Description
In this lecture series, we will see ∞-categorical techniques in action: we will introduce 𝔼ₙ-algebras and factorization algebras in an ∞-category 𝒞 and discuss properties such as gluing and additivity statements. Time permitting, we will discuss applications to topological field theories. In the second half we discuss an application to representation theory: we discuss how one can use (lax) 𝔼ₙ-algebras in higher categories of spans to produce an 𝔼₂-structure on a higher category of representation-theoretic flavor.
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(Scheimbauer) 𝔼ₙ-algebras in an ∞-category 𝒞 are algebras for the operad 𝔼ₙ of little disks; equivalently, they are objects in 𝒞 equipped with n compatible associative multiplications. Examples include n-fold loop spaces, (braided) monoidal categories, and A∞-algebras. Alternatively, they are a special case of a (potentially pointless) factorization algebra, a structure inspired by the observables of a quantum field theory. In this lecture, we will introduce these notions, discuss examples and first properties.
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(Scheimbauer) In the second lecture, we will see how Kan extensions can be used in practice to prove things about factorization algebras. We will discuss gluing of factorization algebras and additivity of 𝔼ₙ-algebras and constructible factorization algebras. Time permitting, we will discuss applications to topological field theories; namely, the relevance for Morita (∞,n)-categories.
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(Stern) We continue our discussion of 𝔼ₙ-algebras in higher categories by defining a variant of the Boardman–Vogt tensor product which allows us to consider lax 𝔼₂-algebras in symmetric monoidal (∞,n)-categories. We will further discuss under which conditions lax 𝔼₂-algebras simplify to strong 𝔼₂-algebras.
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(Stern) A span—in algebraic geometry also called a correspondence—is a diagram of the form X ← F → Y in a higher category 𝒞. These can be viewed as the morphisms in a higher category Span(𝒞), where composition is given by pullback. We discuss (∞,n)-categories of spans, their universal properties, and how we can construct lax 𝔼₂-algebras in such categories. Time permitting, we show how 𝔼₂-monoidal categories (braided monoidal categories) in representation theory can be constructed from lax 𝔼₂-algebras in spans.
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Claudia Scheimbauer (TU Munich)27/07/2026, 14:00