Categories and Functors: An Introduction

This lecture introduces the fundamental concepts of categories, functors, and natural transformations. It begins with the definition of categories, highlighting the importance of objects and morphisms. The instructor explains groups and monoids, emphasizing their role as categories. The concept of invertibility in categories is discussed, illustrating how morphisms can be invertible. The lecture then transitions to the construction of new categories from existing ones, using examples such as posets and digraphs. The instructor elaborates on the relationship between categories and functors, detailing how functors map between categories while preserving structure. The distinction between forgetful functors and inclusion functors is made clear, showcasing their applications in various mathematical contexts. The lecture concludes with a discussion on presheaves and their significance in category theory, providing canonical examples to illustrate these concepts. Overall, this lecture serves as a foundational overview of category theory, essential for further studies in mathematics and related fields.