Kleisli monoids describing approach spaces
| Autor: | Eva Colebunders, K. Van Opdenbosch |
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| Přispěvatelé: | Faculty of Sciences and Bioengineering Sciences, Topological Algebra, Functional Analysis and Category Theory, Mathematics, Vriendenkring VUB |
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
power-enriched monad Mathematics(all) Algebra and Number Theory General Computer Science 010102 general mathematics Structure (category theory) 01 natural sciences Prime (order theory) Theoretical Computer Science 010101 applied mathematics Combinatorics Morphism Monad (non-standard analysis) Approach space Bounded function Mathematics::Category Theory prime functional ideal Ideal (ring theory) Isomorphism Kleisli monoid 0101 mathematics relational algebra Mathematics |
| Popis: | We study the functional ideal monad \(\mathbb {I} = (\mathsf {I}, m, e)\) on Set and show that this monad is power-enriched. This leads us to the category \(\mathbb {I}\)- Mon of all \(\mathbb {I}\)-monoids with structure preserving maps. We show that this category is isomorphic to App, the category of approach spaces with contractions as morphisms. Through the concrete isomorphism, an \(\mathbb {I}\)-monoid (X,ν) corresponds to an approach space \((X, \mathfrak {A}),\) described in terms of its bounded local approach system. When I is extended to Rel using the Kleisli extension \(\check {\mathsf {I}},\) from the fact that \(\mathbb {I}\)- Mon and \((\mathbb {I},2)\)- Cat are isomorphic, we obtain the result that App can be isomorphically described in terms of convergence of functional ideals, based on the two axioms of relational algebras, reflexivity and transitivity. We compare these axioms to the ones put forward in Lowen (2015). Considering the submonad \(\mathbb {B}\) of all prime functional ideals, we show that it is both sup-dense and interpolating in \(\mathbb {I}\), from which we get that \((\mathbb {I},2)\)- Cat and \((\mathbb {B},2)\)- Cat are isomorphic. We present some simple axioms describing App in terms of prime functional ideal convergence. |
| Databáze: | OpenAIRE |
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