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The Theory of Topos-Theoretic 'Bridges' – A Conceptual Introduction (2016)

Mathematics is divided into several distinct areas: geometry, number theory, algebra, analysis, mathematical logic, etc. Each of these areas has evolved throughout the years by developing its own ideas and techniques, and by now has reached a remarkable degree of specialization.

More precisely, I imagined that the possibility of representing toposes in multiple ways could be exploited for building ‘bridges’ interconnecting different theories and allowing a transfer of information between them. Throughout the past years, I have developed a set of interdisciplinary methods and techniques for effectively using toposes as unifying ‘bridges,’ and, in doing this, I have uncovered a number of connections between different mathematical theories that were previously hidden and, in many cases, even unsuspected. Technical flexibility: Toposes are mathematical universes that are very rich in terms of internal structure; moreover, they have a very well-behaved representation theory, which makes them extremely effective computationally when considered as ‘bridges.’

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