2-Dimensional Categories

In this chapter, further 2-dimensional categorical structures are presented and discussed. These include monoidal bicategories, as one-object tricategories, along with braided monoidal bicategories, sylleptic monoidal bicategories, and symmetric monoidal bicategories. The rest of the chapter discusses the Gray tensor product on 2-categories, Gray monoids, double categories, and monoidal double categories.

In the last decade quantum field theory and string theory have strongly impacted many areas of mathematics, especially the geometry and topology of low dimensional manifolds. In particular, a wealth of intriguing mathematical structures were discovered to be inherent to so called topological quantum field theories (TQFT’s) and conformal field theories (CFT’s). Originally, these notions refer to a class of concrete physical quantum field theories, among which three dimensional Chern- Simons theory and two dimensional rational conformal field theory are some of the most prominent ones. It was soon realized that the abstract setting of category theory makes it possible to efficiently organize the zoo of data and structures of these field theories. Eventually, TQFT’s evolved into purely mathematical notions, defined axiomatically in the language of categories and functors. Axiomatic TQFT’s and similar theories are, therefore, in nature rather similar to other functors in algebraic topology, such as homology. Atiyah was the first mathematician to cast the notion of TQFT’s into an axiomatic framework in his seminal work [Ati88]. Independently and at about the same time G. Segal [Seg88] formulates a mathematical definition of CFT’s, which very similarly based on categories and functors. The notion of extended TQFT’s that we will introduce here and on which our constructions will be based involves higher category theory, namely double categories and double functors. It thus contains both Atiyah’s notion of a TQFT in dimension three and Segal’s notion of CFT as special cases, though they appear on different categorical levels. The definition will not only be a natural and conceptual unification of previous theories, but further abstractions will allow us to construct new classes of TQFT’s, namely non-semisimple TQFT’s, that are manifestly different from other combinatorially defined ones and in some cases describe TQFT’s based on classical gauge theories.KeywordsHopf AlgebraWilson LineConformal Field TheoryMapping Class GroupAbelian CategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We extend some classical results – such as Quillen’s Theorem A, the Grothendieck construction, Thomason’s theorem and the characterisation of homotopically cofinal functors – from the homotopy theory of small categories to polynomial monads and their algebras. As an application we give a categorical proof of the Dwyer–Hess and Turchin results concerning the explicit double delooping of spaces of long knots.

The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories and indexed categories and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of fibrations into the context of tangent categories and proved that the fibres of a tangent fibration inherit a tangent structure from the total tangent category. The main goal of this paper is to provide a Grothendieck construction for tangent fibrations. Our first attempt will focus on providing a correspondence between tangent fibrations and indexed tangent categories, which are collections of tangent categories and tangent morphisms indexed by the objects and morphisms of a base tangent category. We will show that this construction inverts Cockett and Cruttwell’s result, but it does not provide a full equivalence between these two concepts. In order to understand how to define a genuine Grothendieck equivalence in the context of tangent categories, inspired by Street’s formal approach to monad theory we introduce a new concept: tangent objects. We show that tangent fibrations arise as tangent objects of a suitable $2$ -category and we employ this characterisation to lift the Grothendieck construction between fibrations and indexed categories to a genuine Grothendieck equivalence between tangent fibrations and tangent indexed categories.

In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of monoids. We provide an introduction to the generalised Grothendieck construction and apply it to recover classifications of certain classes of monoid extensions (including Schreier and weakly Schreier extensions in particular).

Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading

Rewriting in higher dimensional linear categories and application to the affine oriented Brauer category

This chapter provides an introduction to the interaction between category theory and mathematical logic. Category theory describes properties of mathematical structures via their transformations, or ‘morphisms’. On the other hand, mathematical logic provides languages for formalizing properties of structures directly in terms of their constituent parts—elements of sets, functions between sets, relations on sets, and so on. It might seem that the kind of properties that can be described purely in terms of morphisms and their composition would be quite limited. However, beginning with the attempt of Lawvere [1964; 1966; 1969; 1970] to reformulate the foundations of mathematics using the language of category theory, the development of categorical logic over the last three decades has shown that this is far from true. Indeed it turns out that many logical constructs can be characterized in terms of relatively few categorical ones, principal among which is the concept of adjoint functor. In this chapter we will see such categorical characterizations for, amongst other things, the notions of variable, substitution, prepositional connectives and quantifiers, equality, and various type-theoretic constructs. We assume that the reader is familiar with some of the basic notions of category theory, such as functor, natural transformation, (co)limit, and adjunction: see Poigné’s [1992] chapter on Basic Category Theory in Vol. I of this handbook, or any of the several available introductions to category theory slanted towards computer science, such as [Barr and Wells, 1990] and [Pierce, 1991]. There are three recurrent themes in the material we present. Categorical semantics. Many systems of logic can only be modelled in a sufficiently complete way by going beyond the usual set-based structures of classical model theory. Categorical logic introduces the idea of a structure valued in a category C, with the classical model-theoretic notion of structure [Chang and Keisler, 1973] appearing as the special case when C is the category of sets and functions. For a particular logical concept, one seeks to identify what properties (or extra structure) are needed in an arbitrary category to interpret the concept in a way that respects given logical axioms and rules. A well-known example is the interpretation of simply typed lambda calculus in cartesian closed categories.

Fibrations and Yoneda's lemma in an ∞-cosmos

Coherence theorems are fundamental to how we think about monoidal categories and their generalizations.In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction.This perspective makes the approach of Mac Lane's proof very amenable to generalization.We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors.Contents 1 Introduction 328 2 Diagrams of cliques 331 3 Presentations of categories 334 4 Coherence for bicategories 339 5 Symmetric monoidal categories 346 6 Shadowed bicategories 349 7 Coherence for functors of bicategories 351 8 Symmetric monoidal functors 362 9 Shadow functors 366

"Basic Category Theory for Computer Scientists," by Benjamin C. Pierce, is an excellent introduction of the branch of mathematics called category theory, and its application to computer science. As the title suggests, this book covers the basics of category theory, and presupposes no prior knowledge of it. The author motivates the development of category theory with examples from computer science, rather than the mathematical examples found in the standard mathematical treatises on category theory.

Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos which allows a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is essentially self-contained except that the authors presuppose a familiarity with basic category theory and topos theory.

Information search and retrieval interactions usually involve information content in the form of document collections, information retrieval systems and interfaces, and the user. To fully understand information search and retrieval interactions between users' cognitive space and the information space, researchers need to turn to cognitive models and theories. In this article, the authors use one of these theories, the basic level theory. Use of the basic level theory to understand human categorization is both appropriate and essential to user‐centered design of taxonomies, ontologies, browsing interfaces, and other indexing tools and systems. Analyses of data from two studies involving free sorting by 105 participants of 100 images were conducted. The types of categories formed and category labels were examined. Results of the analyses indicate that image category labels generally belong to superordinate to the basic level, and are generic and interpretive. Implications for research on theories of cognition and categorization, and design of image indexing, retrieval and browsing systems are discussed.

Formal Hopf algebra theory, II: Lax centres

Category theory of symbolic dynamics