![]() |
![]() |
Sketches for arithmetic universesAdd to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Uday Reddy. Grothendieck says toposes are generalized topological spaces. Then any geometric theory T can be understood as a space, the points being the models of T, and then the topos is its classifying topos S[T]. A map from T_1 to T_2 is a geometric morphism from S[T_1] to S[T_2], and the universal property of classifying toposes says this is equivalent to a model of T_2 in S[T_1]. This means that, in the context of a generic model of T_1, and using the geometric maths of colimits and finite limits, we construct a model of T_2: hence to describe a map we use the points very explicitly, despite this being a point-free approach, and continuity relies on the geometricity of the construction. In many ways that is a successful logical approach to topology, but it is hard to make a formal system for it because of the infinitary disjunctions and coproducts that are needed, for instance for real analysis. The available infinities are supplied externally as the objects of a chosen base topos S by which the whole programme is parameterized. Long ago I conjectured that Joyal’s Arithmetic Universes (AUs), pretoposes with parametrized list objects, might supply enough internal infinities (e.g. a natural numbers object) to be a finitary substitute, with no dependence on choice of base S, that is good enough in practice. In this talk I outline a 2-category Con that is intended to be the category of generalized spaces in these AU terms. I have a first draft of a detailed paper that I am now polishing up for submission. Each object T describes a geometric theory, and presents by universal algebra an AU AU that stands in for S[T]. Con embeds fully and faithfully in the 2-category of AUs and strict AU-functors.
This talk is part of the Lab Lunch series. This talk is included in these lists:Note that ex-directory lists are not shown. |
Other listsWhat's on in Physics? Centre for Systems Biology Coffee Mornings Pure DétoursOther talksKolmogorov-Smirnov type testing for structural breaks: A new adjusted-range based self-normalization approach Towards the next generation of hazardous weather prediction: observation uncertainty and data assimilation EIC (Title TBC) Harness light-matter interaction in low-dimensional materials and nanostructures: from advanced light manipulation to smart photonic devices The Holographic Universe TBA |