The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified, Treumann, based on unpublished ideas of MacPherson, constructed an ‘exit category’ (in the terminology of this paper, the ‘fundamental category’) which classifies constructible sheaves, equivalently stratified etale covers. This pap...

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic suucient criterion for the existence of a covariant action of the (universal covering of) the Poincar e group. In particular this gives, together with our previous results, an intrinsi...

In my last talk I presented a poset-theoretic perspective on the Elder rule, which associates a barcode (persistence diagram) to a persistent set. In this talk, I will describe a quiver and representation-theoretic perspective on persistence that emerges from a classification theorem for constructible cosheaves originally described by MacPherson and proved in various versions by Shepard, Treuma...

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincaré group. In particular this gives, together with our previous results, an intrins...

Background and Motivation 1 1. Sheaves and Stacks 3 1.1. Sheaves 3 1.2. Stacks 4 2. Sheaves on Stacks 5 2.1. The Site of a Stack 5 2.2. The site (Mell)ét 6 2.3. Homotopy Limits and Sheaves of Spectra 7 2.3.1. Limits and Homotopy Limits 7 2.3.2. Derived Limits and Corrected Homotopy Limits 9 2.3.3. Sheaves of Orthogonal Spectra, Symmetric Spectra, and S-Modules 13 3. The Descent Spectral Sequenc...

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic suucient criterion for the existence of a covariant action of the (universal covering of) the Poincar e group. In particular this gives, together with our previous results, an intrinsi...

The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functi...