Explore the concept of topological complexity using arbitrary covers in this 55-minute lecture by José Manuel García-Calcines. Delve into the main aspects of topological complexity of a topological space, focusing on sectional category as the primary tool. Examine generalized sectional category and its applications through key examples. Investigate absolute neighborhood retracts (ANRs) and two crucial results related to them. Discover the main result of the talk and its implications. Learn about monoidal topological complexity and its significance. Analyze relative category in the sense of Doeruene-El Haouari and its generalized form. Conclude by understanding generalized monoidal TC and its relevance in the field of applied algebraic topology.