Presentations for Topology

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# Presentations for Topology - PowerPoint PPT Presentation

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

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 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
• 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?
• 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
• 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:

(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

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
• 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

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
• Axiomatization of a category of spaces
• Compact Open duality
• Duality between compact Hausdorff and discrete. End
What I didn’t tell you about
• Coverage Theorem
• Categorical interpretation of Coverage Theorem.