Coherence in Three-Dimensional Category Theory

Coherence in Three-Dimensional Category Theory
Author: Nick Gurski
Publisher: Cambridge University Press
Total Pages: 287
Release: 2013-03-21
Genre: Mathematics
ISBN: 1107034892

Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.

Coherence in Three-Dimensional Category Theory

Coherence in Three-Dimensional Category Theory
Author: Nick Gurski
Publisher:
Total Pages: 288
Release: 2014-05-14
Genre: Tricategories
ISBN: 9781107336896

Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.

Category Theory in Context

Category Theory in Context
Author: Emily Riehl
Publisher: Courier Dover Publications
Total Pages: 273
Release: 2017-03-09
Genre: Mathematics
ISBN: 0486820807

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

2-Dimensional Categories

2-Dimensional Categories
Author: Niles Johnson
Publisher: Oxford University Press, USA
Total Pages: 636
Release: 2021-01-31
Genre: Mathematics
ISBN: 0198871376

2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory.

Coherence for Tricategories

Coherence for Tricategories
Author: Robert Gordon
Publisher: American Mathematical Soc.
Total Pages: 94
Release: 1995
Genre: Mathematics
ISBN: 0821803441

This work defines the concept of tricategory as the natural 3-dimensional generalization of bicategory. Trihomomorphism and triequivalence for tricategories are also defined so as to extend the concepts of homomorphism and biequivalence for bicategories.

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory
Author: Donald Yau
Publisher: American Mathematical Society
Total Pages: 555
Release: 2024-10-08
Genre: Mathematics
ISBN: 1470478099

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the general title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories?this book, Volume II: Braided Bimonoidal Categories with Applications, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book proves in detail Laplaza's two coherence theorems and May's strictification theorem of symmetric bimonoidal categories, as well as their bimonoidal analogues. This part includes detailed corrections to several inaccurate statements and proofs found in the literature. Part 2 proves Baez's Conjecture on the existence of a bi-initial object in a 2-category of symmetric bimonoidal categories. The next main theorem states that a matrix construction, involving the matrix product and the matrix tensor product, sends a symmetric bimonoidal category with invertible distributivity morphisms to a symmetric monoidal bicategory, with no strict structure morphisms in general.

Simplicial Methods for Higher Categories

Simplicial Methods for Higher Categories
Author: Simona Paoli
Publisher: Springer
Total Pages: 353
Release: 2019-06-03
Genre: Mathematics
ISBN: 3030056740

This monograph presents a new model of mathematical structures called weak n-categories. These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic. While strict n-categories are easily defined in terms associative and unital composition operations they are of limited use in applications, which often call for weakened variants of these laws. The author proposes a new approach to this weakening, whose generality arises not from a weakening of such laws but from the very geometric structure of its cells; a geometry dubbed weak globularity. The new model, called weakly globular n-fold categories, is one of the simplest known algebraic structures yielding a model of weak n-categories. The central result is the equivalence of this model to one of the existing models, due to Tamsamani and further studied by Simpson. This theory has intended applications to homotopy theory, mathematical physics and to long-standing open questions in category theory. As the theory is described in elementary terms and the book is largely self-contained, it is accessible to beginning graduate students and to mathematicians from a wide range of disciplines well beyond higher category theory. The new model makes a transparent connection between higher category theory and homotopy theory, rendering it particularly suitable for category theorists and algebraic topologists. Although the results are complex, readers are guided with an intuitive explanation before each concept is introduced, and with diagrams showing the interconnections between the main ideas and results.

Slenderness

Slenderness
Author: Radoslav Milan Dimitric
Publisher: Cambridge University Press
Total Pages: 330
Release: 2019
Genre: Mathematics
ISBN: 110847442X

A leading expert presents a unified concept of slenderness in Abelian categories, with numerous open problems and exercises.

Elements of ∞-Category Theory

Elements of ∞-Category Theory
Author: Emily Riehl
Publisher: Cambridge University Press
Total Pages: 782
Release: 2022-02-10
Genre: Mathematics
ISBN: 1108952194

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

Basic Category Theory

Basic Category Theory
Author: Tom Leinster
Publisher: Cambridge University Press
Total Pages: 193
Release: 2014-07-24
Genre: Mathematics
ISBN: 1107044243

A short introduction ideal for students learning category theory for the first time.