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Categorical Geometry and Integration Without Points

Publikace na Matematicko-fyzikální fakulta |
2014

Tento text není v aktuálním jazyce dostupný. Zobrazuje se verze "en".Abstrakt

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy.

Such an approach was suggested and initiated by Segal in his pioneering article (Segal, Bull Am Math Soc 71:419-489, 1965). In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology.

First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical sigma-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function mu with values in [0,1] can be extended to a measure on an abstract sigma-algebra; this correspondence is functorial and yields uniqueness.

As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.