Mathematics

LATTICE THEORY AND TOPOLOGY SEMINAR

Participants: Dr B. Bartlett Prof D.B. Holgate Dr Z. Janelidze Dr P. OuwehandProf I.M. Rewitzky , Dr G. Boxall, D.J. Basson, W.T.W. Cloete, J. Masuret, F.I Michael, A.D. Razafindrakoto.

OCTOBER 2010 – MAY 2011

TOPOS THEORY SEMINAR (coordinated by Bruce Bartlett)

As the lattice theory and topology group, we’ll be learning some topos theory and its applications using the following book as our main reference:

• `Maclane and Moerdijk, Sheaves in Geometry and Logic - A first introduction to topos theory. `

Other useful (and freely available online) materials include:

One clear goal of the seminar is to develop enough understanding of topos theory so as to understand the topos-theoretic formulation of Cohen’s proof of the independence of the continuum hypothesis. The rough plan of talks is as follows:

• Introduction to topos theory, basic ideas, definitions and examples (Bruce Bartlett, 3 talks). This covers Maclane and Moerdijk I 1-7.
• Heyting algebras (Jacques Masuret and Ingrid Rewitzky, 1 talk each). This covers Maclane and Moerdijk I 7-8 and also IV.8.
• Introduction to sheaves with topos theory in mind (Dirk Basson, 2 talks). This basically covers Maclane and Moerdijk II, just emphasizing the basic important material; section (a) from Ross Street’s notes and page 19 of Moerdijk and Van Oosten show what is important.
• The equivalence between Grothendieck topologies, Lawvere-Tierney topologies, and universal closure operators. (Pages 19-21 of Moerdijk and Van Oosten’s notes, culminating in Theorem 2.5, but only the first three equivalences). This is a condensed version of the material in Maclane and Moerdijk V 1-4.  (David Holgate and Ando Razafindrakato, 2 talks).
• Sheafification; and characterizing the category of sheaves as a full subcategory of the category of presheaves whose inclusion functor has a left adjoint preserving finite limits (Zurab Janelidze and Gareth Boxall, 2 talks).
• Topos-theoretic version of the independence of the continuum hypothesis. This covers Maclane and Moerdijk VI 1-3. (Peter Ouwehand, 2 talks).

Talks so far in the topos theory seminar

• Monday, 25th October 15h00. Introduction to topos theory I, Bruce Bartlett. Scanned notes.

The basic idea of a topos as a space of “variable sets”. Sheaves. Definition of elementary topos.  Yoneda lemma. Main examples of topoi:  category of sets, presheaves on a category, and sheaves on a topological space/site.

Homework: try to browse through the book and/or the notes of Street/Moerdijk and van Oosten and identify parts which interest you, so that we can chart a longer-term course of action next week.

• Monday, 1st November 15h00. Introduction to topos theory II, Bruce Bartlett.

More on the Yoneda lemma. The category of right G-sets as an illustration. The Yoneda lemma implies Cayley’s theorem! Review definition of limits and colimits in a category. Internal and external definitions. Definition of exponentials and subobject classifiers. Working out what exponentials are in the category of right G-sets (Maclane and Moerdijk exercise I. 5b). Checking that the three examples mentioned in the first lecture indeed satisfy the axioms of an elementary topos.

Homework: Maclane and Moerdijk, exercise I. 5 (a).

• Monday, 8th November 15h00. Introduction to topos theory III, Bruce Bartlett.

Sieves and subfunctors. Finishing the proof that our three main examples of topoi satisfy the axioms.

• Monday, 15th November, 15h00. Crash course in Heyting algebras, Ingrid Rewitzky.

DECEMBER VACATION

• Thursday, 3 March, 10h30. Heyting algebras II, Jacques Masuret. Scanned notes.

How the sub-presheaves of a given presheaf form a Heyting algebra.  Definition of the exponential, and the “not” operation. The topos of right G-sets as an example. Sketch of the general result in an arbitrary topos.