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# Starting Category Theory

eBook | 8 April 2024

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One of the central highlights of this work is the exploration of the Yoneda lemma and its profound implications, during which intuitive explanations are provided, as well as detailed proofs, and specific examples. This book covers aspects of category theory often considered advanced in a clear and intuitive way, with rigorous mathematical proofs. It investigates universal properties, coherence, the relationship between categories and graphs, and treats monads and comonads on an equal footing, providing theorems, interpretations and concrete examples. Finally, this text contains an introduction to monoidal categories and to strong and commutative monads, which are essential tools in current research but seldom found in other textbooks.

Starting Category Theory serves as an accessible and comprehensive introduction to the fundamental concepts of category theory. Originally crafted as lecture notes for an undergraduate course, it has been developed to be equally well-suited for individuals pursuing self-study. Most crucially, it deliberately caters to those who are new to category theory, not requiring readers to have a background in pure mathematics, but only a basic understanding of linear algebra.

Contents:

• Preface

• Acknowledgments

• Basic Concepts:

• Categories
• Mono and Epi
• Functors
• Natural Transformations
• Studying Categories by Means of Functors
• The Yoneda Lemma:

• Representable Functors and the Yoneda Embedding Theorem
• Statement and Proof of the Yoneda Lemma
• Universal Properties
• Limits and Colimits:

• General Definitions
• Particular Limits and Colimits
• Functors, Limits and Colimits
• Limits and Colimits of Sets

• General Definitions
• Unit and Counit
• The Adjoint Functor Theorem for Preorders

• Monads as Extensions of Spaces
• Monads as Theories of Operations
• Comonads as Processes on Spaces
• Monoidal Categories:

• General Definitions
• Monoids and Comonoids
• Monoidal Functors
• Closed Monoidal Categories
• Conclusion

• Bibliography

• Index

Readership: This book is primarily targeted towards undergraduate and graduate students in mathematics and related fields (physics, computer science, statistics, engineering), and is suitable for either course adoption for category theory and discrete mathematics, or for self-study. More broadly, this book can appeal to researchers in related fields and professionals working in technology (machine learning, etc.).
Key Features:

• This book combines intuitive explanations and motivation for the abstract formalism with detailed and rigorous mathematical proofs
• It covers several crucial aspects of category theory, often considered difficult and advanced while providing lots of intuition
• Rare in that it cover much material whilst still providing intuitive interpretations for each concept, some of which are impossible to find outside of research papers or advanced manuals such as comonads, strong and commutative monads, closed monoidal categories and monads on them, categories of graphs and their products
• This book seems to be the only one to date that talks about category theory in full detail (for example, proving the Yoneda lemma) without requiring the readers to have a background in pure mathematics