## The Coalgebraic Dual Of Birkhoff's Variety Theorem (2000)

Citations: | 11 - 0 self |

### BibTeX

@TECHREPORT{Awodey00thecoalgebraic,

author = {Steve Awodey and Jesse Hughes},

title = {The Coalgebraic Dual Of Birkhoff's Variety Theorem},

institution = {},

year = {2000}

}

### OpenURL

### Abstract

We prove an abstract dual of Birkho's variety theorem for categories E of coalgebras, given suitable assumptions on the underlying category E and suitable : E ## E . We also discuss covarieties closed under bisimulations and show that they are denable by a trivial kind of coequation { namely, over one "color". We end with an example of a covariety which is not closed under bisimulations. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott.

### Citations

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(Show Context)
Citation Context ...) are studied in [GS98]. We generalize this work and provide a natural example of a covariety which is not behavioral. A dual to Birkho's variety theorem for coalgebras over Set wassrst mentioned in [=-=Rut96-=-]. This result was further developed in [GS98], where behavioral covarieties weresrst studied (under the name complete covarieties). Behavioral covarieties also arise (under the name sinks) in [Ros00]... |

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Citation Context ...e category theoretic notion of-algebras for an endofunctor . This, in turn, leads to the dual notion of-coalgebras. Coalgebras have proven useful in modeling processes and objects in computer science =-=[JR97]. -=-It is natural to ask whether such a basic theorem as Birkho's variety theorem can be dualized to a \Co-Birkho Covariety Theorem" for coalgebras. In order to dualize the Birkho Variety Theorem, we... |

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54 |
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(Show Context)
Citation Context ...ped for varieties in [BH76], which we discovered after proving the results herein. Alexander Kurz simultaneously and independently took the same approach for covarieties and proved similar results in =-=[Kur00-=-]. This work forms part of the second author's doctoral dissertation, written under the supervision of Professor Dana S. Scott. We both thank him for suggesting we consider the dual of Birkho's theore... |

53 |
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(Show Context)
Citation Context ... that U is comonadic). Then the category E of -coalgebras is also co-Birkho. Proof. It is well known that the coalgebraic forgetful functor has a right adjoint i it is comonadic. See, for instance, [T=-=ur96]-=- for details. One shows directly that U re ects regular monos and so inherits the epi-regular mono factorizations from E . This implies that E is regularly well-powered. Because HA is injective if A i... |

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Citation Context ...f Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1. Introduction One of the earliest theorems in universal algebra is Garrett Birkho's Variety Theorem [=-=Bir35-=-]. It states that a class V of algebras is closed under homomorphic images, subalgebras and products just in case V is the collection of all algebras satisfying some set of equations. The classical de... |

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Citation Context ...We take the perspective here that a -algebra is an algebra for a polynomial endofunctor, and equational denability is a special kind of orthogonality condition. Alternatively, in the terminology of [A=-=R-=-94], a class of algebras is equationally denable just in case it is an injectivity class for an appropriate collection of morphisms. In more details, given a signature , the category of -algebras is i... |

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Citation Context ...ality, and can be regarded as a condition on the possible \colorings" of a coalgebra. The covarieties over Set which are closed under bisimulations (here called behavioralscovarieties) are studie=-=d in [GS98-=-]. We generalize this work and provide a natural example of a covariety which is not behavioral. A dual to Birkho's variety theorem for coalgebras over Set wassrst mentioned in [Rut96]. This result wa... |

13 | A study of categories of algebras and coalgebras
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- 2001
(Show Context)
Citation Context ...iple hhR; i; r 1 ; r 2 i such that the maps r 1 : hR; i ## hA; i ; r 2 : hR; i ## hB;si are jointly monic. For a more traditional denition suitable for coalgebras over co-Birkho categories, see [Hug01=-=-=-]. In [GS98], a complete covariety is dened as one closed under total bisimulations; we adopt the term behavioral covariety instead. A total bisimulation is one such that each projection is epi (in E ... |

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(Show Context)
Citation Context ...er the name sinks) in [Ros00]. We take a more general approach, which yields a covariety theorem for coalgebras over a wide class of categories. The basic approach wassrst developed for varieties in [=-=BH76]-=-, which we discovered after proving the results herein. Alexander Kurz simultaneously and independently took the same approach for covarieties and proved similar results in [Kur00]. This work forms pa... |

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