Towards Directed Model Categories

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For many years mathematicians have studied topology, which refers to shapes and spaces that are invariant under continuous deformation. Intuitively, this amounts to treating shapes as strechable and contractible, but without tearing them.Topology is supposed to capture geometry "as if everything is made from rubber." However, even that is not strong enough. For various reasons we often want to think of topology up to homotopy. This means that shapes like a mug and a doughnut should be treated as the same. Homotopy theory has seen incredible development over the past century.In the last 15 years researchers have discovered connections between homotopy theory and the field of formal logic. Somewhat surprisingly, they have been able to take advantage of these connections to enrich various pieces of software that is used to verify the correctness and reliability of mission-critical computer systems (e.g. medical equipment, power grids).In an unrelated stream of work, researchers have also developed a version of topology that is directed. This means that we still study geometrical shapes, but we can only "walk" on them in a particular direction. For example, imagine a circle on which one can only move clockwise. This theory has found remarkable applications in verifying the good behaviour of concurrent systems, i.e. software in which more than one thing is happening at once.However, directed topology does not have its own "directed homotopy theory." Such a theory would enable connections with logic, which can in turn be used to understand and study the behaviour of non-reversible transformations, including the function of concurrent computer systems. This project aims to lay the foundational stone in developing such a theory. It will do this by attempting to adapt the fundamental technical notion of "model category" to directed topology.