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Monads and their applications SemesterWiSe 2019 / 20 Lecturer Daniel Schäppi Type of course (Veranstaltungsart) Vorlesung Contents Monads provide a categorical approach to universal algebra. They make it possible to describe objects endowed with additional structure, subject to various axioms, over any base category. A key feature of this situation is that we can freely add the additional structure and that we can forget it: any kind of structure which can be described by a monad comes with a “free-forgetful” adjunction. For example, the free group on a set or the free algebra on a vector space are of this form. Beck’s monadicity theorem gives a characterisation of such adjunctions. This has important applications in various areas of mathematics. For example, faithfully flat descent in algebraic geometry, the Tannakian recognition theorem, and the classification of covering spaces can all be proved using the monadicity theorem. The theory of monads has been particularly fruitful in the systematic study of categories endowed with additional structure (for example, various types of monoidal categories, or categories which have a certain class of limits or colimits). There are also generalisations of the monadicity theorem to the context of infinity categories. Jacob Lurie has used this approach to prove various descent theorems for infinity categories. In this course, we will develop the classical theory of monads and explain the monadicity theorem with a view towards applications. In the second half of the course, we will move up the “categorical ladder” and present two-dimensional monad theory. Time permitting, we will conclude with the generalisation of Beck’s monadicity theorem to the world of infinity categories. Recommended previous knowledge Familiarity with the language of categories, functors, and natural transformations Time/Date Wednesday, 12-14; Friday, 14-16 Location M 101 Registration
There will be a weekly exercise session, Tuesday 10-12, M 009 Modules MV, MArGeo, MGAGeo ECTS 9 |