Adjunctions and Limits: Exploring Functors and Co-limits

This lecture discusses the main theorems related to adjunctions and limits, focusing on how to determine whether functors are adjoints and how to compute (co)limits. The instructor begins with a warm-up session, summarizing key slogans that encapsulate the week's material. The discussion progresses to formal definitions and theorems, emphasizing the preservation properties of left and right adjoints. The instructor introduces a consequence related to Fubini's theorem, illustrating how to manipulate small categories and their adjoints. The lecture also covers practical applications, such as using these results to ascertain the adjoint status of functors. The importance of products and equalizers in constructing all limits is highlighted, culminating in a proof that these concepts are foundational in category theory. The instructor engages with the audience, encouraging questions and clarifications throughout the session, ensuring a comprehensive understanding of the material presented.