11 research outputs found
Sheaf Theory through Examples
An approachable introduction to elementary sheaf theory and its applications beyond pure math. Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more. With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms. Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas
\'Etale structures and the Joyal-Tierney representation theorem in countable model theory
An \'etale structure over a topological space is a continuous family of
structures (in some first-order language) indexed over . We give an
exposition of this fundamental concept from sheaf theory and its relevance to
countable model theory and invariant descriptive set theory. We show that many
classical aspects of spaces of countable models can be naturally framed and
generalized in the context of \'etale structures, including the Lopez-Escobar
theorem on invariant Borel sets, an omitting types theorem, and various
characterizations of Scott rank. We also present and prove the countable
version of the Joyal-Tierney representation theorem, which states that the
isomorphism groupoid of an \'etale structure determines its theory up to
bi-interpretability; and we explain how special cases of this theorem recover
several recent results in the literature on groupoids of models and functors
between them.Comment: 41 page
Toposes of monoid actions
openWe study toposes of actions of monoids on sets. We begin with ordinary actions, producing a class of presheaf toposes which we characterize. As groundwork for considering topological monoids, we branch out into a study of supercompactly generated toposes (a class strictly larger than presheaf toposes). This enables us to efficiently study and characterize toposes of continuous actions of topological monoids on sets, where the latter are viewed as discrete spaces. Finally, we refine this characterization into necessary and sufficient conditions for a supercompactly generated topos to be equivalent to a topos of this form.openInformatica e matematica del calcoloRogers, Morga