## The Constructive Lift Monad (1995)

Venue: | Informix Software, Inc |

Citations: | 5 - 0 self |

### BibTeX

@TECHREPORT{Kock95theconstructive,

author = {Anders Kock and Anders Kock},

title = {The Constructive Lift Monad},

institution = {Informix Software, Inc},

year = {1995}

}

### OpenURL

### Abstract

ut by applying T to some poset (namely the original poset less the bottom). Both these properties fail to hold constructively, if the lift monad is interpreted as "adding a bottom"; see Remark below. If, on the other hand, we interpret the lift monad as the one which freely provides supremum for each subset with at most one element (which is what we shall do), then the first property holds; and we give a necessary and sufficient condition that the second does. Finally, we shall investigate the lift monad in the context of (constructive) locale theory. I would like to thank Bart Jacobs for guiding me to the litterature on Z-systems; to Gonzalo Reyes for calling my attention to Barr's work on totally connected spaces; to Steve Vickers for some pertinent correspondence. I would like to thank the Netherlands Science Organization (NWO) for supporting my visit to Utrecht, where a part of the present research was carried out, and for various travel support from

### Citations

75 |
Stone spaces
- Johnstone
- 1982
(Show Context)
Citation Context ...eory of KZ-monads [14] can be seen to seen to apply). The most well known of these submonads Z is the monad Idl, where Idl(C) is the set of filtering lower subsets of C (ideals, in the terminology of =-=[7]-=-). This special case led Thatcher, Wagner, Wright, Venugopalan and others to generalize a family of notions and terminology from continuous lattice theory to general Z: Z-continuous and Z-algebraic po... |

67 |
Semantics of weakening and contraction
- Jacobs
- 1994
(Show Context)
Citation Context ...sets is Cartesian closed, the more specific notions of [12] apply. We claim that the equivalent conditions of loc.cit. Proposition 2.2 hold (so that T is a relevant monad in the sense of Bart Jacobs, =-=[5]-=-). Here it just means that the two canonical maps T (A \Theta B) - T (A) \Theta T (B) - / T (A \Theta B) compose to the identity map on T (A \Theta B). In fact, let W ` A \Theta B be quasi-principal. ... |

33 | Monads for which structures are adjoint to units
- KOCK
- 1972
(Show Context)
Citation Context ...nto the context of order completion monads in general, in the sense extensively studied in the literature, cf. e.g. [1], [11], [17], [21], [23], and in partiular put it into the context of KZ-monads, =-=[14]-=-. Recall that the full cocompletion monad J on the category POS of posets is the construction which to a poset C associates the set J(C) of all lower subsets of C, i.e. subsets D ` C such that if bsa ... |

29 |
Proper maps of toposes
- Moerdijk, Vermeulen
(Show Context)
Citation Context ...int to units; but recall that the direction of the 2-cells is governed by the inverse image maps (frame maps)). 15 The following result is basically known (except possibly for the condition 2)), from =-=[10]-=-, (cf. also [9], [4]), and is included for completeness, and for comparison with the previous theorem on the lift monad T on LOC Theorem 14 Let C be a locale. Then t.f.a.e. 1) C admits a structure for... |

26 | Fibrations and Yoneda’s lemma in a 2-category - Street - 1974 |

19 |
Open locales and exponentiation
- Johnstone
- 1984
(Show Context)
Citation Context ...boolean situation, an element is positive iff it is not the bottom element; the present positive way of expressing this property is related to a notion of "positive elements in locales" cons=-=idered in [8]-=- and [18]. In fact, for any poset with a top element, an element a is T-compact iff every cover of a is inhabited.) For general reasons, we have that b 0sbsasa 0 implies b 0sa 0 , and thatssatisfies a... |

18 |
Bilinearity and cartesian closed monads
- Kock
- 1971
(Show Context)
Citation Context ...ical preorder, and in this context, the partial map classifier is in fact a KZ-monad, as proved in loc. cit. Remark 2. Let us remark that T is a commutative monad in the sense of the author, cf. e.g. =-=[12]-=- and the references therin; for, it is a submonad of J , which is commutative. Since the category POS of posets is Cartesian closed, the more specific notions of [12] apply. We claim that the equivale... |

11 | Locales are not pointless
- Vickers
- 1995
(Show Context)
Citation Context ... composite TA \Theta TB - / T (A \Theta B) -sTA \Theta TB can be shown to besthe identity map on TA \Theta TB, and sosa /, meaning that T is of the interesting kind of KZ monads studied by Vickers in =-=[22]-=-. Remark 3. Proposition 1 and 3, for X = 1, i.e. for the lift monad T posets, can easily be upgraded to the statement that there is a distributive law of the monad Idl over the lift monad T , so that ... |

