Papers
Status
Hilsum-Skandalis maps as Frobenius adjunctions with application to geometric morphisms
Preprint
When are enriched strong monads double exponential monads?
Bulletin of the Belgian Mathematical Society 23 (2016), 311–319
Stability of Properties of Locales Under Groups
Applied Categorical Structures. (2016)
Principal bundles as Frobenius adjunctions with application to geometric morphisms
Math. Proc. Camb. Phil. Soc. 159(03) (2015), 433-444
A localic proof of the localic groupoid representation of Grothendieck toposes
Proc. Amer. Math. Soc. 142 (2014), 859-866
Representing geometric morphisms using power locale monads
Appl. categ. Struct. 21(1), 15-47 (2013)
Aspects of slice stability in Locale Theory
Georgian Mathematical Journal. Vol. 19, Issue 2, 317–374 (2012)
A representation theorem for geometric morphisms
Appl. Categ. Struct. 18(6), 573–583 (2010)
An axiomatic account of weak triquotient assignments in locale theory
Journal of Pure and Applied Algebra,
Volume 214, Issue 6, June 2010, Pages 729-739
A categorical account of the localic closed subgroup theorem
Comment.Math.Univ.Carolin. 48,3 (2007) 541-553.
The patch construction is dual to algebraic DCPO representation
Appl. Categ. Struct. 19(1), 61–92 (2011)
A Categorical proof of the equivalence of Local
Compactness and Exponentiability in Locale Theory
Cahiers de Top. et Geom. Diff. Cat. Volume XLVII-3
A categorical account of the Hofmann-Mislove theorem
Math. Proc. Camb. Philos. Soc. 139, No.3, 441-455 (2005).
Axiomatic Characterization of the Category of Locales
PREPRINT. November 2003
Scott is Natural between Frames
Topology Proceedings 29 No. 2 (2005), pp. 613-640
An Axiomatic account of Weak Localic Triquotient Assignments - DRAFT
Early Draft
A Universal Characterization of the Double Powerlocale
(Joint with S. Vickers)
Theoretical Computer Science 316 (2004) 297-321
Presenting Locale Pullback via Directed Complete Partial Orders
Theoretical Computer Science 316 (2004) 225-258
On the Parallel between the Suplattice and Preframe
approaches to Locale Theory
Annals of Pure and Applied Logic, Volume 137,
Numbers 1-3 (2006) 391-412
Localic Priestley Duality
Journal of Pure and Applied Algebra 116 (1997) 323-335
Preframe Techniques in Constructive Locale Theory
PhD Thesis, Department of Computing, Imperial College, 1996.
Generalized Coverage Theorem
PREPRINT. February 1996
Hausdorff Systems
PREPRINT. January 1996

ABSTRACTS FROM PAPERS


"Hilsum-Skandalis maps as Frobenius adjunctions with application to geometric morphisms"
 


Preprint

Abstract

Hilsum-Skandalis maps, from differential geometry, are studied in the context of a cartesian category. It is shown that Hilsum-Skandalis maps can be represented as stably Frobenius adjunctions. This leads to a new and more general proof that Hilsum-Skandalis maps represent a universal way of inverting essential equivalences between internal groupoids. To prove the representation theorem, a new characterisation of the connected components adjunction of any internal groupoid is given. The characterisation is that the adjunction is covered by a stable Frobenius adjunction that is a slice and whose right adjoint is monadic. Geometric morphisms can be represented as stably Frobenius adjunctions. As applications of the study we show how it is easy to recover properties of geometric morphisms, seeing them as aspects of properties of stably Frobenius adjunctions.

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"When are enriched strong monads double exponential monads?"
 


Bulletin of BMS (2016)

Abstract

Some categorical conditions are given that are sufficient to show that an enriched monad with a strength is a double exponential monad.>
The conditions hold for the first double power locale monad (enriched over posets) and so as an application it is shown that >
the double power locale monad is a double exponential monad. A benefit is that this result about the double power locale monad >
can be established without the need for any detailed discussion of frame presentations or topos theory.

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"Stability of properties of locale theory under groups"
 


Applied Categorical Structures (2016)

Abstract

Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales,
we show that if C is a category that satisfies the axioms then so too is the category [G,C] of G-objects, for any internal group G.
To achieve this we prove a general categorical result: if an object S is double exponentiable
in a category with finite products then so is its associated trivial G-object (S, π 2:G × S → S).
The result holds even if S is not exponentiable. An example is given of a category C
that satisfies the axioms, but for which there is no elementary topos E such that C is the category of locales over E.
It is shown, in outline, how the results can be extended from groups to groupoids.

