Synthetic braided geometry. I (Q1303461)

Synthetic treatments of supergeometry have been discussed by the author [Int. J. Theor. Phys. 37, 2803-2822 (1998; Zbl 0942.58011)] and \textit{D. N. Yetter} [Cah. Topologie Géom. Différ. Catégoriques 29, No. 2, 87-108 (1988; Zbl 0649.18010)]. The author refers to [\textit{R. Lavendhomme}, Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht (1996; Zbl 0866.58001)] as a good introduction to Synthetic differential geometry and to [\textit{Yu. I. Manin}, Gauge Field Theory and Complex Geometry, Springer Verlag, Heidelberg (1988; Zbl 0641.53001) as a good introduction to supergeometry. The principal objective of this paper is to give a synthetic treatment of symmetric braided geometry along the lines of the former. (The braided geometry in the Majid's and Marcinek's sense is considered as an ``elegant and far-reaching generalization of supergeometry''. Indeed the category of vector spaces is replaced by a braided monoidal category).