Weak homomorphisms of coalgebras

Date
2007-01-01
Authors
Chung, Key
Major Professor
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Jonathan D.H. Smith
Committee Member
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Altmetrics
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Mathematics
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Mathematics
Abstract

Although coalgebras have primarily been used to model various structures in theoretical computer science, it has been observed that they may also model mathematical structures. For example, there is a natural way to turn graphs into coalgebras obtained from the finite powerset functor. There is also a natural way to turn topological spaces into coalgebras for the filter functor. We observe that when coalgebras are used to model these mathematical structures, standard coalgebraic notion of a homomorphism is too strict. In this thesis, we propose a elaxation of the condition for the definition of a homomorphism and we show that our weak version induces the proper level homomorphisms between mathematical structures.;Based on this appropriate relaxation of the concept of coalgebra homomorphism, we demonstrate the finite completeness and non-cocompleteness of the category of locally finite graphs (including loops) and graph homomorphisms by using existing results from the theory of coalgebras. We also prove the equivalence between the usual category of topological spaces and the category of coalgebras obtained from topological spaces. Because of our relaxation, we gains and loses some aspects of coalgebras properties. We illustrate it by giving an example of complete category of all coalgebras for the powerset functor having a simple construction with our relaxation. We give another example of noncocomplete category of all coalgebras for the powerset functor with our relaxation.

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