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"Principal bundles as Frobenius adjunctions with application to geometric morphisms"
 


Math. Proc. Camb. Phil. Soc. 159(03) (2015), 433-444

Abstract

Using a suitable notion of principal G-bundle, defined relative to an arbitrary cartesian category,
it is shown that principal bundles can be characterised as adjunctions that stably satisfy Frobenius reciprocity.
The result extends from internal groups to internal groupoids. Since geometric morphisms can be described
as certain adjunctions that are stably Frobenius, as an application it is proved that
all geometric morphisms, from a localic topos to a bounded topos, can be characterised as principal bundles.

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"A localic proof of the localic groupoid representation of Grothendieck toposes"
 


Proc. Amer. Math. Soc. 142 (2014), 859-866

Abstract

It is known that each Grothendieck topos is the category of G-equivariant sheaves for some
localic groupoid G. A simple proof of this is given which relies on the recently observed fact that
the pullback adjunction between locales induced by any geometric morphism satisfies
Frobenius reciprocity

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"Representing geometric morphisms using power locale monads"
 


Appl. categ. Struct. 21(1), 15-47 (2013)

Abstract

It is shown that geometric morphisms between elementary toposes can be represented
as adjunctions between the corresponding categories of locales. These adjunctions are
characterised as those that preserve the order enrichment, commute with the double
power locale monad and whose right adjoints preserve finite coproduct. They are also
characterised as those adjunctions that preserve the order enrichment and commute
with both the upper and the lower power locale monads.

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"Aspects of slice stability in Locale Theory"
 


Georgian Mathematical Journal. Vol. 19, Issue 2, 317–374 (2012)

Abstract

It is shown that a particular categorical axiomatisation of the category of locales is slice stable.
This localic slice stability can be used to recover the fundamental theorem of topos theory.
A categorical account of the ideal completion of a preorder is developed and is used to give
a new proof of Joyal and Tierney's result on the slice stability of locales.

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"A representation theorem for geometric morphisms"


Appl. Categ. Struct. 18(6), 573–583 (2010)

Abstract

It is shown that geometric morphisms between elementary toposes can be represented as certain
adjunctions between the corresponding categories of locales. These adjunctions are characterized by (i) they
preserve the order enrichment and the Sierpinski locale, and (ii) they satisfy Frobenius reciprocity.

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"‘An axiomatic account of weak triquotient assignments in locale theory"

Journal of Pure and Applied Algebra, Volume 214, Issue 6, June 2010, Pages 729-739

Abstract

‘In locale theory, weak triquotient assignments on a map f : X -> Y can be
represented as the points of the double power locale of f relative to the topos of
sheaves over Y. A categorical proof of this representation theorem is given
based on a categorical account of the Sierpinski locale.

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"A Categorical Account of the Localic Closed Subgroup Theorem"

Comment.Math.Univ.Carolin. 48,3 (2007) 541-553.

Abstract

Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem
is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has
an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.

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"The patch construction is dual to algebraic dcpo representation"

Appl. Categ. Struct. 19(1), 61–92 (2011)

Abstract

Using the parallel between the preframe and the suplattice approach to locale theory it is shown that the patch construction,
as an action on topologies, is the same thing as the process of recovering a discrete poset from its algebraic dcpo (ideal completion).

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"A Short Proof of the Equivalence of Local Compactness and Exponentiability in Locale Theory"

Cahiers de Top. et Geom. Diff. Cat. Volume XLVII-3

Abstract

A well known result in locale theory shows that a locale is locally compact if and only if it is
exponentiable. A recent result of Vickers and Townsend represents dcpo homomorphisms between the
opens of locales in terms of natural transformations. Here we use this representation theorem to give
a short proof that a locale is locally compact if and only if it is exponentiable.

MSC Classifications: Primary 06D22; Secondary 54B30, 54C35, 03G30, 06B23.

Keywords and Phrases: Locally compact, locales, exponentiability, continuous lattice, directed complete partial order.

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"A categorical account of the Hofmann-Mislove theorem"

Math. Proc. Camb. Philos. Soc. 139, No.3, 441-455 (2005).

