ABSTRACTS FROM TALKS

A category of 'spaces' that is not a category of locales

Topos ŕ l'IHES (23 au 27 novembre 2015)

Description

We describe a short list of categorical axioms that make a category behave like the category of locales. In summary the axioms assert that the category has an object that behaves like the Sierpinski space and this object is double exponentiable. A number of the usual results of locale theory can be derived using the axioms: the (weakly) closed subgroup theorem proved, closed and proper surjections are of effective descent, parallel theories of discrete and compact Hausdorff spaces emerge. An example is given of a category that satisfies the axioms but which is not the category of locales for any topos. We show how to embed the category of elementary toposes into the category whose objects are categories that satisfy the axioms.


.pdf version of notes


YouTube video of the talk



Geometric morphisms as structure preserving maps

Category Theory - CT2014 - Cambridge July 2014

Description

The `structure' which makes a cartesian category into a topos is the power set functor, which classifies relations. However the morphisms between toposes (geometric morphisms) are not structure preserving maps; they are instead adjunctions for which the left adjoint preserves finite limits. Relative to any topos there is its category of locales. Some attempts to provide a categorical axiomatization of locales hinge on assuming the existence of a double power locale functor, characterized categorically as a double exponential. The talk will provide a very brief overview of this categorical axiomatization of locales. The focus of the talk will be on describing how geometric morphisms can be represented as morphisms between categories of locales (so axiomatized). It turns out that geometric morphisms correspond to adjunctions that commute with the double power locale functor (and associated monad). In this way geometric morphisms can be re-interpreted as 'structure preserving maps'.

Reference

Townsend, C.F. Representing Geometric Morphisms Using Power Locale Monads. Appl.Categ. Struct. Volume 21, Issue 1, pp 15-47 (2013)

.pdf version of notes



‘’Towards a localic proof of the localic groupoid representation of Grothendieck toposes’’

Theoretical Computer Science Seminar, University of Birmingham, 2nd September 2011.

Description

Given that geometric morphisms can be represented as adjunctions between categories of locales, it feels natural to ask whether the representation of bounded geometric morphisms (i.e. Grothendieck toposes) via localic groupoids can be carried out using only locale theory. This could allow the representation theorem to be be proved without having to deploy some of the heavy machinery of topos theory (notably, sites and the pullback stability of various properties of geometric morphisms). Although I have not been able to complete such a proof I would like to present a strategy which will give some insight into how the localic view of geometric morphisms can be used in practice. The strategy relies on having a localic characterization of bounded geometric morphisms; although it is clear what this localic characterization should look like (for the theorem to work that is!) proving that the characterization is correct appears to be technically challenging.

The talk will provide background on Joyal and Tierney’s representation of Grothendieck toposes using localic groupoids and so will provide some understanding of how this important theorem works.

.pdf version of notes



‘’Categorical aspects of Locale Theory’’

School of Computer Science. University of Birmingham. (Part of ‘Theory Seminars’ series.) 16th November 2007

Description

The purpose of the talk is to try and describe aspects of locale theory that are available using only categorical reasoning.
Some basic 'facts about locales' will be recalled and then we will cover various results from locale theory using only these
'facts about locales' as categorical assumptions. The plan is to cover the results: pullback stability of (maps with) weak
triquotient assignments, the Hofmann-Mislove theorem, the closed subgroup theorem, the patch construction, Hyland's
result and a representation theorem for geometric morphisms. Only outlines will be given. It will be assumed that the audience
is familiar with what the category of locales is; for example knowing what the upper and lower power locale monads are.
It will also be assumed that the audience has some knowledge of Compact/Open duality in locale theory; i.e., broadly, that
the theory of proper maps is the same thing as the theory of open maps but with preframe homomorphisms in the place
of suplattice homomorphisms and with the order enrichment reversed. So, open subobject is dual to closed subobject
and, further, the notion of 'compact object' is dual to the notion of 'open object' under Compact/Open duality.

.pdf version from slides



“A categorical account of the localic closed subgroup theorem”

Algebraic and Topological Methods in Non-Classical Logics III. Mathematical Institute. University of Oxford. 5-9 August, 2007.

Description

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.

.pdf version from slides



"A Representation Theorem for Geometric Morphisms"

83rd Peripatetic Seminar on Sheaves and Logic, Glasgow, 6&7th May 2006

Description

We  show that geometric morphisms between elementary toposes can be represented as adjunctions
between the corresponding categories of locales.These adjunctions are characterized as those that preserve the
order enrichment and have right adjoints commuting with the power locale constructions. They can also be
characterized in terms of a modified Frobenius condition.

As application a new categorical account is given of the theory of geometric morphisms for which the pullback
stability of localic geometric morphisms and the hyperconnected-localic factorization results are recovered.

.pdf version from slides



"The Patch Construction is the same as Algebraic DCPO Representation"

Mathematical Institute, Oxford, 7th March 2005, Analytic Topology in Mathematics and Computer Science series

Description

By use of the parallel between the suplattice approach and the preframe approach to locale theory it can be shown that the
patch construction, as an action on topologies, is the same thing as the process of backing out a poset from its ideal completion.

.pdf version from slides


"Axiomatic Localic Relational Composition"

19th ‘Summer’ Conference on General Topology and its Applications. University of Cape Town, South Africa, July 5-9 2004.

