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Toposes as 'bridges' for unifying Mathematics

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In the paper “The unification of Mathematics via Topos Theory” I introduced a new point of view on the concept of Grothendieck topos, namely the idea of a topos as a ‘bridge’ which can be effectively used for transferring information between distinct mathematical theories. The topos-theoretic techniques resulting from an implementation of this idea have already proved themselves to be very fruitful in Mathematics; indeed, they have generated a great number of non-trivial applications in distinct mathematical fields including Algebra, Topology, Algebraic Geometry, Model Theory and Proof Theory. This naturally stimulates a wider reflection on the reasons why toposes are so effective in allowing a transfer of knowledge between distinct fields. In the talk I will present substantial specific evidence for this, by identifying several crucial features of the concept of topos which are responsible for its technical effectiveness with respect to the goal of ‘unifying Mathematics’; I shall also put the technique ‘toposes as bridges’ in a broader perspective by extracting the real essence of the idea of ‘bridge’ and discussing other incarnations of the concept both in Mathematics and in different scientific fields. The analysis will be complemented by analogies with Astronomy, Linguistics and Genetics.

This talk is part of the Category Theory Seminar series.

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