## Constructive complete distributivity IV (1994)

Venue: | Appl. Cat. Struct |

Citations: | 7 - 5 self |

### BibTeX

@ARTICLE{Rosebrugh94constructivecomplete,

author = {Robert Rosebrugh and N. B. Canada and R. J. Wood},

title = {Constructive complete distributivity IV},

journal = {Appl. Cat. Struct},

year = {1994},

volume = {2},

pages = {2--119}

}

### OpenURL

### Abstract

A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from down-closed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and sup-preserving arrows. There is a restriction to order ideals and "totally algebraic" lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram package. AMS Subject Classification Primary: 06D10 Secondary 18B35, 03G10. Keywords: completely distributive, adjunction, projective, nuclear Introduction Idempotents do not split in the category of rel...

### Citations

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(Show Context)
Citation Context ...CCD), [6], so it is reasonable to expect that nuclearity is constructively equivalent to (CCD) --- of course over an arbitrary topos, E. The monoidal structure of sup = sup(E) was used extensively in =-=[10]-=-. We recall that the unit object is\Omega so that L is [L; \Omega\Gamma while L op is isomorphic to [L;\Omega op ]. There is a comparison L \Gamma! L op in sup, namely [L; :] , where : is negation for... |

116 |
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Citation Context ...ing, survives but careful clarification of this point will provide a deeper analysis of the main result of this paper. For X an ordered object we write QX for the Cauchy completion of X as defined in =-=[11] and-=- studied explicitly in this context in [3]. It is defined as a representing object for map(idl)(\Gamma; X) : ord op \Gamma! ord. (So for each Y we have ord(Y; QX) �� = map(idl)(Y; X).) It is a cen... |

44 | A 2-categorical approach to change of base and geometric morphisms I, Cahiers Topologie Geom. DiHerentielle Categoriques 32
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Citation Context ...es 19 and 24 restrict accordingly. Write lex:ord for the locally-full sub-ord-category of ord determined by the finitely complete objects and arrows which preserve finite meets. In the terminology of =-=[2]-=-, the objects of lex:ord are the cartesian objects of ord. Note that to ask for a left adjoint in lex:ord is to ask for a left adjoint in ord and further require that it preserve finite meets. For exa... |

32 |
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Citation Context ...r a general f there need be no transformation (containment) of which the "=" above expresses the invertibility. This might seem inconsequential but many of our results generalize to Yoneda s=-=tructures [17] on 2-cate-=-gories that are not locally ordered. In that case "commutativity " is obviously too strong but we cannot simply replace equality by an isomorphism because the mates of both such an isomorphi... |

17 |
A subdirect-union representation for completely distributive complete lattices
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Citation Context ...ymmetry. We point this out as we go. We are greatly indebted to Steve Vickers for stimulating correspondence on complete distributivity and for reminding us of Raney's work on idempotent relations in =-=[12]-=-. It is difficult to overestimate Raney's insights into complete distributivity. We have attempted in Sections 1 and 2 to acknowledge those results which, in non-categorical form, first appeared in [1... |

16 | Constructive complete distributivity
- Fawcett, Wood
- 1990
(Show Context)
Citation Context ...ary elementary topos that serves as the base for the entire discusssion and of course the result is proved for "constructively" completely distributive lattices, the subject of this series o=-=f papers: [6]-=-, [13], [14]. Now the categories of the equivalence mentioned above are both 2-categories, more preciselysord(= ord(E))-categories, so it is obvious that one should make this an ord-result --- if it i... |

15 | Information systems for continuous posets
- Vickers
- 1993
(Show Context)
Citation Context ...ork of Guitart and Riguet, [5], who have proved a constructive, but non-ord-enriched, version of Theorem 17 using methods quite different from ours. Vickers too has proved a version of our Theorem 17 =-=[18]-=-. The results here are fundamentally indebted to Eilenberg and Mac Lane, for without the language of categories Proposition 11, which separates our contributions from those of Raney, is difficult to s... |

11 |
Order ideals in categories
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(Show Context)
Citation Context ... of kar(rel) and we adopt this name. We refer to the arrows as modules. For Vickers they are the lower approximable semimappings. If X and A are orders then a module is an (order-)ideal as studied in =-=[3]-=-. In the present context the conditions on R : X - A simplify to aRysx implies aRx and asbRx implies aRx. We have full and locally full containments rel - - idl - - kar(rel) where for the first we reg... |

