  We study idempotents in intensional Martin-L\"of type theory, and in
particular the question of when and whether they split. We show that in the
presence of propositional truncation and Voevodsky's univalence axiom, there
exist idempotents that do not split; thus in plain MLTT not all idempotents can
be proven to split. On the other hand, assuming only function extensionality,
an idempotent can be split if and only if its witness of idempotency satisfies
one extra coherence condition. Both proofs are inspired by parallel results of
Lurie in higher category theory, showing that ideas from higher category theory
and homotopy theory can have applications even in ordinary MLTT.
  Finally, we show that although the witness of idempotency can be recovered
from a splitting, the one extra coherence condition cannot in general; and we
construct "the type of fully coherent idempotents", by splitting an idempotent
on the type of partially coherent ones. Our results have been formally verified
in the proof assistant Coq.
