Seminar on Higher Homotopical Structures
Tuesday 8th June 2021
Celebration in honour of Carles Casacuberta on the occasion of his 60th birthday (online Carlesfest)
Tuesday 8th June 2021
Celebration in honour of Carles Casacuberta on the occasion of his 60th birthday (online Carlesfest)
Speaker: Ieke Moerdijk
Title: What is a profinite (completion of an) operad? (Lecture on the occasion of Carles Casacuberta’s 60th birthday)
Abstract: The category of profinite spaces can be modelled by simplicial objects in the category of profinite sets, and supports a well-behaved homotopy theory and profinite completion functor from the homotopy theory of simplicial sets (Fabien Morel, Gereon Quick). This profinite completion does not preserve products, so it is not immediately clear whether there exists a workable theory of profinite completion of operads, although profinite operads have been shown to be relevant in the context of the Grothendieck–Teichmüller group (Horel, Boavida–Horel–Robertson). In this talk, we will sketch a general method for defining a good category of profinite infinity-operads, supporting a Quillen model structure, for which profinite completion can be interpreted as a left Quillen functor from (infinity or simplicial) operads. The talk is based on joint work with Thomas Blom.
Speaker: Ieke Moerdijk
Title: What is a profinite (completion of an) operad? (Lecture on the occasion of Carles Casacuberta’s 60th birthday)
Abstract: The category of profinite spaces can be modelled by simplicial objects in the category of profinite sets, and supports a well-behaved homotopy theory and profinite completion functor from the homotopy theory of simplicial sets (Fabien Morel, Gereon Quick). This profinite completion does not preserve products, so it is not immediately clear whether there exists a workable theory of profinite completion of operads, although profinite operads have been shown to be relevant in the context of the Grothendieck–Teichmüller group (Horel, Boavida–Horel–Robertson). In this talk, we will sketch a general method for defining a good category of profinite infinity-operads, supporting a Quillen model structure, for which profinite completion can be interpreted as a left Quillen functor from (infinity or simplicial) operads. The talk is based on joint work with Thomas Blom.
Speaker: Imma Gálvez Carrillo
Title: Objective combinatorial bialgebras through decomposition spaces
Speaker: Imma Gálvez Carrillo
Title: Objective combinatorial bialgebras through decomposition spaces
Abstract: We report on joint work in progress (with Joachim Kock and Andrew Tonks, and also with Iván Chercoles), envisaging to deepen the understanding of the relation between higher homotopy structures as encoded in decomposition spaces (also known as 2-Segal spaces), and bialgebra structures both in their combinatorial objective and purely algebraic and numerical realisations. Many of these bialgebras and Hopf algebras have a long history in mathematics, and their interest continues to grow, as they appear in new areas, providing new connections, calling for a more unified treatment. Among them, we will single out bialgebras related to symmetric functions and to trees and forests. Time permitting, we will examine some base changes, Galois connections, and duality constructions.
Abstract: We report on joint work in progress (with Joachim Kock and Andrew Tonks, and also with Iván Chercoles), envisaging to deepen the understanding of the relation between higher homotopy structures as encoded in decomposition spaces (also known as 2-Segal spaces), and bialgebra structures both in their combinatorial objective and purely algebraic and numerical realisations. Many of these bialgebras and Hopf algebras have a long history in mathematics, and their interest continues to grow, as they appear in new areas, providing new connections, calling for a more unified treatment. Among them, we will single out bialgebras related to symmetric functions and to trees and forests. Time permitting, we will examine some base changes, Galois connections, and duality constructions.
Speaker: Jiří Rosický
Title: Combinatorial and accessible model categories
Abstract: We will relate cofibrantly generated and accessible weak factorization systems on locally presentable categories. In particular, we will demonstrate the importance of full images of accessible functors. At the end, we will deal with combinatorial and accessible model categories.
Abstract: We will relate cofibrantly generated and accessible weak factorization systems on locally presentable categories. In particular, we will demonstrate the importance of full images of accessible functors. At the end, we will deal with combinatorial and accessible model categories.
Speaker: Federico Cantero
Title: Steenrod squares in Khovanov homology
Speaker: Federico Cantero
Title: Steenrod squares in Khovanov homology
Abstract: Seven years ago, Lipshitz and Sarkar refined Khovanov homology to an invariant of links valued in spectra. From the computational approach, this improvement immediately led to new invariants of links coming from the action of the Steenrod algebra on the Khovanov homology of each link. Nonetheless, the chain complex of Khovanov homology is not the chain complex of any of the classical models of spectra, and therefore the classical formulas by Steenrod did not apply in that context. Lipshitz and Sarkar found a formula for the second Steenrod square, and found that the action of the Steenrod algebra on Khovanov homology is a strictly finer invariant than Khovanov homology itself. In this talk we will present recent formulas for the higher Steenrod squares.
Abstract: Seven years ago, Lipshitz and Sarkar refined Khovanov homology to an invariant of links valued in spectra. From the computational approach, this improvement immediately led to new invariants of links coming from the action of the Steenrod algebra on the Khovanov homology of each link. Nonetheless, the chain complex of Khovanov homology is not the chain complex of any of the classical models of spectra, and therefore the classical formulas by Steenrod did not apply in that context. Lipshitz and Sarkar found a formula for the second Steenrod square, and found that the action of the Steenrod algebra on Khovanov homology is a strictly finer invariant than Khovanov homology itself. In this talk we will present recent formulas for the higher Steenrod squares.
Seminar Coordinators
Seminar Coordinators
Joana Cirici (Universitat de Barcelona)
Joachim Kock (Universitat Autònoma de Barcelona and CRM)
Bruno Vallette (Université Sorbonne Paris Nord)