TITLE: Degeneracies for semi-simplicial Kan complexes
QUESTION [16 upvotes]: By a semi-simplicial set I mean a simplicial set without degeneracies.  In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn.  Unlike when degeneracies are present, we have to include 1-dimensional horns: having fillers for 1-dimensional horns means that every vertex is both the source of some 1-simplex and the target of some 1-simplex.
I have been told that it is possible to choose degeneracies for any semi-simplicial Kan complex to make it into an ordinary simplicial Kan complex.  For instance, to obtain a degenerate 1-simplex on a vertex $x$, we first find (by filling a 1-dimensional 1-horn) a 1-simplex $f\colon x\to y$, then we fill a 2-dimensional 2-horn to get a 2-simplex $f g \sim f$, and we can choose $g\colon x\to x$ to be degenerate.  But obviously there are many possible choices of such a $g$.
I have three questions:

Where can I find this construction written down?
Is the choice of degeneracies unique in some "up to homotopy" sense?  Ideally, there would be a space of choices which is always contractible.
Does a morphism of semi-simplicial Kan complexes necessarily preserve degeneracies in some "up to homotopy" sense?  (A sufficiently positive answer to this would imply a corresponding answer to the previous question, by considering the identity morphism.)

REPLY [4 votes]: Kan wrote a very short note while the paper by Rourke and Sanderson (mentioned in the other answers) was in the publishing process:
Daniel M. