## Functors for Coalgebras

Venue: | Algebra Universalis |

Citations: | 21 - 5 self |

### BibTeX

@ARTICLE{Gumm_functorsfor,

author = {H. Peter Gumm},

title = {Functors for Coalgebras},

journal = {Algebra Universalis},

year = {},

volume = {45},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor F which associates a set X with the set F(X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as F- coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated F-coalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler [AM89] to model various types of transition systems. Reichel [Rei95], and Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut96], J.J.M.M. Rutten develops the a fundamental theory of "universal coalgebra" along the lines of univers...

### Citations

1035 | Categories for the Working Mathematician - Lane - 1999 |

146 | Terminal coalgebras in well-founded set theory - Barr - 1993 |

94 |
An approach to object semantics based on terminal co–algebras
- Reichel
- 1995
(Show Context)
Citation Context ...need not be a coalgebra, and 1-generated F-coalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler [AM89] to model various types of transition systems. Reichel =-=[Rei95], and-=- Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut96], J.J.M.M. Rutten develops the a fundamental theory of "univ... |

56 | Initial algebra and final coalgebra semantics for concurrency - Rutten, Turi - 1994 |

51 |
Universal coalgebra: a theory of systems, Theor
- Rutten
(Show Context)
Citation Context ...AM89] to model various types of transition systems. Reichel [Rei95], and Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In =-=[Rut96], J.J.M.M.-=- Rutten develops the a fundamental theory of "universal coalgebra" along the lines of universal algebra ([Gra79, Coh81, Ihr93]). Various other authors have contributed further details to the... |

39 | Elements of the general theory of coalgebras - Gumm - 1999 |

24 | Terminal sequences for accessible endofunctors - Worrell |

20 | Functors for coalgebras. Algebra universalis 45 - Gumm - 2001 |

18 | Coalgebraic structure from weak limit preserving functors - Gumm, Schröder - 2000 |

14 | Covarieties and complete covarieties - Gumm, Schröder - 1998 |

12 | On descriptive classification of set-functors I - Trnková - 1971 |

9 |
Objects and Classes, co-algebraically, Object-Orientation with Parallelism and Book Persistence, B.Freitag et
- Jacobs
- 1996
(Show Context)
Citation Context ...bra, and 1-generated F-coalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler [AM89] to model various types of transition systems. Reichel [Rei95], and Jacobs =-=[Jac96] show that-=- coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut96], J.J.M.M. Rutten develops the a fundamental theory of "universal coalgebra" alo... |

8 | Equational and implicational classes of coalgebras
- Gumm
- 1999
(Show Context)
Citation Context ... to obtain a rich structure theory along the lines of universal algebra. In order to develop a satisfactory theory of cofree coalgebras, of covarieties and of coequational or coimplicational classes (=-=[Gum98]-=-), one additionally requires that T should be bounded, which amounts to saying that the cardinality of 1-generated T -coalgebras should not exceed a fixed cardinality. From the above functors, only th... |

7 |
A final coalgebra theorem, Proceedings category theory and computer science
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ... a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated F-coalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler =-=[AM89]-=- to model various types of transition systems. Reichel [Rei95], and Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut9... |

5 |
Covarieties and complete covarieties, Coalgebraic Methods
- Gumm, Schroeder
- 1998
(Show Context)
Citation Context ... subcoalgebras is not a subcoalgebra. Even the intersection of all subcoalgebras containing a given point x need not be a subcoalgebra, so the notion of "1-generated subcoalgebra, which is centra=-=l in [GS98]-=- and also in [Gum98] is not available, in spite of the fact that the type functor F preserves weak pullbacks. We do, of course, have that the intersection of finitely many subcoalgebras is a subcoalge... |

3 | Toposes of coalgebras and hidden algebras, Coalgebraic Methods in Computer Science - Worrell - 1998 |

1 | Limits in categories of coalgebras - Kurz - 1999 |