What is a Higher Level Set?

Structuralist foundations of mathematics aim for an “invariant” conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. In this paper I argue in favor of the former over the latter. First, I explain why to pick between them we need to ask the question of what is the correct “categorified” version of a set. Second, I argue in favor of groupoids over categories as “categorified” sets by introducing a pre-formal understanding of groupoids as abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics

The effect of decerebrate rigidity on intracranial pressure in man and animals

Patients with decerebrate rigidity frequently also show intracranial hypertension. The factors responsible for this effect and their inter -relationships were explored in cats and in patients with head injuries.Animals: The factors examined, separately and in combination, were elevation of central venous, intrathoracic, intra- abdominal and systemic arterial pressures. The baselines thus established were used for the investigation of the effects of these factors on the intracranial pressure (ICP) in cats which had been rendered decerebrate by focal stereotactic mesencephalic lesions.Little or no change occurred in the ICP when: 1) Rigidity was mainly unilateral. 2) Bilateral limb rigidity was extreme.Persistent elevation of ICP occurred when 1) Truncal rigidity resulted in the simultaneous elevation of the intrathoracic and intra- abdominal pressures 2) Elevation of the systemic arterial pressure occurred in the presence of defective cerebrovascular homeostasis.Human: The dynamics and management of the complex clinical problem posed by decerebrate rigidity were investigated in patients with head injuries who exhibited well -developed bi- lateral rigidity under conditions of altered cerebral elastance.Rigidity was quantified by measuring the resonant frequency of the wrist induced by a printed- circuit motor. The brain elastance, ICP, intrathoracic and blood pressures were measured throughout the study. The effect of pharmacological muscle paralysis on the ICP and rigidity was examined.It appeared that well- developed decerebrate rigidity increased the ICP. The relationship was direct; the greater the rigidity or cerebral elastance, the greater the rise in ICP and vice versa. The two factors mainly responsible were muscle hypertonicity and cerebral elastance. The rises in ICP were caused by the rigidity and although it may not always be possible to reduce the abnormally increased elastance, the rigidity can certainly be abolished. As long as the cerebral vascular homeostatic mechanisms were intact, spontaneous waning of the rigidity or its abolition by muscle relaxants returned the ICP to its previous resting level. Pancuronium produced much deeper and more lasting relaxation than either diazepam or chlorpromazine.During the period of mechanical ventilation, alterations in ICP were of prognostic value as regards the outcome of the injuries

Foundations and Philosophy

The Univalent Foundations (UF) of mathematics take the point of view that spatial notions (e.g. “point” and “path”) are fundamental, rather than derived, and that all of mathematics can be encoded in terms of them. We will argue that this new point of view has important implications for philosophy, and especially for those parts of analytic philosophy that take set theory and first-order logic as their benchmark of rigor. To do so, we will explore the connection between foundations and philosophy, outline what is distinctive about the logic of UF, and then describe new philosophical theses one can express in terms of this new logic

Foundations and Philosophy

The Univalent Foundations (UF) of mathematics take the point of view that spatial notions (e.g. “point” and “path”) are fundamental, rather than derived, and that all of mathematics can be encoded in terms of them. We will argue that this new point of view has important implications for philosophy, and especially for those parts of analytic philosophy that take set theory and first-order logic as their benchmark of rigor. To do so, we will explore the connection between foundations and philosophy, outline what is distinctive about the logic of UF, and then describe new philosophical theses one can express in terms of this new logic

A Meaning Explanation for HoTT

The Univalent Foundations (UF) offer a new picture of the foundations of mathematics largely independent from set theory. In this paper I will focus on the question of whether Homotopy Type Theory (HoTT) (as a formalization of UF) can be justified intuitively as a theory of shapes in the same way that ZFC (as a formalization of set-theoretic foundations) can be justified intuitively as a theory of collections. I first clarify what I mean by an “intuitive justification” by distinguishing between formal and pre- formal “meaning explanations” in the vein of Martin-Löf. I then explain why Martin-Löf’s original meaning explanation for type theory no longer applies to HoTT. Finally, I outline a pre-formal meaning explanation for HoTT based on spatial notions like “shape”, “path”, “point” etc. which in particular provides an intuitive justification of the axiom of univalence. I conclude by discussing the limitations and prospects of such a project

What is a Higher Level Set?

Structuralist foundations of mathematics aim for an “invariant” conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. In this paper I argue in favor of the former over the latter. First, I explain why to pick between them we need to ask the question of what is the correct “categorified” version of a set. Second, I argue in favor of groupoids over categories as “categorified” sets by introducing a pre-formal understanding of groupoids as abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics

The Univalence Principle

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk's completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.Comment: A short version of this book is available as arXiv:2004.06572. v2: added references and some details on morphisms of premonoidal categorie