FYI ________________________________ Da: Tamas Szamuely <tamas.szamuely@unipi.it> Inviato: domenica, gennaio 11, 2026 5:05 PM A: saga@lists.dm.unipi.it <saga@lists.dm.unipi.it> Oggetto: [Saga] Seminario di geometria algebrica e aritmetica, 15/01 Cari tutti, il primo seminario di 2026 si terr�� gioved�� 15/01 in Aula Magna del dipartimento. Programma: -------------- 15:00--16:00 Francesca Pratali (Utrecht University): Localization of ��-operads Operads are combinatorial gadgets encoding categories of algebras. Introduced by May and Boardman�CVogt in the 1970s to study iterated loop spaces, they have since found applications in algebraic topology, mathematical physics, and algebraic geometry. In modern homotopy theory, categories are studied up to localization at weak equivalences (e.g. spaces up to weak homotopy equivalence, chain complexes up to quasi-isomorphism). Operads are replaced by homotopy-coherent versions: these are Lurie��s ��-operads, and Cisinski-Moerdijk-Weiss dendroidal ��-operads. In homotopical settings, ��-operads often arise via localization, the process of freely inverting a class of morphisms, as in the cases of little disks operads and factorization homology. In the first part of the talk, I will review operads and their algebras, and explain how they provide a unified framework for the homotopy theory of diverse algebraic structures. In a second part, I will focus on operadic localization. The main result I will show is that every ��-operad is weakly equivalent to the localization of a discrete operad, generalizing Joyal��s delocalization theorem for ��-categories. If time permits, I will briefly sketch the combinatorial proof using dendroidal methods and highlight some open questions. Molti saluti, Tam��s Szamuely