. Gravitation 43, No. 9, 2335--2392 (2011; Zbl 1225.83001)], [\textit{J. C. Baez} and \textit{U. Schreiber}, Contemp. Math. 431, 7--30 (2007; Zbl 1132.55007)], [\textit{J. F. Martins} and \textit{R. Picken}, Differ. Geom. Appl. 29, No. 2, 179--206 (2011; Zbl 1217.53049)], [\textit{J. F. Martins} and \textit{R. Picken}, Adv. Math. 226, No. 4, 3309--3366 (2011; Zbl 1214.53043)], [\textit{U. Schreiber} and \textit{K. Waldorf}, J. Homotopy Relat. Struct. 12, No. 3, 617--658 (2017; Zbl 1404.18015); Theory Appl. Categ. 28, 476--540 (2013; Zbl 1279.53024)], [\textit{A. J. Parzygnat}, Theory Appl. Categ. 30, 1319--1428 (2015; Zbl 1341.53078)], [\textit{D. Bambusi}, in: Multiscale methods in quantum mechanics. Theory and experiment. Papers from the meeting, Rome, Italy, December 16--20, 2002. Boston, MA: Birkhäuser. 23--39 (2004; Zbl 1322.81046)], [\textit{E. Soncini} and \textit{R. Zucchini}, J. Geom. Phys. 95, 28--73 (2015; Zbl 1322.81064); J. High Energy Phys. 2014, No. 10, Paper No. 079, 64 p. (2014; Zbl 1333.81272)], [\textit{W. Wang}, J. Math. Phys. 55, No. 4, 043506, 32 p. (2014; Zbl 1300.81065); Theory Appl. Categ. 30, 1999--2047 (2015; Zbl 1353.18006)], [\textit{K. Waldorf}, Trans. Am. Math. Soc. 365, No. 8, 4393--4432 (2013; Zbl 1277.53024)]. The authors' motivation is more differential geometric than category theoretic, and the typical examples are decorated bundles (\S 2.14) and twisted-product bundels (\S 3) appearing to find their due place only in the authors' approach. A categorical connection (\S 2.11) in the authors' approach involves several distinct \(1\)-forms and \(2\)-forms that are to be used to prescribe parallel transport.
scientific article; zbMATH DE number 6948941
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