Description
In this lecture series, we want to build basic language for the use of (∞,1)-categories and also (∞,n)-categories, both in other talks and in mathematical 'real life'. In a model, a lot of this was done by Joyal, Lurie, Riehl and Verity, whereas a model-independent approach is currently being developed by Cisinski, Cnossen, Nguyen and Walde.
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Introduction to (∞,1)-categories. Why are ordinary categories not enough? What is actually the difference between (∞,1)-categories and just categories? What things go through essentially the same and which require more work? We will start addressing these questions in the first talk.
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Examples. How to construct examples of (∞,1)-categories? We will devote this talk to several ways of doing so. To relate to previously done 'homotopy-coherent mathematics', we discuss a presentation of (∞,1)-categories given by model categories; this allows for further examples.
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Introduction to (∞,n)-categories. After building a basic understanding of (∞,1)-categories, we want to turn our attention to changing the parameter 1 to an arbitrary natural number n. In this talk, we will discuss the axiomatization by Barwick–Schommer-Pries and some examples. If time permits, we will also look into models of (∞,n)-categories.
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Complicial sets as a model of (∞,n)-categories. After basics of (∞,n)-categories, I will speak about a particular implementation referred to as 'complicial sets' based on work of Verity, Riehl, Rovelli and myself, and further insights by Loubaton. I will discuss advantages and drawbacks of this model.
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Viktoriya Ozornova (MPIM Bonn)27/07/2026, 11:00