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Presentations for Topology. Dr Christopher Townsend (Open University). Common Language. We have the following assumptions for the talk: (a) What a presentation is. I.e. what A=Alg<G R> means

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Presentations for topology

Presentations for Topology

Dr Christopher Townsend

(Open University)


Common language
Common Language

  • We have the following assumptions for the talk:

  • (a) What a presentation is. I.e. what A=Alg<G R> means

  • (b) The category of frames (Objects: complete Heyting algebras. Morphisms: preserve all joins and finite meets)

  • (c) The category of Locales (Loc) = the opposite of the category of frames. Power locales.

  • (d ) Locales are important…

    • a good framework for topology

    • a first step towards generalised spaces (toposes)


Frame presentations
Frame Presentations

  • Objectives for the Talk

    • 1) Show that frame presentations come in different flavours

    • 2) Show that frame presentations commute with maps between toposes (frames do not)

    • 3) Show how the ‘flavours’ correspond to well known power locale constructions

  • Thereafter: Applications to describing Loc


Why frame presentations
Why Frame Presentations?

  • Usually, not interesting objects… but we have different flavours emerging

  • Each example is a ‘finitary’ (Geometric…) object

  • The relationship between flavours corresponds to power locale constructions

  • They are stable when moving from one topos to another… leading to arguments about universality of the double power locale.

  • IN SHORT: careful arguments about presentations correspond to important topological constructions.


Frame presentations 4 flavours
Frame Presentations: 4 Flavours

  • A presentation is a pair (G,R) where G is a set and R are relations… OR G is a lattice and R is a relation…

(DLat, )

‘Qua Dlat’

(/\SLat, +\/)

(\/SLat, +/\)

(Poset, +\/+/\)

Every Frame can be presented by any node...


Moving between presentations
Moving Between Presentations:

(DLat, )

P_L

P_U

Forget

(/\SLat, +\/)

(\/SLat, +/\)

P_U

P_L

(Poset, +\/+/\)

I.e. power locale is action of forgetting structure on presentation


Geometric stability
Geometric Stability

f*

f:EE’ a geometric morphism, f* the inverse image then:

(DLat, )

P_L

P_U

Forget

(/\SLat, +\/)

(\/SLat, +/\)

P_U

P_L

(Poset, +\/+/\)

... in E.


Pullback stability
Pullback Stability

  • f:E E’a geometric morphism; so any locale X (in E’) can be pulled back to a locale in E, denoted f*X :

  • (Crucial Fact): If X is presented by (G,R) then f*X is presented by f*(G,R) - presentations are pullback stable.

  • Not only the objects (locales), but the power constructions are pullback stable via the constructions on the corresponding presentations. (Known; widely?)


Consequences of stability
Consequences of Stability

In practice: f:E E’is just a continuous map between locales. I.e. f : SX  SY, the topos of sheaves over X,Y respectively.Since

LocSX=Loc/X (Joyal and Tierney)

we can argue in Set and then pullback to Loc/X to obtain more general results.

  • Broad geometric techniques for arguing about locales. (Locales have points again…)

  • Extend to Ideal Completion of a poset

  • Double Power Locale Result: PPX=$^($^X)

E.g.


Consequences of double power result
Consequences of Double Power Result

  • Axiomatization of a category of spaces

  • Compact Open duality

    • Duality between compact Hausdorff and discrete. End


What i didn t tell you about
What I didn’t tell you about

  • Coverage Theorem

  • Categorical interpretation of Coverage Theorem.


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