This preprint is now available: The topology of critical processes, I (Processes and models) M. Grandis https://arxiv.org/abs/2309.01991 [math.AT] 23 pages Abstract. This article belongs to a subject, Directed Algebraic Topology, whose general aim is including non-reversible processes in the range of topology and algebraic topology. Here, as a further step, we also want to cover `critical processes', indivisible and unstoppable. This introductory article is devoted to fixing the new framework and representing processes of diverse domains, with minimal mathematical prerequisites. The fundamental category and singular homology in the present setting will be dealt with in a sequel. Regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]