On coalgebras and type transformations

H. Peter Gumm

Discussiones Mathematicae - General Algebra and Applications (2007)

  • Volume: 27, Issue: 2, page 187-197
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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H. Peter Gumm. "On coalgebras and type transformations." Discussiones Mathematicae - General Algebra and Applications 27.2 (2007): 187-197. <http://eudml.org/doc/276913>.

@article{H2007,
abstract = {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.},
author = {H. Peter Gumm},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {coalgebra; endofunctor; filter functor; cartesian transformation; crisp},
language = {eng},
number = {2},
pages = {187-197},
title = {On coalgebras and type transformations},
url = {http://eudml.org/doc/276913},
volume = {27},
year = {2007},
}

TY - JOUR
AU - H. Peter Gumm
TI - On coalgebras and type transformations
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2007
VL - 27
IS - 2
SP - 187
EP - 197
AB - 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.
LA - eng
KW - coalgebra; endofunctor; filter functor; cartesian transformation; crisp
UR - http://eudml.org/doc/276913
ER -

References

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  1. [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. [2] S. Awodey, Category Theory, Oxford University Press (2006). 
  3. [3] H.P. Gumm, Birkhoff's variety theorem for coalgebras, Contributions to General Algebra 13 (2000), 159-173. Zbl0987.08004
  4. [4] H.P. Gumm, Functors for coalgebras, Algebra Universalis 45 (2001), 135-147. Zbl0982.08003
  5. [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. Zbl1151.18001
  6. [6] H.P. Gumm and T. Schröder, Coalgebras of bounded type, Math. Struct. in Comp. Science 12 (2001), 565-578. Zbl1011.08009
  7. [7] H.P. Gumm and T. Schröder, Types and coalgebraic structure, Algebra Universalis 53 (2005), 229-252. Zbl1086.08002
  8. [8] E.G. Manes, Implementing collection classes with monads, Math. Struct. in Comp. Science 8 (1998), 231-276. Zbl0916.68016
  9. [9] J.J.M.M. Rutten, Universal coalgebra: a theory of systems, Theoretical Computer Science 249 (2000), 3-80. Zbl0951.68038
  10. [10] J.D.H. Smith, Permutation representations of left quasigroups, Algebra Universalis 55 (2006), 387-406. Zbl1117.20051

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