Combinatorics of past-similarity in higher dimensional transition systems

The key notion to understand the left determined Olschok model category of star-shaped Cattani-Sassone transition systems is past-similarity. Two states are past-similar if they have homotopic pasts. An object is fibrant if and only if the set of transitions is closed under past-similarity. A map is a weak equivalence if and only if it induces an isomorphism after the identification of all past-similar states. The last part of this paper is a discussion about the link between causality and homotopy.