Notes on Category Theory with examples from basic mathematics (1912.10642v7)

Abstract: These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field. Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. From the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Finally, monads and comonads are treated on an equal footing, differently to most literature in which comonads are often overlooked as "just the dual to monads". Theorems, interpretations and concrete examples are given for monads as well as for comonads. This work, thoroughly revised and expanded, is now a book, with an extra section on monoidal categories.

• The paper presents a comprehensive introduction to category theory by clarifying foundational constructs like limits, functors, and adjunctions.
• The paper employs methodical proofs and a variety of practical examples from mathematics to illustrate abstract theoretical concepts.
• The paper highlights the significance of universal constructions such as the Yoneda Lemma, bridging abstract theory with practical applications.

An Overview of "Notes on Category Theory" by Paolo Perrone

Paolo Perrone's "Notes on Category Theory" provides an extensive introduction to the field of category theory, designed for a broad audience ranging from mathematicians to scientists in disciplines such as computer science and chemistry. It serves as both a primer and a detailed exploration of fundamental concepts through a collection of lecture notes initially developed for a diverse course at the Max Planck Institute of Leipzig.

Essence and Approach

The notes aim to introduce category theory from the ground up, presuming only a foundational knowledge of linear algebra. What distinguishes this material is its focus on elucidating the abstract principles of category theory through rigorous explanations, detailed proofs, and a multitude of examples from various mathematical domains. The material is organized around themes critical to the understanding and application of category theory, such as limits, colimits, functors, natural transformations, and adjoints, all presented with precision but also with an eye towards practical utility across scientific disciplines.

Key Highlights

1. Foundational Constructs

The document begins with a precise formalization of categories, defining them as collections of objects and morphisms satisfying specified axioms. This provides a universal language for mathematical abstraction, enabling a wide range of structures to be studied under a unified framework. Perrone employs examples from fields like topology, group theory, and graph theory to elucidate these foundational constructs, ensuring that readers can grasp these concepts in the context of familiar mathematical objects.

2. The Yoneda Lemma and Representable Functors

A cornerstone of the notes is the thorough treatment of the Yoneda Lemma, one of the most essential results in category theory. It asserts the representability of functors and establishes a profound correspondence between objects and sets of morphisms to these objects. The lemma demonstrates that objects in a category can be fully characterized by the morphisms they accommodate, providing a highly general method for probing their structure. Perrone emphasizes the intuition behind this lemma, presenting it as a tool that enables category theorists to reason about objects by considering their relationships within the category.

3. Limits and Colimits

Limits and colimits are introduced as universal constructions that generalize concepts like products, coproducts, intersections, and unions. Perrone explains these constructs through explicit diagrams and examples, aiding readers in visualizing complex relationships within categories. The discussion of completeness (existence of all small limits) and cocompleteness (existence of all small colimits) showcases the depth and richness of category theoretical constructs, tying these notions back to more tangible mathematical frameworks.

4. Adjunctions and Universal Properties

Adjunctions are another focal point, encapsulating a form of duality ubiquitous in mathematics. The treatment of adjunctions in the notes wraps around natural transformations and the concept of free-forgetful adjunctions, which are well-illustrated through examples involving familiar categories like sets and groups. Perrone carefully unfolds the notion of unit and counit in an adjunction, tying it to universal mapping properties and enriching the reader’s understanding of these nuanced interactions.

Implications and Future Directions

The material in these notes offers implications both theoretical and practical. Theoretically, category theory provides the language and tools to explore and define constructs abstractly and uniformly across mathematical disciplines, encouraging new ways of thinking about structure and symmetry. Practically, it aids in the formalization and understanding of complex systems in fields like computer science, where it underpins the semantics of programming languages and the design of type systems.

Going forward, further exploration in category theory might focus on its implications for emerging fields such as quantum computing and artificial intelligence. By providing a rigorous basis for discussing transformations and objects, category theory could help formalize and unify concepts across these cutting-edge domains.

In conclusion, Paolo Perrone's notes are not just an introduction but an invitation to explore category theory, with a wealth of examples and rich exposition that speaks to both the uninitiated and the expert. These notes lay the groundwork for further paper, making the powerful abstraction of category theory accessible and applicable to a wide range of mathematical and scientific pursuits.