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Online seminář z algebry - 15.4.2021 PDF Tisk

Další seminář z algebry se koná 15.4.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Mark Kamsma

Independence Relations in Abstract Elementary Categories

In Shelah's classification of first-order theories we classify theories using combinatorial properties. The most well-known class is that of stable theories, which are very well-behaved. Simple theories are more general, and then even more general are the NSOP_1 theories. We can characterise those classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP_1-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Aktualizováno Úterý, 13 Duben 2021 08:37
Online seminář z algebry - 8.4.2021 PDF Tisk

Další seminář z algebry se koná 8.4.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Michael Ching

Tangent ∞-categories and Goodwillie calculus

In 1984 Rosický introduced tangent categories in order to capture axiomatically some properties of the tangent bundle functor on the category of smooth manifolds and smooth maps. Starting in 2014 Cockett and Cruttwell have developed this theory in more detail to emphasize connections with cartesian differential categories and other contexts arising from computer science and logic.

In this talk I will discuss joint work with Kristine Bauer and Matthew Burke which extends the notion of tangent category to ∞-categories. To make this generalization we use a characterization by Leung of tangent categories as modules over a symmetric monoidal category of Weil-algebras and algebra homomorphisms. Our main example of a tangent ∞-category is based on Lurie's model for the tangent bundle to an ∞-category itself. Thus we show that there is a tangent structure on the ∞-category of (differentiable) ∞-categories. This tangent structure encodes all the higher derivative information in Goodwillie's calculus of functors, and sets the scene for further applications of ideas from differential geometry to higher category theory.

Aktualizováno Pondělí, 05 Duben 2021 20:10