## Exponentiable morphisms: posets, spaces, locales (2000)

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Venue: | and Grothendieck toposes, Theory and Applications of Categories 8 |

Citations: | 8 - 5 self |

### BibTeX

@INPROCEEDINGS{Niefield00exponentiablemorphisms:,

author = {Susan Niefield},

title = {Exponentiable morphisms: posets, spaces, locales},

booktitle = {and Grothendieck toposes, Theory and Applications of Categories 8},

year = {2000},

pages = {16--32}

}

### OpenURL

### Abstract

ABSTRACT. Inthis paper, we consider those morphisms p: P − → B of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes. In particular, we show that a necessary condition is that the induced map p ↓ : P ↓ − → B ↓ is exponentiable in the category of topological spaces, where P ↓ is the space whose points are elements of P and open sets are downward closed subsets of P. Along the way, we show that p ↓ : P ↓ − → B ↓ is exponentiable if and only if p: P − → B is exponentiable in the category of posets and satisfies an additional compactness condition. The criteria for exponentiability of morphisms of posets is related to (but weaker than) the factorization-lifting property for exponentiability of morphisms in the

### Citations

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- 1977
(Show Context)
Citation Context ...ntiability. This question arose when the author was working with Marta Bunge on [BN]. Since it is well-known that this functor preserves products and exponentiable objects, though not equalizers (see =-=[J1]-=-), we did not expect that exponentiable morphisms would be preserved, but we made little progress towards an answer. Subsequently, the author realized that this problem might be solved by first restri... |

98 |
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- 1959
(Show Context)
Citation Context ...th the definition of continuous posets based on the Hofmann and Stralka [HS] characterization of continuous lattices as those for which the map � :Idl(A) −→ A has a left adjoint. Using Grothendieck’s =-=[Gr]-=- notion of the ind-completion Ind(E) of a locally small category E, they defined E to be a continuous category if it has filtered colimits and the functor −→ lim: Ind(E) −→ Ehas a left adjoint. They t... |

77 | Categories for the Working - Lane, S - 1971 |

32 | Topology: A First - MUNKRES - 1975 |

20 |
Méthode de la descente
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- 1964
(Show Context)
Citation Context ...considerable interest in exponentiability in Cat. It has long been known that Cat is cartesian closed but not locally cartesian closed. The exponentiable morphisms of Cat were characterized by Giraud =-=[Gi]-=- (and later rediscovered by Conduché [C]) as those satisfying a certain factorization lifting property. In [BN], Bunge and Niefield introduced the notion of a locally closed subcategory and showed tha... |

16 |
General topology. Graduate Texts
- Kelley
- 1975
(Show Context)
Citation Context ... For non-Hausdorff spaces, there were two nonequivalent definitions of local compactness under consideration in the 1970’s. One hypothesized the existence of a compact neighborhood of each point (see =-=[K]-=-) while the other required arbitrarily small such neighborhoods (see [M]). In 1978, Hofmann and Lawson [HL] showed that the sober spaces satisfying the latter definition are those for which O(X) is co... |

14 | On Topologies for Function Spaces - FOX - 1945 |

14 |
Cartesianness: Topological spaces, uniform spaces and affine varieties
- Niefield
- 1982
(Show Context)
Citation Context ...d Applications of Categories, Vol. 8, No. 2 17 Aff affine schemes and morphism of affine schemes This paper concerns the first five categories listed above. For the latter two, we refer the reader to =-=[N3]-=-. We began by asking to what extent the presheaf functor PSh: Cat −→ GTop preserves and reflects exponentiability. This question arose when the author was working with Marta Bunge on [BN]. Since it is... |

13 |
The spectral theory of distributive continuous lattices
- Hofmann, Lawson
- 1978
(Show Context)
Citation Context ...on in the 1970’s. One hypothesized the existence of a compact neighborhood of each point (see [K]) while the other required arbitrarily small such neighborhoods (see [M]). In 1978, Hofmann and Lawson =-=[HL]-=- showed that the sober spaces satisfying the latter definition are those for which O(X) is continuous, and hence, precisely the sober spaces which are exponentiable in Top. Thus, the second definition... |

