Simplicial and Cosimplicial Objects: Examples and Applications

This lecture discusses the concepts of simplicial and cosimplicial objects within category theory. It begins with a warm-up on the category of simplicial objects, explaining how these objects are constructed from functors. The instructor provides examples of simplicial objects derived from groups and cosimplicial objects from sets, illustrating the relationships between these structures. The lecture emphasizes the role of comonads in generating simplicial objects and monads in producing cosimplicial objects. The instructor also highlights the importance of understanding morphisms and their compositions in these categories. Throughout the lecture, various diagrams and notations are introduced to clarify the relationships between different objects and morphisms. The discussion culminates in the application of these concepts to real-life mathematical scenarios, showcasing the relevance of simplicial and cosimplicial objects in broader mathematical contexts. The lecture concludes with a preview of upcoming topics related to nerve and geometric realization, encouraging students to engage with the material actively.