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Presentations for TopologyPowerPoint Presentation

Presentations for Topology

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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

- 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:EE’ 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.

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