Abstract

A categorical account is given of the Hofmann-Mislove theorem, describing the Scott open filters on a frame.
The account is stable under an order duality and so is shown to also cover Bunge and Funk's constructive description
of the points of the lower power locale. The categorical axioms offered are based on a representation theorem for dcpo
homomorphism between frames in terms of certain natural transformations; this allows for a categorical account to be
given of dcpo homomorphisms. This specializes to give a new categorical description of the upper and lower power
locale constructions. MSC Classifications: 06D22, 06D; 03G30; 54B20, 54B30; 03F55, 03F65; 18B30. Keywords:
locale; power locale; categorical logic; Scott open

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  "Axiomatic Characterization of the Category of Locales"

PREPRINT. November 2003

Abstract

An axiomatic characterization of the category of locales is given. The main idea is to introduce the ideal completion
locale as a categorical tensor with the terminal object. The Sierpinski locale is then the ideal completion of 2. The spectral
locales are axiomatised as abstract logarithms base Sierpinski (following a suggestion by Vickers). The axiomatisation
is not elementary since the category is required to be enriched over directed complete partial orders.

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"Scott is Natural between Frames"

Topology Proceedings 29 No. 2 (2005), pp. 613-640

Abstract

This paper is not presenting new material, but is rewriting the original work contained in "A Universal Characterization
of the Double Powerlocale". The main result is a representation theorem for the Scott continuous maps between frames in
terms of natural transformations between functors indexed by frames. The result specializes to frame homomorphisms,
thereby giving a representation theorem for all locale maps (i.e. continuous maps as defined in locale theory). The
purpose of the rewrite is to emphasis the lattice theoretic side of the work.

Keywords: frame, complete lattice, Scott continuity, locale, topos.

MSC 2000 codes 06D22; 54B30, 54C35, 03G30, 06B23.

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"An Axiomatic account of Weak Localic Triquotient Assignments - DRAFT "

Early Draft

Abstract

Localic triquotient maps were studied by Plewe as a natural generalization of open and proper localic surjections.
Using a weaker notion of triquotient assignment, introduced by Vickers, arbitrary proper and open locale maps can be
studied. This paper gives an axiomatic account of the theory of weak triquotient assignments, recovering what is known
localically: triquotient surjections are of effective descent, proper and open maps are pullback stable. Triquotient inclusions
are also characterized. The axiomatic account covers compact Hausdorff and discrete objects, and it is shown that
these form regular categories, a fact observed by Taylor using a different axiomatization.

MSC Classifications: 03G30; 06D, 54B30; 16B50, 03F55; 18B30

Keywords: triquotient map, locale, dcpo, proper, open, effective descent.

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"A Universal Characterization of the Double Powerlocale"

Joint paper with Dr Steven Vickers. Theoretical Computer Science 316 (2004) 297-321.

Abstract

The double powerlocale PP(X) (found by composing, in either order, the upper and lower powerlocale constructions PU
and PL) is shown to be isomorphic in [Locop,Set] to the double exponential $^($^X), where $ is the Sierpinski locale. Further
PU(X) and PL(X) are shown to be the subobjects of PP(X) comprising of, respectively, the meet semilattice and join semilattice
homomorphisms. A key lemma shows that, for any locales X and Y, natural transformations from $X (the presheaf Loc(_´ X,$))
to $Y (i.e. Loc(_´ Y,$)) are equivalent to dcpo morphisms from the frame WX to WY. It is also shown that $X has a localic
reflection in [Locop,Set] whose frame is dcpo(WX, W ). The reasoning is constructive in the sense of topos validity.

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"Presenting Locale Pullback via Directed Complete Partial Orders"

Theoretical Computer Science 316 (2004) 225-258

Description

This paper shows how to describe the pullbacks of directed complete poset (dcpo) presentations along geometric
morphisms. This extends Joyal and Tierney's original results on the pullbacks of suplattices. It is then shown how to
treat every frame as a dcpo and so locale pullback is described in this way. Applications are given describing triquotient
assignments in terms of internal dcpo maps, leading to pullback stability results for triquotient maps. The main application here
shows how dcpo maps between frames can be described in terms of certain external natural transformations.

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"On the Parallel between the Suplattice and Preframe approaches to Locale Theory"

Annals of Pure and Applied Logic, Volume 137,
Numbers 1-3 (2006) 391-412

Description

This paper uses the locale theory approach to topology. Two descriptions are given of all locale limits, the first
description using suplattice constructions and the second preframe constructions. The symmetries between these two
approaches to locale theory are explored. Given an informal assumption that open locale maps are parallel to proper maps
(an assumption hinted at by the underlying finitary symmetry of the lattice theory but not formally proved) we argue that various
pairs of locale theory results are “parallel”, that is, identical in structure but prove facts about proper maps on one side of the
pair and about open maps on the other. The pairs of results are: pullback stability of proper/open maps, regularity of the
category of compact Hausdorff/discrete locales, and theorems on information systems. Some remarks are included
on a possible formalization of this parallel as a duality.

