We establish a Myhill-Nerode type theorem for higher-dimensional automata
(HDAs), stating that a language is regular precisely if it has finite prefix
quotient. HDAs extend standard automata with additional structure, making it
possible to distinguish between interleavings and concurrency. We also
introduce deterministic HDAs and show that not all HDAs are determinizable,
that is, there exist regular languages that cannot be recognised by a
deterministic HDA. Using our theorem, we develop an internal characterisation
of deterministic languages