The topology of critical processes, II (The fundamental category)

Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The first part of this series introduced controlled spaces, examining how they can model critical processes in various domains, from the change of state in a memory cell to the action of a thermostat or a siphon. We now construct the fundamental category of these spaces. [The previous version of this Preprint was too long and has been split in Parts II and III.]