Halliday & Matthiessen (1999: 452-3):
According to the first position, a material nuclear process is interpreted as if it was a participant in a relational process. This gives a logical analysis of the form “Subject ‘be’ Predicate”: Socrates ran is analysed as “Subject: ‘Socrates’ ‘be’ Predicate: ‘running’”. In terms of grammatical classes, a verb is a copula plus an adjective.