Related papers: Concentration structures on categories and horizon…
Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these…
Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical…
Colimits are a fundamental construction in category theory. They provide a way to construct new objects by gluing together existing objects that are related in some way. We introduce a complementary notion of anticolimits, which provide a…
Structures based on polarities have been used to provide relational semantics for propositional logics that are modelled algebraically by non-distributive lattices with additional operators. This article develops a first order notion of…
Following Eilenberg-Steenrod axiomatic approach we construct the universal ordinary homology theory for any homological structure on a given category by representing ordinary theories with values in abelian categories. For a convenient…
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…
Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous…
This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical…
We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal…
Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…
This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories…
Based on the monoid classifier, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a…
We describe quantizations on monoidal categories of modules over finite groups. They are given by quantizers which are elements of a group algebra. Over the complex numbers we find these explicitly. For modules over S3 and A4 we are given…
We present a common framework to study varieties in great generality from a categorical point of view. The main application of this study is in the setting of algebraic categories, where we introduce Birkhoff varieties which are essentially…
By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In…
We develop a unified categorical framework for gauging both continuous and finite symmetries in arbitrary spacetime dimensions. Our construction applies to geometric categories i.e. categories internal to stacks. This generalizes the…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…