12 |
Exponentiability and Single Universes
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- 1966
(Show Context)
Citation Context ...he reader to [N3]. We began by asking to what extent the presheaf functor PSh: Cat −→ GTop preserves and reflects exponentiability. This question arose when the author was working with Marta Bunge on =-=[BN]-=-. Since it is well-known that this functor preserves products and exponentiable objects, though not equalizers (see [J1]), we did not expect that exponentiable morphisms would be preserved, but we mad... |

12 |
Function space in the category of locales
- Hyland
- 1981
(Show Context)
Citation Context ...nd locales) that we shall follow in this paper. In the decade following the Day/Kelly paper, the relationship between exponentiability and local compactness was generalized to locales and toposes. In =-=[H]-=-, Hyland showed that a locale is exponentiable in Loc if and only if it is locally compact. Then Johnstone and Joyal generalized this result to Grothendieck toposes. They began with the definition of ... |

12 |
Continuous categories and exponentiable toposes
- Johnstone, Joyal
- 1982
(Show Context)
Citation Context ... morphism p: PSh(P ) −→ PSh(B) is exponentiable in GTop. Then using the equivalence PSh(P ) � Sh(P ↓ ), we know that p:Sh(P ↓ ) −→ Sh(B ↓ )is exponentiable as well, and since the Johnstone/Joyal work =-=[JJ]-=- is constructive and hence applies to the relative case, it follows that p∗(Ω P ↓) is locally compact in the category Loc(Sh(B ↓ )) of internal locales in Sh(B ↓ ). But, Loc(Sh(B ↓ )) � Loc/O(B ↓ ) (s... |

11 |
General function spaces, products and continuous lattices
- Isbell
- 1986
(Show Context)
Citation Context ...showed that continuity of O(X) coincides with local compactness of X for Hausdorff spaces. For a thorough treatment of exponentiability in Top and related function space problems see Isbell’s article =-=[I]-=- on the influence of the Day and Kelly paper. For non-Hausdorff spaces, there were two nonequivalent definitions of local compactness under consideration in the 1970’s. One hypothesized the existence ... |

9 |
Factorization and pullback theorems for localic geometric morphisms, Univ. Cath. de Lovain, Sém. do math. pure. Rapport no
- Johnstone
- 1979
(Show Context)
Citation Context ...ctive and hence applies to the relative case, it follows that p∗(Ω P ↓) is locally compact in the category Loc(Sh(B ↓ )) of internal locales in Sh(B ↓ ). But, Loc(Sh(B ↓ )) � Loc/O(B ↓ ) (see [JT] or =-=[J2]-=-), and so using the results from [N4], it follows that p ↓ : P ↓ −→ B ↓ is exponentiable in Top. Thus, we are led to considering the effect of the functor ( ) ↓ : Pos −→ Top on exponentiability. We be... |

7 |
An Extension of the Galois Theory
- Joyal, Tierney
- 1984
(Show Context)
Citation Context ... constructive and hence applies to the relative case, it follows that p∗(Ω P ↓) is locally compact in the category Loc(Sh(B ↓ )) of internal locales in Sh(B ↓ ). But, Loc(Sh(B ↓ )) � Loc/O(B ↓ ) (see =-=[JT]-=- or [J2]), and so using the results from [N4], it follows that p ↓ : P ↓ −→ B ↓ is exponentiable in Top. Thus, we are led to considering the effect of the functor ( ) ↓ : Pos −→ Top on exponentiabilit... |

5 |
sujet de l’existence d’adjoints à droite aux foncteurs “image réciproque” dans la catégorie des catégories
- Conduché, Au
- 1972
(Show Context)
Citation Context ...of posets is related to (but weaker than) the factorization-lifting property for exponentiability of morphisms in the category of small categories (considered independently by Giraud [G] and Conduché =-=[C]-=-). 1. Introduction Let A be a category with finite limits. Recall that an object X of A is called exponentiable if the functor −×X: A −→ A has a right adjoint (denoted by ( ) X ). The category A is ca... |

