An ELL post uses a term "past infinitive"
If the speaker uses the past infinitive here (to have started) they are setting the temporal focus later than the time the lesson should have started.
I am aware of the concept of past simple, present perfect, etc. I am also aware of the concept of infinitive.
I understand that the "past infinitive" refer to "to have started" in the original discussion. I just did't know the term before, and I googled it. top results link to something like French grammar.
Is "past infinitive" a canonical name in English grammar?