Related papers: Category Theory with Stratified Set Theory
New Foundations ($\mathrm{NF}$) is a set theory obtained from naive set theory by putting a stratification constraint on the comprehension schema; for example, it proves that there is a universal set $V$. $\mathrm{NFU}$ ($\mathrm{NF}$ with…
This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…
This paper is primarily concerned with assessing a set-theoretical system, $S^*$, for the foundations of category theory suggested by Solomon Feferman. $S^*$ is an extension of NFU, and may be seen as an attempt to accommodate unrestricted…
A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences…
We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our…
We introduce a notion of $\Theta$-categories, which is a refinement of the notion of symmetric monoidal $\infty$-categories. We use this notion to prove a Tannakian duality statement, relating $\Theta$-categories with fpqc-stacks by means…
We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an…
The study of abstraction and composition - the focus of category theory - naturally leads to sophisticated diagrams which can encode complex algebraic semantics. Consequently, these diagrams facilitate a clearer visual comprehension of…
Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…
In this paper, we first introduce a technique that we call "Yoneda representation of flat functors", based on ideas from indexed category theory; then we provide applications of this technique to the theory of classifying toposes.…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces…
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
A hierarchy of type universes is a rudimentary ingredient in the type theories of many proof assistants to prevent the logical inconsistency resulting from combining dependent functions and the type-in-type rule. In this work, we argue that…
Many definitions of weak and strict $\infty$-categories have been proposed. In this paper we present a definition for $\infty$-categories with strict associators, but which is otherwise fully weak. Our approach is based on the existing type…
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
We use the terms $\infty$-categories and $\infty$-functors to mean the objects and morphisms in an $\infty$-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects.…
In this paper, we present a proof of the consistency of the New Foundations set theory ($\mathit{NF}$). $\mathit{NF}$'s main idea is to permit very large sets (including the Universal Set) by restricting set formation to stratified…