4 |
On topological quotients preserved by pullback or products
- Day, Kelly
- 1970
(Show Context)
Citation Context ... and only if it is locally compact. The characterization of exponentiable spaces was finally achieved by Day and Kelly in their 1970 paper “On topological quotients preserved by pullback or products” =-=[DK]-=-, where they proved that, for a space X, the functor −×X: Top −→ TopsTheory and Applications of Categories, Vol. 8, No. 2 18 preserves quotients if and only if the lattice O(X) ofopensetsofX is (what ... |

3 |
Cartesian inclusions: locales and toposes
- Niefield
- 1981
(Show Context)
Citation Context ...of T , then the inclusion X −→ T is exponentiable in Top if and only if X is locally closed (i.e., the intersection of an open and a closed subspace of T ). The latter was extended to Loc and GTop in =-=[N2]-=-. The results of [N3] were alsosTheory and Applications of Categories, Vol. 8, No. 2 19 used by the author in [N4] to show that for sober spaces X and T (in which points of T are locally closed), a co... |

3 |
Cartesian spaces over T and locales over Ω(T ), Cahiers Topologie Géom. Différentielle 23
- Niefield
- 1982
(Show Context)
Citation Context ...d sets are of the form {A} and {F }, whereF⊆Ais finite. Thus, we see that if p: X −→ T is exponentiable in Sob, then it is exponentiable in Top. With a further assumption on T , we can do even better =-=[N4]-=-. In particular, we need not assume that X is sober, only that ˜p: � X −→ T is exponentiable in Sob to get that p: X −→ T is exponentiable in Top,where� : Top −→ Sob is the reflection of Top in Sob, i... |

2 |
Email: niefiels@union.edu This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/n5/n5.{dvi,ps} THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate article
- Scott
- 1972
(Show Context)
Citation Context ... −→ TopsTheory and Applications of Categories, Vol. 8, No. 2 18 preserves quotients if and only if the lattice O(X) ofopensetsofX is (what is now known as) a continuous lattice, in the sense of Scott =-=[S]-=-, i.e., O(X) satisfies V = � {U|U <<V}, where U<<Vif every open cover of V has a finite subfamily that covers U. Since−×X preserves coproducts in any case, preservation of quotients (or equivalently, ... |

1 |
Local Compactness, lecture at University of Sussex Category Theory Conference
- Eilenberg
- 1982
(Show Context)
Citation Context ...(X) is continuous, and hence, precisely the sober spaces which are exponentiable in Top. Thus, the second definition seemed to be the appropriate one for non-Hausdorff spaces. Then in 1982, Eilenberg =-=[E]-=- proposed that continuity of O(X) be taken to be the definition of local compactness for a general space X. For this to make sense, he suggested that we need only rename the relation << by saying “U i... |

1 |
The algebraic theory of Lawson semilattices
- Hofmann, Stralka
- 1976
(Show Context)
Citation Context ...oc if and only if it is locally compact. Then Johnstone and Joyal generalized this result to Grothendieck toposes. They began with the definition of continuous posets based on the Hofmann and Stralka =-=[HS]-=- characterization of continuous lattices as those for which the map � :Idl(A) −→ A has a left adjoint. Using Grothendieck’s [Gr] notion of the ind-completion Ind(E) of a locally small category E, they... |

1 |
Open locales and exponentiation, Contemp
- Johnstone
(Show Context)
Citation Context ...(T )whose valueatanopensetG of T is the locale O(p −1 (G)), and for G ′ ⊆ G, the restriction map O(p −1 (G)) −→ O(p −1 (G ′ )) is given by V | G ′ = V ∩ p −1 (G ′ )(see[JT]or[J2]). Following [J2] and =-=[J4]-=-, local compactness in Sh(T ) can by described as follows. Recall that for an ideal I of p∗(ΩX) defined over an open set G of T ,wehave � I = � {U|U ∈ I(G ′ ), for some G ′ ⊆ G} Thus, to show that p∗(... |