Participants: Dr B. Bartlett , Prof D.B. Holgate , Dr Z. Janelidze , Dr P. Ouwehand, Prof 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:
Talks so far in the topos theory seminar
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.
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).
Sieves and subfunctors. Finishing the proof that our three main examples of topoi satisfy the axioms.
DECEMBER VACATION
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.