Sheaves for Systems

The problem of analyzing 'emergent behaviour' in compositional system is currently of central interest in the Applied Category Theory community. Emergent behaviour concerns the study of those situations for which composing two open systems gives rise to a new system whose behaviour is richer than what you'd expect. A proposed formalization of this fact was given by [Hed19] (but already present in the folklore) through the notion of lax pseudofunctors C -> Rel, where C is a category of 'boundaries' and 'open systems' between them. In sheaf theory, the opposite happens: the 'behaviour' of bits of a space is richer than that of their union. Here the sheaf condition ensures compatible local sections glue into global ones, and sheaf cohomology probes how demanding this compatibility condition is. We believe something analogous lurks behind emergent behaviours, that is one may be able to study emergent phenomena as cohomology of a 'behaviour sheaf'. The details of this construction, which is going to be somewhat complicated given the level of generality of the setting and forms the core of the proposed research.