Basic Concepts of Enriched Category Theory
Author: Gregory Maxwell Kelly
Publisher: CUP Archive
Published: 1982-02-18
Total Pages: 260
ISBN-13: 9780521287029
DOWNLOAD EBOOK →Lanterna Rally
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Categorical Homotopy Theory
Author: Emily Riehl
Publisher: Cambridge University Press
Published: 2014-05-26
Total Pages: 371
ISBN-13: 1139952633
DOWNLOAD EBOOK →This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.
Basic Category Theory
Author: Tom Leinster
Publisher: Cambridge University Press
Published: 2014-07-24
Total Pages: 193
ISBN-13: 1107044243
DOWNLOAD EBOOK →A short introduction ideal for students learning category theory for the first time.
Kan Extensions in Enriched Category Theory
Author: Eduardo J. Dubuc
Publisher: Springer
Published: 2006-11-15
Total Pages: 190
ISBN-13: 3540363076
DOWNLOAD EBOOK →The original purpose of this paper was to provide suitable enriched completions of small enriched categories.
Elements of ∞-Category Theory
Author: Emily Riehl
Publisher: Cambridge University Press
Published: 2022-02-10
Total Pages: 782
ISBN-13: 1108952194
DOWNLOAD EBOOK →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.
Category Theory in Context
Author: Emily Riehl
Publisher: Courier Dover Publications
Published: 2017-03-09
Total Pages: 272
ISBN-13: 0486820807
DOWNLOAD EBOOK →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.
From Categories to Homotopy Theory
Author: Birgit Richter
Publisher: Cambridge University Press
Published: 2020-04-16
Total Pages: 402
ISBN-13: 1108847625
DOWNLOAD EBOOK →Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.
An Invitation to Applied Category Theory
Author: Brendan Fong
Publisher: Cambridge University Press
Published: 2019-07-18
Total Pages: 351
ISBN-13: 1108482295
DOWNLOAD EBOOK →Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
(Co)end Calculus
Author: Fosco Loregian
Publisher: Cambridge University Press
Published: 2021-07-22
Total Pages: 331
ISBN-13: 1108746128
DOWNLOAD EBOOK →This easy-to-cite handbook gives the first systematic treatment of the (co)end calculus in category theory and its applications.
Category Theory for Programmers (New Edition, Hardcover)
Author: Bartosz Milewski
Publisher:
Published: 2019-08-24
Total Pages:
ISBN-13: 9780464243878
DOWNLOAD EBOOK →Category Theory is one of the most abstract branches of mathematics. It is usually taught to graduate students after they have mastered several other branches of mathematics, like algebra, topology, and group theory. It might, therefore, come as a shock that the basic concepts of category theory can be explained in relatively simple terms to anybody with some experience in programming.That's because, just like programming, category theory is about structure. Mathematicians discover structure in mathematical theories, programmers discover structure in computer programs. Well-structured programs are easier to understand and maintain and are less likely to contain bugs. Category theory provides the language to talk about structure and learning it will make you a better programmer.
