Exploring 2-Category Theory: Resources and Concepts

2-category theory is a fascinating area of mathematics that builds upon the foundational concepts of category theory, introducing a layer of depth and complexity by considering morphisms of morphisms. This theory provides a framework to understand and analyze various mathematical structures in a more nuanced way. To get started with 2-category theory, it's essential to explore a variety of resources that provide detailed explanations, examples, and insights.

Origins and Definitions

Charles Ehresmann first introduced the concept of a 2-category in his work on enriched categories in 1965, as part of his broader contributions to the field of category theory. According to the article on 2-categories in Wikipedia, Ehresmann's work laid the groundwork for this field, focusing on the categorical structures enriched over other categories (Ehresmann, 1965).

Several decades later, Jean Bénabou discovered the general concept of bicategories or weak 2-categories in 1968. In his seminal work, published in the Reports of the Midwest Category Seminar, Bénabou introduced a more flexible approach to category theory, where the associativity of morphisms is only required up to a 2-isomorphism (Bénabou, 1967).

Understanding 2-Category Theory

The term 'bicategory' is often used interchangeably with 'weak 2-category' and refers to a category where the composition of morphisms is not strictly associative but associative up to natural isomorphisms. This weakening of the strict associativity condition makes bicategories a powerful tool in areas of mathematics such as algebraic topology, homotopy theory, and theoretical computer science.

Key Concepts in 2-Category Theory

2-Categories: A 2-category is a category enriched in the category of categories. It consists of objects, morphisms, and 2-morphisms, where the composition of morphisms is strictly associative and the composition of 2-morphisms is associative but only up to a 2-isomorphism. Bicategories: As mentioned, a bicategory is a generalization of a 2-category where the composition of morphisms is associative only up to a 2-isomorphism. This allows for a more flexible and nuanced treatment of categorical structures. Enriched Categories: Enriched categories extend the concept of a category by replacing the hom-sets with objects from another category. This allows for a richer structure and more sophisticated ways of composing morphisms.

Online Resources Further Reading

Theoretical and practical knowledge of 2-category theory can be expanded by exploring various online resources. Here are a few recommended sources:

2-Categories by Ross Tate - Part of the Cornell University CS 6117 course, this resource provides a comprehensive introduction to 2-categories, making it ideal for both beginners and advanced learners. 2-Categories from the Columbia University Stacks Project - This resource is part of a larger project that offers detailed explanations and examples of various mathematical concepts, including 2-categories. A 2-Category Companion by Stephen Lack - This companion provides an in-depth look at 2-categories and bicategories, making it a valuable resource for those looking to deepen their understanding. What are 2-Categories by Katerina Hristova - This concise overview offers a clear and accessible introduction to 2-categories, suitable for beginners and intermediate learners.

Conclusion

2-category theory is a rich and intricate field of study that extends the capabilities of category theory. By exploring the concepts and resources mentioned above, you can gain a deeper understanding of this fascinating area of mathematics. Whether you're a student, researcher, or simply curious about advanced mathematical concepts, 2-category theory offers a unique and powerful framework for understanding and analyzing various structures.

Note: Ensure to cite these resources appropriately and refer to the original publications for detailed insights and advanced topics in 2-category theory.