11 |
Intuitionistic algebra and representations of rings
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Citation Context ...eals. The semi-continuous reals is the "real numbers object" constructed from lower Dedekind cuts on the rationals and is isomorphic to the sheaf of lower semi-continuous functions in a spat=-=ial topos [4]. 4 -=-Totally algebraic lattices For any i : X \Gamma! L in ord, i is said to be dense if 1L is the left extension of i along i. In terms of elements this means that for all a in L, a �� = W fi(x) j x i... |

8 |
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Citation Context ...canonical arrow L \Omega M \Gamma! [L; M ] is an isomorphism, where L is [L; I], I being the unit object. The author noted that it suffices for the condition to hold for M = L. In a subsequent paper, =-=[7]-=-, the condition was explored in the symmetric monoidal closed category sup = sup(set) and it was shown that the nuclear objects are precisely the completely distributive lattices. If the axiom of choi... |

5 |
Projective and supercoherent frames
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(Show Context)
Citation Context ...lus of adjunctions --- and such idempotents characterize, in a way which we make precise, the lattices which were called "stably supercontinuous frames" in a recent paper by Banaschewski and=-= Niefield [1]-=-. Indeed this is one of the corollaries of the main result mentioned above. We consider that the point of view on idempotents presented here is likely to be useful elsewhere and while we do not burden... |

5 |
Constructive complete distributivity
- Rosebrugh, R, et al.
- 1994
(Show Context)
Citation Context ...lementary topos that serves as the base for the entire discusssion and of course the result is proved for "constructively" completely distributive lattices, the subject of this series of pap=-=ers: [6], [13]-=-, [14]. Now the categories of the equivalence mentioned above are both 2-categories, more preciselysord(= ord(E))-categories, so it is obvious that one should make this an ord-result --- if it is one ... |

3 | Constructive complete distributivity III - Rosebrugh, Wood - 1992 |

3 |
Proarrows 1. Cahiers de Topologie et Géometrie Différentielle
- Wood
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(Show Context)
Citation Context ...Gamma! kar(rel). By construction, ( ) + is the identity on objects, locally fully faithful and every arrow in inf gives an adjunction in kar(rel). By definition therefore, ( ) + is proarrow equipment =-=[19]-=- and an object of the 3-category IF studied in [2]. It restricts to ( ) + : ord \Gamma! idl, well known to be an object of IF which as such contains ( )s: E \Gamma! rel. As we said in the Introduction... |

2 |
Continuous posets and adjoint sequences
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(Show Context)
Citation Context ...X for X a sup lattice and for X a (CCD) lattice in terms of extra adjoints. We will refer to the following situation: L DL oe W - # - + oe n W ? ? ? This configuration was considered by R-E. Hoffmann =-=[8]-=- for continuous posets. There in place of down-closed subsets he considered down-closed and up-directed subsets. Some of our proofs below are similar to his but a more systematic use of adjointness ma... |

2 |
Proarrows 2. Cahiers de topologie et g'eometrie diff'erentielle cat'egoriques
- Wood
- 1985
(Show Context)
Citation Context ... observations ensure that ( ) + : inf \Gamma! kar(rel) shares many 2-categorical properties with ( ) + : ord \Gamma! idl. In fact ( ) + : inf \Gamma! kar(rel) satisfies the stronger Axioms 4 and 5 of =-=[20]-=-. For reference later, note that if f : X \Gamma! A and u : A \Gamma! X in inf then f a u if and only if f + = u+ . This follows immediately from the definition of the ord-structure of inf . We denote... |

1 |
Envelopes karoubiennes de cat'egories de kleisli. Cahiers de topologie et g'eometrie diff'erentielle cat'egoriques, XXX:261
- Riguet
- 1992
(Show Context)
Citation Context ...plete distributivity. We have attempted in Sections 1 and 2 to acknowledge those results which, in non-categorical form, first appeared in [12]. We should also mention the work of Guitart and Riguet, =-=[5]-=-, who have proved a constructive, but non-ord-enriched, version of Theorem 17 using methods quite different from ours. Vickers too has proved a version of our Theorem 17 [18]. The results here are fun... |