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2007 | 27 | 2 | 187-197
Tytuł artykułu

On coalgebras and type transformations

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Abstrakty
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We show that for an arbitrary Set-endofunctor T the generalized membership function given by a sub-cartesian transformation μ from T to the filter functor 𝔽 can be alternatively defined by the collection of subcoalgebras of constant T-coalgebras. Sub-natural transformations ε between any two functors S and T are shown to be sub-cartesian if and only if they respect μ. The class of T-coalgebras whose structure map factors through ε is shown to be a covariety if ε is a natural and sub-cartesian mono-transformation.
Twórcy
  • Philipps-University Marburg, Faculty of Mathematics and Computer Science, Hans-Meerwein-Straße 35032 Marburg, Germany
Bibliografia
  • [1] P. Aczel and N. Mendler, A final coalgebra theorem, pp. 357-365 in: D.H. Pitt et al., eds, Proceedings Category Theory and Computer Science, Lecture Notes in Computer Science, Springer 1989.
  • [2] S. Awodey, Category Theory, Oxford University Press (2006).
  • [3] H.P. Gumm, Birkhoff's variety theorem for coalgebras, Contributions to General Algebra 13 (2000), 159-173.
  • [4] H.P. Gumm, Functors for coalgebras, Algebra Universalis 45 (2001), 135-147.
  • [5] H.P. Gumm, From{T-coalgebras to filter structures and transition systems, pp. 194-212 in: D.H. Fiadeiro et al., eds, Algebra and Coalgebra in Computer Science, vol 3629 of Lecture Notes in Computer Science, Springer 2005.
  • [6] H.P. Gumm and T. Schröder, Coalgebras of bounded type, Math. Struct. in Comp. Science 12 (2001), 565-578.
  • [7] H.P. Gumm and T. Schröder, Types and coalgebraic structure, Algebra Universalis 53 (2005), 229-252.
  • [8] E.G. Manes, Implementing collection classes with monads, Math. Struct. in Comp. Science 8 (1998), 231-276.
  • [9] J.J.M.M. Rutten, Universal coalgebra: a theory of systems, Theoretical Computer Science 249 (2000), 3-80.
  • [10] J.D.H. Smith, Permutation representations of left quasigroups, Algebra Universalis 55 (2006), 387-406.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1126
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