TITLE: Important open exposition problems?
QUESTION [39 upvotes]: Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it totally perspicuous." 
What are some open exposition problems in your field, particularly ones that you think would help interested mathematicians break into it? 
For instance, there is no shortage of references on chromatic homotopy theory -- Ravenel's Green and Orange books, Lurie's course notes, and Hopkins' COCTALOS notes. Nonetheless, there does not seem to be a complete, smooth, and carefully-put-together exposition of the chromatic story that assumes neither too much homotopy theory nor algebraic geometry.

REPLY [4 votes]: I have two such open exposition questions which I hope to address some day (alas, I have a long list of both mathematical research and pedagogical problems which are all worthy of attention - but this kind of question is fun because it is at the intersection of these areas).  One is giving a cochain-level treatment of Poincaré duality through intersection theory and then using that to give geometric, cochain level refinements of much of "intermediate" algebraic topology: wrong-way maps, characteristic classes, Thom isomorphism, Steenrod operations, Eilenberg-Moore spectral sequence.  A second is to treat loop spaces and classifying spaces in a more unified way, from scratch (course title: "Loop, de-loop.")  This would expand on my little expository paper on Koszul duality.