Related papers: Yoneda representations of flat functors and classi…

We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal…

This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent…

The tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces…

We study (vertically) normal lax double functors valued in the weak double category $\mathbb{C}\mathrm{at}$ of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for…

We present a study on the Yoneda-Dress construction of biset functors of linear representations over a field of characteristic zero. We give a characterization of their lattices of ideals and we provide a criterion of vanishing for their…

We construct classifying $\infty$-topoi by showing that the $(\infty,2)$-category of topoi has weighted limits. We show that several prestacks of interest have a classifying topos, including the prestack of spectra.

We regard a geometric theory classified by a topos as a syntactic presentation for the topos and develop tools for finding such presentations. Extensions of geometric theories, which can add axioms, symbols and sorts, are treated as objects…

We use quotients of span categories to introduce the language of a topos. We also study the logical relations and the quotients of span categories derived from them. As an application we show that the category of Boolean toposes is a…

We 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…

The basic notion of how topoi can be utilized in physics is presented here. Topos and category theory serve as valuable tools which extend our ordinary set-theoretical conceptions, can further the study of quantum logic and give rise to new…

As the prototypical category, $\mathbf{Set}$ has many properties which make it special amongst categories. From the point of view of mathematical logic, one such property is that $\mathbf{Set}$ has enough structure to "properly" formalise…

We show that, for a right exact functor from an abelian category to abelian groups, Yoneda's isomorphism commutes with homology and, hence, with functor derivation. Then we extend this result to semiabelian domains. An interpretation in…

We define the notion of sheaf in the context of doctrines. We prove the associate sheaf functor theorem. We show that grothendieck toposes and toposes obtained by the tripos to topos construction are instances of categories of sheaves for a…

This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…

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 homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many aspects of homotopy type theory, which also includes…

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…

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…

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models…

We explain the use of category theory in describing certain sorts of anyons. Yoneda's lemma leads to a simplification of that description. For the particular case of Fibonacci anyons, we also exhibit some calculations that seem to be known…