Uniform fields vs. fibrewise uniform spaces: An adjoint situation (Q485006)

Using an existence theorem and the localization process, the author establishes an adjunction functor between the category of Hausdorff uniform fields and the category of Hausdorff fibrewise uniform spaces. The most interesting differences between these two categories are also pointed out. The paper also contains some fundamentals of the theory of uniform fields, results due to \textit{J. Varela} [Rev. Colomb. Mat. 29, No. 2, 95--101 (1995; Zbl 0868.55013)] on the localization process (a tool to obtain uniform fields), results on fibrewise topology{/}fibrewise uniform spaces due to \textit{I. M. James} [Fibrewise topology, Cambridge Tracts in Mathematics 91. Cambridge: Cambridge University Press (1989; Zbl 0671.55001)], \textit{Y. Konami and T. Miwa} [Acta Math. Hung. 122, No. 1-2, 1--28 (2009; Zbl 1224.55005)] and the construction of the adjoint functor \(G\) between the two categories (uniform fibrewise spaces vs. uniform fields). The author studies the existing generalizations of uniform structures to categories of functions. He finds a relationship between fibrewise uniform structures and uniform fields. Examples are also given.