MSC Classifications: 03G30; 06D, 54B30; 16B50, 03F55; 18B30

Keywords: locale, suplattice, preframe, proper map, open map, information system.

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"Localic Priestley Duality"

Journal of Pure and Applied Algebra 116 (1997) 323-335.

Description

Given the category OSoneSp of ordered Stone spaces (as introduced by Priestley, 1970) and the category CohSp of
coherent spaces (= spectral spaces) one can construct a pair of functors between them which Priestley (1970) has shown,
using the prime ideal theorem, define an equivalence. In this paper we define ordered Stone locales. These are classically just the
ordered Stone spaces. It is well known that the localic analogue of the coherent spaces is the category of coherent locales.
We prove, entirely constructively, that the category of coherent locales is equivalent to the category of ordered Stone locales.

.pdf version

Please
e-mail an address to me and a copy of the published article will be sent.

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"Preframe Techniques in Constructive Locale Theory"

PhD Thesis, Department of Computing, Imperial College, 1996. 158 pages.

Description

Our work is entirely constructive; none of the mathematics developed uses the excluded middle or any choice axioms.
No use is made of a natural numbers object. We get a glimpse of the parallel between the preframe approach and the SUP-lattice
approach to locale theory by developing the preframe coverage theorem and the SUP-lattice coverage theorem side by
side and as examples of a generalized coverage theorem.

Proper locale maps and open locale maps are defined and seen to be parallel. We argue that the compact regular locales
are parallel to the discrete locales. It is an examination of this parallel that is the driving force behind the thesis.

We proceed with examples: relational composition in Set can be expressed as a statement about discrete locales; we then
appeal to our parallel and examine relational composition of closed relations of compact regular locales. A technical achievement
of the thesis is the discove ry of a preframe formula for this relational composition.

We use this formula to investigate ordered compact regular locales (where the order is required to be closed). We find that
Banaschewski and Bruemmer's compact regular biframes (Stably continuous frames [Math. Proc. Camb. Phil. Soc. (1988) 104 7-19])
are equivalent to the compact regular posets with closed partial order. We also find that the ordered Stone locales are equivalent
to the coherent locales. This is a localic, and so constructive, version of Priestley's duality.

Further, using this relational composition, we can define the Hausdorff systems as the proper parallel to Vickers' continuous
information systems (Information systems for continuous posets [Theoretical Computer Science 114 (1993) 201-229]) The category
of continuous information systems is shown by Vickers to be equivalent to the (constructively) completely distributive lattices; we
prove the proper parallel to this result which is that the Hausdorff systems are equivalent to the stably locally compact locales.
This last result can be viewed as an extension of Priestley's duality.

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"Generalized Coverage Theorem"

PREPRINT. February 1996

Abstract

We express both Johnstone's original coverage theorem and its preframe version as facts about the creation of
coequalizers in the category of frames. The former constructs a coequalizer from a particular coequalizer in the underlying
cateogry of SUP-lattices, the latter does the same but uses the underlying category of preframes. We see that both these
coverage theorems have the same form applied to two different symmetric monoidal closed categories. We state and prove
a coverage result for any symmetric monoidal closed category and then look at examples. These cover all the well known
coverage results but yield somethings new in the case when the underlying symmetric monoidal closed category is that
of directed complete partial orders. The coverage theorem then tells us that preframe presentations present.

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  "Hausdorff Systems"

PREPRINT. January 1996

Abstract

The open/proper parallel is stated. Hausdorff systems (HausSys) are introduced as the proper parallel to
Vickers' continuous information systems (Infosys) [Vic93] and just as Infosys corresponds to the completely distributive
lattices we prove that there is a well known class of locales which correspond to the Hausdorff systems: they are the
stably locally compact locales. We proves this fact, essentially by manipulating Banaschewski and Brummers proof
that stably locally compact locales are just compact regular biframes [BB88].

Since ordered Stone locales are examples of Hausdorff systems and coherent locales are examples of stably
locally compact locales it is natural to ask whether the equivalence of HausSys and stably locally compact locales is an
extension of localic Priestley duality i.e. of the equivalence between ordered Stone locales and coherent locales [Tow97].
Some work needs to be done on defining the maps between Hausdorff systems before we can be sure of this.

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