About: Categorical Geometry and Integration Without Points

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://dx.doi.org/10.1007/s10485-012-9295-2
Description	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. 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. (en)
Title	Categorical Geometry and Integration Without Points Categorical Geometry and Integration Without Points (en)
skos:prefLabel	Categorical Geometry and Integration Without Points Categorical Geometry and Integration Without Points (en)
skos:notation	RIV/00216208:11320/14:10286605!RIV15-MSM-11320___
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(1M0545), Z(MSM0021620838)
http://linked.open...iv/cisloPeriodika	1
http://linked.open...vai/riv/dodaniDat	2015
http://linked.open...aciTvurceVysledku	Pultr, Aleš
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	6318
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/14:10286605
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Segal space; Locales; Categorical geometry; Boolean rings; Point free measures (en)
http://linked.open.../riv/klicoveSlovo	Boolean rings Categorical geometry Locales Point free measures Segal space
http://linked.open...odStatuVydavatele	NL - Nizozemsko
http://linked.open...ontrolniKodProRIV	[3150FF92B71A]
http://linked.open...i/riv/nazevZdroje	Applied Categorical Structures
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Institute for Theoretical Computer Science
http://linked.open...UplatneniVysledku	2014
http://linked.open...v/svazekPeriodika	22
http://linked.open...iv/tvurceVysledku	Pultr, Aleš Kriz, Igor
http://linked.open...ain/vavai/riv/wos	000331045200005
http://linked.open...n/vavai/riv/zamer	Moderní metody, struktury a systémy informatiky
issn	0927-2852
number of pages	19 (xsd:int)
http://bibframe.org/vocab/doi	10.1007/s10485-012-9295-2
http://localhost/t...ganizacniJednotka	11320

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