Higher Segal spaces I

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen's S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups.Comment: 221 page

Some observations and results concerning submeasures on Boolean algebras

We investigate submeasures on Boolean algebras in the context of Maharam's problem and its solution. We generalise results that were originally proved for measures, to cases where additivity is not present. We investigate Talagrand's construction of a pathological exhaustive submeasure, attempting to give a more explicit description of this submeasure and we also consider some of its forcing properties. We consider the forcing consisting of submeasures that have as their domain a �nite subalgebra of the countable atomless Boolean algebra. We �nd and investigate a linear association between the real vector space of all real-valued functionals on the countable atomless Boolean algebra, which includes the collection of all submeasures, and the space of all signed �nitely additive measures on this Boolean algebra

Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications