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, ) (/\SLat, +\/) (\/SLat, +/\) (Poset, +\/+/\) Every Frame can be presented by any node... ‘Qua Dlat’

Moving Between Presentations: (DLat, ) (/\SLat, +\/) (Poset, +\/+/\) (\/SLat, +/\) Forget P_L P_U P_UP_L I.e. power locale is action of forgetting structure on presentation

Geometric Stability (DLat, ) (/\SLat, +\/) (Poset, +\/+/\) (\/SLat, +/\) Forget P_L P_U P_UP_L f:EE’ f:E  E’ a geometric morphism, f* the inverse image then: f* E... in E.

Pullback Stability f:E E’ E’Ef: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 Broad geometric techniques for arguing about locales. (Locales have points again…) Extend to Ideal Completion of a poset Double Power Locale Result: PPX=$^($^X) f:E E’ 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 Loc SX = Loc/X(Joyal and Tierney) we can argue in Set and then pullback to Loc/X to obtain more general results. 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.