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Presentations for Topology

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

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  1. Presentations for Topology Dr Christopher Townsend (Open University)

  2. 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)

  3. 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

  4. 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.

  5. 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...

  6. 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

  7. 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.

  8. 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?)

  9. 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.

  10. Consequences of Double Power Result • Axiomatization of a category of spaces • Compact Open duality • Duality between compact Hausdorff and discrete. End

  11. What I didn’t tell you about • Coverage Theorem • Categorical interpretation of Coverage Theorem.

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