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A Categorical Approach to Probability Theory

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Abstract

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications.

Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one.

Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications.

Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovak Republic

    Roman Frič

  2. Department of Mathematics, Faculty of Education, Catholic University in Ružomberok, Hrabovská cesta 1, 034 01, Ružomberok, Slovak Republic

    Roman Frič & Martin Papčo

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  1. Roman Frič
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  2. Martin Papčo
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Correspondence to Roman Frič.

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Frič, R., Papčo, M. A Categorical Approach to Probability Theory. Stud Logica 94, 215–230 (2010). https://doi.org/10.1007/s11225-010-9232-z

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