University of Birmingham > Talks@bham > Theoretical computer science seminar > Towards a localic proof of the localic groupoid representation of Grothendieck toposes

Towards a localic proof of the localic groupoid representation of Grothendieck toposes

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If you have a question about this talk, please contact Paul Levy.

Given that geometric morphisms can be represented as adjunctions between categories of locales, it feels natural to ask whether the representation of bounded geometric morphisms (i.e. Grothendieck toposes) via localic groupoids can be carried out using only locale theory. This could allow the representation theorem to be be proved without having to deploy some of the heavy machinery of topos theory (notably, sites and the pullback stability of various properties of geometric morphisms). Although I have not been able to complete such a proof I would like to present a strategy which will give some insight into how the localic view of geometric morphisms can be used in practice. The strategy relies on having a localic characterization of bounded geometric morphisms; although it is clear what this localic characterization should look like (for the theorem to work that is!) proving that the characterization is correct appears to be technically challenging.

The talk will provide background on Joyal and Tierney’s representation of Grothendieck toposes using localic groupoids and so will provide some understanding of how this important theorem works.

This talk is part of the Theoretical computer science seminar series.

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