## Mongruences and Cofree Coalgebras (1995)

Venue: | Algebraic Methods and Software Technology, number 936 in Lect. Notes Comp. Sci |

Citations: | 30 - 10 self |

### BibTeX

@INPROCEEDINGS{Jacobs95mongruencesand,

author = {Bart Jacobs},

title = {Mongruences and Cofree Coalgebras},

booktitle = {Algebraic Methods and Software Technology, number 936 in Lect. Notes Comp. Sci},

year = {1995},

pages = {245--260},

publisher = {Springer}

}

### OpenURL

### Abstract

. A coalgebra is introduced here as a model of a certain signature consisting of a type X with various "destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in object-oriented programming: they provide access to the type (or state) X. We show that the category of such coalgebras and structure preserving functions is comonadic over sets. Therefore we introduce the notion of a `mongruence' (predicate) on a coalgebra. It plays the dual role of a congrence (relation) on an algebra. An algebra is a set together with a number of operations on this set which tell how to form (derived) elements in this set, possibly satisfying some equations. A typical example is a monoid, given by a set M with operations 1 ! M , M \Theta M ! M . Here 1 = f;g is a singleton set. In mathematics one usually considers only single-typed algebras, but in computer science one more naturally uses many-typed algebras like 1 ! list(A), A \Theta l...

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Citation Context ...e is a relation which is closed under the algebra operations. We use these mongruences in particular to construct cofree coalgebras satisfying certain equations. In Mac Lane's book on category theory =-=[14] (but see -=-also [15]) there is a section VI, 8 called "Algebras are T -algebras". It is shown ? To appear in Proceedings of AMAST 1995, to be published as a Springer Lecture Notes in Computer Science. ... |

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Citation Context ... account. It is on the wrong level: in the "base" category instead of in the "total" category of the fibration, which is where the predicates live. The definition of bisimulation b=-=y Aczel and Mendler [2]-=- is not on the right level for the very same reasons. (These bisimulations should be what we call coinduction assumptions, see Example 4 below.) And the same applies to the definition of congruence in... |

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Citation Context ... 8 Adding powerset One possible extension of the above-used notion of polynomial functor is to add the powerset functor P. This may be finite subsets only, so that more terminal coalgebras exist, see =-=[3]-=-. One adds a fifth formation rule in Definition 1 (for C = Sets), (v) if T : Sets ! Sets is a polynomial functor, then so is X 7! P(T (X)). Then we can consider functors like X 7! P(A \Theta X), coalg... |

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Citation Context ... set together with operations "going out" of the set; they tell how to "deconstruct" elements of the set. These kind of structures naturally arise within object-oriented programmin=-=g---as explained in [16, 10], but see also [13],-=- or [5] (where the "destructors" are called "observers"). A possible (set-theoretic) semantics for objects is to see them as pairs hx 2 X; c: X ! T (X)i where c is a coalgebra of a... |

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Citation Context ...oing out" of the set; they tell how to "deconstruct" elements of the set. These kind of structures naturally arise within object-oriented programming---as explained in [16, 10], but see=-= also [13], or [5] (where the "de-=-structors" are called "observers"). A possible (set-theoretic) semantics for objects is to see them as pairs hx 2 X; c: X ! T (X)i where c is a coalgebra of a functor T : Sets ! Sets in... |

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Citation Context ...ch is closed under the algebra operations. We use these mongruences in particular to construct cofree coalgebras satisfying certain equations. In Mac Lane's book on category theory [14] (but see also =-=[15]) there is-=- a section VI, 8 called "Algebras are T -algebras". It is shown ? To appear in Proceedings of AMAST 1995, to be published as a Springer Lecture Notes in Computer Science. there that algebras... |

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Citation Context ...r C = Sets), (v) if T : Sets ! Sets is a polynomial functor, then so is X 7! P(T (X)). Then we can consider functors like X 7! P(A \Theta X), coalgebras of which are Alabelled transition systems, see =-=[1, 17]-=-. In our approach we first lift this powerset functor to predicates and relations. Definition 13. Define the lifted powerset functor Pred(P): Pred ! Pred by (Q ` I) 7! (fa j 8i 2 a: Q(i)g ` P(I)): And... |

