Mathematics Colloquium

Le colloquium de mathématiques est une activité scientifique mensuelle de l'IRMP. Il consiste en un exposé d'une heure au cours duquel l'orateur présente un thème lié à la recherche en mathématiques. Le public cible est généraliste, il inclut l'ensemble des membres de l'IRMP: doctorants, post-docs et professeurs de tous les domaines de recherche. Les étudiants de master sont également les bienvenus.

Responsable du Colloquium : Prof. Pierre-Emmanuel Caprace

8 septembre 2022 à 16h15 
Prof. Emily Riehl (Johns Hopkins) : Path induction and the indiscernibility of identicals


Mathematics students learn a powerful technique for proving theorems about an arbitrary natural number: the principle of mathematical induction.

This talk introduces a closely related proof technique called "path induction," which can be thought of as an expression of Leibniz's "indiscernibility of identicals": if x and y are identified, then they must have the same properties, and conversely. 

What makes this interesting is that the notion of identification referenced here is given by Per Martin-Löf's intensional identity types, which encode a more flexible notion of sameness than the traditional equality predicate in that an identification can carry data, for instance of an explicit isomorphism or equivalence.

The nickname "path induction" for the elimination rule for identity types derives from a new homotopical interpretation of type theory, in which the terms of a type define the points of a space and identifications correspond to paths. In this homotopical context, indiscernibility of identicals is a consequence of the path lifting property of fibrations. Path induction is then justified by the fact that based path spaces are contractible.