# Indiana University

 dc.contributor.advisor Moss, Lawrence S. en dc.contributor.author Viglizzo, Ignacio Dario en dc.date.accessioned 2010-05-24T15:09:46Z en dc.date.available 2027-01-24T16:09:46Z en dc.date.available 2010-05-30T02:26:21Z dc.date.issued 2010-05-24T15:09:46Z en dc.date.submitted 2005 en dc.identifier.uri http://hdl.handle.net/2022/7065 en dc.description Thesis (PhD) - Indiana University, Mathematics, 2005 en dc.description.abstract Given an endofunctor T in a category C, a coalgebra is a pair (X,c) consisting of an object X and a morphism c:X ->T(X). X is called the carrier and the morphism c is called the structure map of the T-coalgebra. en The theory of coalgebras has been found to abstract common features of different areas like computer program semantics, modal logic, automata, non-well-founded sets, etc. Most of the work on concrete examples, however, has been limited to the category Set. The work developed in this dissertation is concerned with the category Meas of measurable spaces and measurable functions. Coalgebras of measurable spaces are of interest as a formalization of Markov Chains and can also be used to model probabilistic reasoning. We discuss some general facts related to the most interesting functor in Meas, Delta, that assigns to each measurable space, the space of all probability measures on it. We show that this functor does not preserve weak pullbacks or omega op-limits, conditions assumed in many theorems about coalgebras. The main result will be two constructions of final coalgebras for many interesting functors in Meas. The first construction (joint work with L. Moss), is based on a modal language that lets us build formulas that describe the elements of the final coalgebra. The second method makes use of a subset of the projective limit of the final sequence for the functor in question. That is, the sequence 1 <- T1 <- T 2 1 <-... obtained by iteratively applying the functor to the terminal element 1 of the category. Since these methods seem to be new, we also show how to use them in the category Set, where they provide some insight on how the structure map of the final coalgebra works. We show as an application how to construct universal Type Spaces, an object of interest in Game Theory and Economics. We also compare our method with previously existing constructions. dc.language.iso EN en dc.publisher [Bloomington, Ind.] : Indiana University en dc.rights This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. en dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/ en dc.subject coalgebra en dc.subject beliefs en dc.subject measurable spaces en dc.subject final coalgebra en dc.subject type space en dc.subject probabilities en dc.subject.classification Mathematics en dc.title Coalgebras on Measurable Spaces en dc.type Doctoral Dissertation en
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