10 |
Partial products, bagdomains and hyperlocal toposes
- Johnstone
- 1992
(Show Context)
Citation Context ..., if v 2 S F , we have v 2 V 2 F for suitable V , but since F is # W , v 2 V ` W =# w, so vsw, or v 2# w. This proves the Proposition. (One may alternatively see T as a case of the Fam monad (cf e.g. =-=[9]), namely -=-"families with at most one member" - which, unlike most Fammonads does restrict to a monad on Posets.) 6 Let C be a poset. If X ` C is a subset with at most one element (so X is subterminal,... |

10 |
Continuous fibrations and inverse limits of toposes
- Moerdijk
- 1986
(Show Context)
Citation Context ...situation, an element is positive iff it is not the bottom element; the present positive way of expressing this property is related to a notion of "positive elements in locales" considered i=-=n [8] and [18]-=-. In fact, for any poset with a top element, an element a is T-compact iff every cover of a is inhabited.) For general reasons, we have that b 0sbsasa 0 implies b 0sa 0 , and thatssatisfies an interpo... |

9 |
A uniform approach to inductive posets and inductive closure. Theoret
- Wright, Wagner, et al.
- 1978
(Show Context)
Citation Context ...Definition 2 below. This second description will also put T into the context of order completion monads in general, in the sense extensively studied in the literature, cf. e.g. [1], [11], [17], [21], =-=[23]-=-, and in partiular put it into the context of KZ-monads, [14]. Recall that the full cocompletion monad J on the category POS of posets is the construction which to a poset C associates the set J(C) of... |

6 |
related to de Morgan’s law
- Johnstone
- 1979
(Show Context)
Citation Context ... (finite) infima. Remark. For the simple monad 1 + \Gamma on POS, ("freely adding a bottom element"), the conclusions of Propositions 1 and 2 fail, unless the topos is a deMorgan one (in the=-= sense of [6]-=-). For the 1-point poset 1 is a frame, and (hence) also has filtered sup, but 1+1 cannot be a frame, nor have filtered sup unless the topos is deMorgan. The first statement is clear from [6], and the ... |

5 | Yoneda’s Lemma in a 2-category, in Category Seminar, Sydney 1972/73 - Street - 1986 |

4 |
Limit monads in categories
- Kock
(Show Context)
Citation Context ..., in the sense of Definition 2 below. This second description will also put T into the context of order completion monads in general, in the sense extensively studied in the literature, cf. e.g. [1], =-=[11]-=-, [17], [21], [23], and in partiular put it into the context of KZ-monads, [14]. Recall that the full cocompletion monad J on the category POS of posets is the construction which to a poset C associat... |

3 |
E.Nelson, Completions of partially ordered sets
- Banaschewski
- 1982
(Show Context)
Citation Context ... of C, in the sense of Definition 2 below. This second description will also put T into the context of order completion monads in general, in the sense extensively studied in the literature, cf. e.g. =-=[1]-=-, [11], [17], [21], [23], and in partiular put it into the context of KZ-monads, [14]. Recall that the full cocompletion monad J on the category POS of posets is the construction which to a poset C as... |

3 |
set theory and topos
- Barr
- 1986
(Show Context)
Citation Context ...p D 0 ! D; and C is T -continuous. 4) (If C corresponds to a sober topological space X) X has a maximal point in the specialization ordering (i.e. X is a totally connected space, in the sense of Barr =-=[2]-=-). Proof. Assume 1). A T -structure is a right adjoint (locale-) map for the locale map #, or equivalently, passing to their left adjoints, a left adjoint (frame-) map for the frame map sup : T (C) ! ... |

2 |
A motivation and generalization of Scott’s notion of a continuous lattice
- Markowsky
- 1979
(Show Context)
Citation Context ...t, Venugopalan and others to generalize a family of notions and terminology from continuous lattice theory to general Z: Z-continuous and Z-algebraic posets, and the Zway -below relation. Recall from =-=[16]-=- or [7] that a poset C is called continuous if #: C ! Idl(C) not only has a left adjoint (=supremum formation for filtered lower subset) but this left adjoint in turn has a left adjoint. We shall stud... |

1 |
Lifting as a KZ-doctrine, preprint
- Fiore
- 1995
(Show Context)
Citation Context ... the partialmap -classifier monad, i.e. the lift monad, on sets, cannot be construed as a KZ-monad. But in a certain other world of sets, namely the one described in Synthetic Domain Theory (cf. e.g. =-=[3]-=- and the references therein), every set acquires a canonical preorder, and in this context, the partial map classifier is in fact a KZ-monad, as proved in loc. cit. Remark 2. Let us remark that T is a... |

1 | Constructive theory of the lift monad on posets - Kock - 1992 |

1 |
completion monads, Alg
- Meseguer
(Show Context)
Citation Context ...he sense of Definition 2 below. This second description will also put T into the context of order completion monads in general, in the sense extensively studied in the literature, cf. e.g. [1], [11], =-=[17]-=-, [21], [23], and in partiular put it into the context of KZ-monads, [14]. Recall that the full cocompletion monad J on the category POS of posets is the construction which to a poset C associates the... |

1 | Generalization of continuous posets, Trans.A.M.S - Novak - 1982 |