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Citation Context ... ) and Rel(T ) one also gets these different ways of reasoning about T . For example, one may wish to reason about a base category of metric spaces M with closed subsets X ` M , or with morphisms M ! =-=[0; 1] as predicates on M -=-. This involves different logics, which have their own "logical liftings". The formulation () cannot take such differences into account. It is on the wrong level: in the "base" cat... |

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Citation Context ...ations "going out" of the set; they tell how to "deconstruct" elements of the set. These kind of structures naturally arise within object-oriented programming---as explained in [16=-=, 10], but see also [13], or [5] (where the -=-"destructors" are called "observers"). A possible (set-theoretic) semantics for objects is to see them as pairs hx 2 X; c: X ! T (X)i where c is a coalgebra of a functor T : Sets !... |

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Citation Context ...operties. Then we describe how polynomial functors T : Sets ! Sets can be lifted to polynomial functors Pred(T ): Pred ! Pred and Rel(T ): Rel ! Rel by induction on the structure of T . This is as in =-=[7, 9, 8]-=-. Algebras and coalgebras of these lifted functors Pred(T ) and Rel(T ) give important (logical) information about T . Definition 2. (i) The category Pred has as objects pairs (I; P ) where P ` I is a... |

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Citation Context ...structors from the signature and the term forming operation as associated with finite products and coproducts: tupling and projections and cotupling and coprojections (including empty ones), see e.g. =-=[12, 6]-=-. We use capital letters \Gamma to abbreviate contexts v 1 : oe 1 ; : : : ; v n : oe n of variable declarations. An equation is then a pair of terms with type X in the same context, written as \Gamma ... |

9 | B.: An algebraic view of structural induction
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Citation Context ...operties. Then we describe how polynomial functors T : Sets ! Sets can be lifted to polynomial functors Pred(T ): Pred ! Pred and Rel(T ): Rel ! Rel by induction on the structure of T . This is as in =-=[7, 9, 8]-=-. Algebras and coalgebras of these lifted functors Pred(T ) and Rel(T ) give important (logical) information about T . Definition 2. (i) The category Pred has as objects pairs (I; P ) where P ` I is a... |

8 |
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Citation Context ...operties. Then we describe how polynomial functors T : Sets ! Sets can be lifted to polynomial functors Pred(T ): Pred ! Pred and Rel(T ): Rel ! Rel by induction on the structure of T . This is as in =-=[7, 9, 8]-=-. Algebras and coalgebras of these lifted functors Pred(T ) and Rel(T ) give important (logical) information about T . Definition 2. (i) The category Pred has as objects pairs (I; P ) where P ` I is a... |

8 |
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(Show Context)
Citation Context ... set together with operations "going out" of the set; they tell how to "deconstruct" elements of the set. These kind of structures naturally arise within object-oriented programmin=-=g---as explained in [16, 10], but see also [13],-=- or [5] (where the "destructors" are called "observers"). A possible (set-theoretic) semantics for objects is to see them as pairs hx 2 X; c: X ! T (X)i where c is a coalgebra of a... |

8 |
Parameters and parametrization in specification, using distributive categories
- Jacobs
- 1995
(Show Context)
Citation Context ...structors from the signature and the term forming operation as associated with finite products and coproducts: tupling and projections and cotupling and coprojections (including empty ones), see e.g. =-=[12, 6]-=-. We use capital letters \Gamma to abbreviate contexts v 1 : oe 1 ; : : : ; v n : oe n of variable declarations. An equation is then a pair of terms with type X in the same context, written as \Gamma ... |

1 |
Bisimulation and apartness in coalgebraic specification. Manuscript available from ftp.cwi.nl in /pub/bjacobs
- Jacobs
- 1995
(Show Context)
Citation Context ... that come from destructor signatures. The (carriers of the) terminal coalgebras that one gets in this way are sets of (generally infinite) trees, which can be described in terms of observations, see =-=[11]-=-. We mention a particular case (which is essentially as used in [13, 16]). Proposition 10. The terminal coalgebra of the functor X 7! A \Theta (B ) X) associated with the signature X \Gamma! A, B \The... |