Spectral Decomposition of Ultrametric Spaces and Topos Theory
by
Alex J. Lemin
Moscow State University of Civil Engineering

We consider categories METR and METR_c of metric spaces (of diameter <= c) and non-expanding maps as well as their subcategories ULTRAMETR and  ULTRAMETR_c of ultrametric spaces and the same maps. Given a family of ultrametric spaces, we prove that sums and products, equalizer and co-equalizer, pull-back and push-out, limits of direct and inverse spectra, if exist, are ultrametric. A product and a limit of inverse spectrum of complete metric spaces are complete. A space (X, d) is uniformly discrete if d(x, y) >= \epsilon > 0   for all x,   y in X. This is necessarily complete.
Theorem. Every complete ultrametric space is isometric to a limit of a countable inverse spectrum of uniformly discrete ultrametric spaces (and vise versa) (see [1]).
Corollary 1. Every compact ultrametric space is isometric to a limit of inverse sequence of skeletons of finite dimensional isosceles simplexes lying in Euclidean spaces (see [2]).
Corollary 2. Category ULTRAMETR is a quasi-topos.

1.A.Lemin. On stability of the property of a space being isosceles, - Russ. Math. S. 40:6 (1985).

2.A.Lemin. Isometric imbedding of isosceles (=non- Archimedean) spaces in Euclidean spaces, - Soviet Math. Doklady, 32:3 (1985), 740-744.

Date received: June 15, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacl-81.