1 / 11

# 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Presentations for Topology' - marty

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Presentations for Topology

Dr Christopher Townsend

(Open University)

• 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)

• 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

• 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.

• 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...

(DLat, )

P_L

P_U

Forget

(/\SLat, +\/)

(\/SLat, +/\)

P_U

P_L

(Poset, +\/+/\)

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

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.

• 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?)

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.

• Axiomatization of a category of spaces

• Compact Open duality

• Duality between compact Hausdorff and discrete. End

• Coverage Theorem

• Categorical interpretation of Coverage Theorem.