HoTT/HoTT — pull request #651 by mikeshulman (2014-12-19 06:50:07+00:00)
Title: More idempotents

The main new result is that the type of retracts of `X` is a retract of the type of partially-coherent idempotents on `X` (here called "quasi-idempotents" by analogy with "quasi-inverse"), with the retraction being the splitting construction. The witness `I : f o f == f` can be recovered from the splitting (`f` with this datum is here called a "pre-idempotent"), but not the additional coherence `J` in a quasi-idempotent. A counterexample to the latter is in progress. I'm planning a second blog post to go along with this, but probably not until I have that counterexample formalized too.