This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models. The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action. Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource. It can also serve as a supplemental text for a specialized graduate topics course.

We've all had that conversation-with ourselves or our significant other-the one about what to do with our life. For some it remains just that-a conversation. But for Pat and Ali the simple question had an answer that demanded action. "Why not sail around the world?" Well, for starters, they had never stepped foot on a sailboat before. For another, they were on the fast track to beating out all of the Joneses. But they were bored and getting a little soft around the middle. The idea of moving to the suburbs from the city to become more comfortable made them queasy. Within months they'd bought a 35' catamaran and had set out across the Gulf Stream for the Bahamas, and for the next four years they sailed around the world. They learned that people are good the world over-that every stranger was not out to get them, that even the scariest looking among them-the pirates of our minds eye-are deep down no different from you and I. In fact, they've had that conversation too-What am I going to do with my life?