University of Birmingham > Talks@bham > Lab Lunch > Fibrations and opfibrations of generalized point-free spaces

Fibrations and opfibrations of generalized point-free spaces

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If you have a question about this talk, please contact Dr Steve Vickers.

(Joint work with Sina Hazratpour)

“Point-free” means that the points are defined as models of a geometric theory, not as elements of a set, and “generalized” means the theory may be first-order. (The “ungeneralized” case is for propositional theories, for instance locales.)

There are two well-known topos-theoretic models of generalized spaces: the original Grothendieck toposes (which are relative to classical sets), and a relativized version GTop/S relative to a chosen elementary topos S, the base (which we assume to have nno). Then the generalized spaces are the bounded geometric morphisms from an elementary topos E to S.

Working generically, and using a Chevalley criterion, can often give simple proofs that certain classes of morphisms in GTop/S are fibrations or opfibrations in GTop/S. However, S is largely irrelevant, and Johnstone proved the stronger – and harder – result that they are (op)fibrations in ETop, arbitrary elementary toposes.

The joint work with Sina showed how to get the same results using the simple generic arguments with Chevalley in a third model of generalized spaces, the 2-category Con from my “Sketches for arithmetic universes”.

It was proved by Johnstone that bagtoposes, a topos analogue of powerlocales (or powerdomains) can be characterized as partial products. Some colimits of toposes (e.g. coproducts, lifting) can be then be constructed using bagtoposes. However, in a 2-category the universal characterization depends on the morphism ingredient of the partial product being a fibration or opfibration, and hence it becomes important to be able to analyse those properties.

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