Limits and Colimits: Equalizers and Coequalizers

This lecture discusses the concepts of limits and colimits in category theory, focusing on equalizers and coequalizers. The instructor begins by recalling the definitions of limits and colimits, providing a foundational understanding. They illustrate these concepts with explicit computations, particularly when the target category is Set or Ab. The lecture emphasizes the universal properties associated with equalizers and coequalizers, explaining how these properties relate to morphisms and objects within a category. The instructor also compares equalizers and pullbacks in the category of sets, highlighting the relationships between these concepts. Throughout the lecture, examples are provided to clarify the definitions and properties, ensuring that students grasp the practical applications of limits and colimits in various categories. The session concludes with a discussion on the implications of these concepts in different contexts, reinforcing the importance of understanding the underlying structures in category theory.