Explore the fascinating world of homological programming in this 57-minute conference talk from the Applied Algebraic Topology Network. Delve into complex topics such as coverage problems, pursuit-evasion scenarios, and positive cones in vector spaces. Examine the Alexander dual to o-homology, focusing on how cones in 1-cohomology encode orientation. Investigate positive 1-homology and the concept of Positive Alexander Duality. Learn about decomposability of criterion, directed topology, and ditopology. Discover the intricacies of cubical sets and directed (co)homology, concluding with a practical calculation and a comprehensive dictionary of terms. Gain valuable insights into this advanced mathematical approach to programming and its applications in various fields.