Adjunctions and Push Outs

This lecture introduces the concept of adjunctions between functors in category theory, explaining the bijections between morphisms induced by adjunctions. The instructor discusses the unit of adjunction, natural bijections, and the properties of push outs as colimits. Through examples and detailed explanations, the lecture covers the universal properties of colimits and the construction of free products of groups. The discussion extends to the notion of discrete fibers in coverings, emphasizing the importance of understanding the topology induced by fibers. The lecture concludes with a practical example of constructing a quotient space using identifications and symmetries, highlighting the role of group actions in determining the properties of coverings.