Abstract

It is well known that in any regular category relational composition can be defined. This has application, for example,
in studying the category of compact Hausdorff spaces which is known to be regular by Manes’ theorem. The machinery
of relational composition is therefore automatically available for the study of, say, compact Hausdorff posets (i.e. certain
topological posets). However an assumption of regularity is not sufficient to prove the known representation theorems using
topological lattices (e.g. Priestley duality, continuous posets via informations systems and Banaschewski/Brummers’ representation
of stably locally compact locales). This talk offers a categorical account of topological relational composition that heads
towards an entirely categorical account of these representation theorems.

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"Effective descent via natural transformations"

80th Peripatetic Seminar on Sheaves and Logic (Cambridge, UK), 3-4 April, 2004.

Abstract

Localic triquotient surjections is a class of surjections in the category of locales that covers two familiar classes;
the open surjections and the proper surjections. They are probably the widest class of surjections known to be of
effective descent, as was shown by Plewe. In this talk we show that by representing dcpo homomorphisms between
frames as natural transformations, a categorical description is available of triquotient surjections. We are then able to
give an entirely categorical account of Plewe’s result that triquotient surjections are of effective descent.

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"Scott is Natural between Frames"

Special Session on 'The Many Lives of Lattice Theory II'. AMS/MAA Joint Meeting, Phoenix, Arizona. January 7-10, 2004.

Abstract

Reporting on joint work with S. Vickers. The talk will outline a representation theorem for the Scott continuous maps
between frames: they are equivalent to certain natural transformations. A frame is a poset with arbitrary joins (and
therefore meets) such that finite meets distribute over arbitrary joins; the motivating example being the opens of a
topological space. Scott continuous maps are maps that preserve directed joins. Their study is relevant to theoretical
computer science, but following work by Vermeulen (and Joyal and Tierney) their study can also be related to the topological
study of proper and open maps. The representation theorem that we outline shows how to discuss Scott continuity between
frames via a categorical construction (natural transformations) thus giving a categorical flavour to this infinitary notion.

Time permitting an application will be given in the category of locales (that is, the opposite of the category of frames).
The application is that whilst for any locale, X, the exponential $X only exists if X is locally compact, $^($X) always exists
as a locale provided we embed in the presheaf category [Locop,Set]. Here $ is the Sierpinski locale, i.e. the localic model
of the Sierpinski topological space.

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"Axioms for a category of spaces: Discrete and Compact Hausdorff spaces"

University of Leicester, Computer Science Internal Seminar, October 22, 2003.

Abstract

A detailed account was given of the regularity of the category of compact Hausdorff locales and of discrete
locales. Sufficient categorical axioms were given to show these results to be order dual. The theory of compact
Hausdorff spaces and the theory of discrete spaces (set theory) are therefore the same thing with respect
to any categorical proposition definable using regularity.

No slides.



"Axioms for a category of spaces"

79th Meeting, Peripatetic Seminar on Sheaves and Logic (Doorn, Netherlands), June 28-29 2003.

Abstract

An axiomatic proof of the pullback stability of maps with triquotient assignement was outlined.
It was shown how this specialised to known results about proper and open locale maps.

No slides.



"Axioms for a category of spaces (Birmingham)"

Given as part of the 'Topology Seminar' series at the Computer Science Dept, Birmingham University, 14th March 2003.

Abstract

A recent observation (with Vickers) states that directed join preserving (i.e. Scott continuous/dcpo) maps between frames
can be described as natural transformations in the functor category [Locop,Set]. This observation is taken as the basis for an
axiomatization of a category of spaces. The axioms essentially relate continuous maps between spaces to dcpo maps (that is, natural
transformations) in the same manner that locale maps are related to dcpo maps between frames. In this setting the theory of compact
Hausdorff spaces is order-dual to the theory of discrete spaces and so the theories of these two classes have equal status.

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"Axioms for a category of spaces"

Internal Open University seminar. Pure Mathematics Department, 4th March 2003.

Description

Aimed at a general mathematical audience this talk presents the background ideas from category theory needed to attempt
an axiomatization of the category of topological spaces. An axiom schema is outlined and some consequences discussed.
In this schema the theories of compact Hausdorff spaces and discrete spaces have equal status.

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"Locale Pullback via DCPOs"

The 78th meeting of the Peripatetic Seminar on Sheaves and Logic.
Held at the Institut de Recherche Mathématique Avancé de Strasbourg, 15th-16th February 2003.

Description

This talk shows how to give a description of locale pullback in terms of a construction in the category of
directed complete posets (dcpos). This is done by extending the change of base results of Joyal and Tierney from supremum
lattices to dcpos. An insight is that rather than viewing locales as rings internal to the category supremum lattices,
locales are viewed a sorder-internal distributive lattices in the category of dcpos.

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"Presentations for Topology"

Domains VI conference. Birmingham University. 16th-19th September 2002.

Description

Outlines the various different ways that a frame (and therefore a topological space) can be presented as
a lattice. The forgetful functors which exist between the different forms correspond to the upper and lower
power space constructions. That upper and lower constructions commute is therefore immediate.

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"The Relationship between Topology and Logic"

Open University Seminar, 11th June 2002

Description

This talk is aimed at beginners to category theory. It outlines how a category works as a mathematical universe, and gives the
flavour of the axioms of a topos. It ends with a description of the various types of logic which emerge including geometric logic.

.ppt version