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Conference Papers Year : 2020

Approximate Coalgebra Homomorphisms and Approximate Solutions

Jiří Adámek
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Abstract

Terminal coalgebras $$\nu F$$ of finitary endofunctors F on categories called strongly lfp are proved to carry a canonical ultrametric on their underlying sets. The subspace formed by the initial algebra $$\mu F$$ has the property that for every coalgebra A we obtain its unique homomorphism into $$\nu F$$ as a limit of a Cauchy sequence of morphisms into $$\mu F$$ called approximate homomorphisms. The concept of a strongly lfp category includes categories of sets, posets, vector spaces, boolean algebras, and many others.For the free completely iterative algebra $$\varPsi B$$ on a pointed object B we analogously present a canonical ultrametric on its underlying set. The subspace formed by the free algebra $$\varPhi B$$ on B has the property that for every recursive equation in $$\varPsi B$$ we obtain the unique solution as a limit of a Cauchy sequence of morphisms into $$\varPhi B$$ called approximate solutions. A completely analogous result holds for the free iterative algebra RB on B.
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hal-03232353 , version 1 (21-05-2021)

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Jiří Adámek. Approximate Coalgebra Homomorphisms and Approximate Solutions. 15th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2020, Dublin, Ireland. pp.11-31, ⟨10.1007/978-3-030-57201-3_2⟩. ⟨hal-03232353⟩
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