Speaker
Description
Introduced by May and Boardman–Vogt to study iterated loop spaces, operads are combinatorial gadgets governing homotopy coherent algebraic structures on spaces —and in any other symmetric monoidal category. In homotopy theory, where spaces and objects are considered up to weak equivalence, algebraic structures must be encoded in a homotopy-invariant way. This is achieved using the more flexible formalism of “∞-operads”, which can be thought of as operads up to homotopy.
In the first part of this talk, I will introduce the theory of ∞-operads and their algebras, focusing on how the models of Lurie and Moerdijk–Weiss relate to partition posets and trees. I will then turn to operadic localization, the process of freely inverting a class of morphisms in an operad. This construction has proved to be a fundamental tool in the study of, for instance, factorization algebras and quantum field theory. Nevertheless, the general theory of operadic localization is still not fully understood, and many natural questions remain open. In the second part of the talk, I will present some new results in this direction, along with applications to the categories of algebras.