I think Iβve figured out most parts of the cubical type theory papers; Iβm going to take a shot to explain it informally in the format of Q&As. I prefer using syntax or terminologies that fit better rather than the more standard ones.
Q: What is cubical type theory?
A: Itβs a type theory giving homotopy type theory its computational meaning.
Q: What is homotopy type theory then?
A: Itβs traditional type theory (which refers to Martin-LΓΆf type theory in this Q&A) augmented with higher inductive types and the univalence axiom.