The General Universal Property of the Propositional Truncation

In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B

A Prehistory of n-Categorical Physics

This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open-closed topological string theory, the categorification of quantum groups, Khovanov homology, and Lurie's work on the classification of topological quantum field theories.Comment: 129 pages, 8 eps figure

Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories

This dissertation is an exposition of the work conducted by the author in the later years of graduate school, when two main projects were completed. Both projects concern the application of sheaf-theoretic techniques to construct geometric structures on categories appearing in the mathematical description of the A-model, which are of interest to symplectic geometers and mathematicians working in mirror symmetry. This dissertation starts with an introduction to the aspects of the physics of mirror symmetry that will be needed for the exposition of the techniques and results of these two projects. The first project concerns the construction of Calabi-Yau structures on topological Fukaya categories, using the microlocal model of Nadler and others for these categories. The second project introduces and studies a similar local-to-global technique, this time used to construct Bridgeland stability conditions on Fukaya categories of marked surfaces, extending some results of Haiden, Katzarkov and Kontsevich on the relation between stability of Fukaya categories and geometry of holomorphic differentials

A Panorama Of Physical Mathematics c. 2022

What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1]

Chapter 1. Preface In1963,JohnMilnorputforwardalistofproblemsingeometrictopology.

homeomorphic to S 5? 2. Is simple homotopy type a topological invariant? 3. Can rational Pontrjagin classes be defined as topological invariants? 4. (Hauptvermutung) If two PL manifolds are homeomorphic, does it follow that they are PL homeomorphic? 5. Can topological manifolds be triangulated? 6. The Poincaré hypothesis in dimensions 3, 4. 7. (The annulus conjecture) Is the region bounded by two locally flat n-spheres in (n+1)-space necessarily homeomorphic to S n ×[0,1]? These were presented at the 1963 conference on differential and algebraic topology in Seattle, Washington. A much larger problem set from the conference is published in Ann. Math.81 (1965) pp. 565–591. In the last 30 years, much progress has been made on these problems. Problems 1, 2, 3, and 7 were solved affirmatively by Edwards-Cannon, Chapman, Novikov, and Kirby in the late 1960’s and early 1970’s, while problems 4 and 5 were solved negatively by Kirby-Siebenmann in 1967. Freedman solved the 4-dimensional (TOP) Poincaré Conjecture i