Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) can be proved in ZFC if S is Σ1, while it follows from the existence of a proper class of supercompact cardinals if S is Σ2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is Σn+2 for n≥1. These cardinals form a new hierarchy, and we show that Vopěnka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence of our approach, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This follows from the fact that E∗-equivalence classes are Σ2, where E denotes a spectrum treated as a parameter. In contrast with this fact, E∗-equivalence classes are Σ1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC

Sovereigns, Trustees, Guardians: Private-Law Concepts and the Limits of Legitimate State Power

One major tradition of understanding the powers and duties of sovereigns has particular relevance to arguments for revival and refurbishment of the odious debt doctrine. Here, Purdy and Fielding survey the critical role of private-law concepts in the development of this tradition. In this account, the state is a constructed and purposive legal actor, composed of a set of powers assigned by its subjects for the pursuit of certain human interests and bound by the obligation to secure and respect those interests. Moreover, they narrate that if there are inherent powers in a sovereign, they are only those that are implied by its inherent duties

Foundational, compositional (co)datatypes for higher-order logic: category theory applied to theorem proving

Interactive theorem provers based on higher-order logic (HOL) traditionally follow the definitional approach, reducing high-level specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes. We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a bounded natural functor—an enriched type constructor satisfying specific properties preserved by interesting categorical operations. Our ideas are implemented as a definitional package in Isabelle, addressing a frequent request from users

Spartan Daily, October 29, 1976

Volume 67, Issue 41https://scholarworks.sjsu.edu/spartandaily/6126/thumbnail.jp