Product preserving gauge bundle functors on all principal bundle homomorphisms

Włodzimierz M. Mikulski¹

¹ Institute of Mathematics Jagiellonian University /Lojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 101 (2011) no. 2, pp. 163-207.

Résumé

Let $\mathcal{P}\mathcal{B}$ be the category of principal bundles and principal bundle homomorphisms. We describe completely the product preserving gauge bundle functors (ppgb-functors) on $\mathcal{P}\mathcal{B}$ and their natural transformations in terms of the so-called admissible triples and their morphisms. Then we deduce that any ppgb-functor on $\mathcal{P}\mathcal{B}$ admits a prolongation of principal connections to general ones. We also prove a “reduction” theorem for prolongations of principal connections into principal ones by means of Weil functors. We observe that there exist plenty of such prolongations. In Appendix, we classify the natural operators lifting vector-valued $1$-forms (or vector-valued maps) to vector-valued $1$-forms on Weil bundles.

Détail

DOI : 10.4064/ap101-2-6

Keywords: mathcal mathcal category principal bundles principal bundle homomorphisms describe completely product preserving gauge bundle functors ppgb functors mathcal mathcal their natural transformations terms so called admissible triples their morphisms deduce ppgb functor mathcal mathcal admits prolongation principal connections general prove reduction theorem prolongations principal connections principal means weil functors observe there exist plenty prolongations appendix classify natural operators lifting vector valued forms vector valued maps vector valued forms weil bundles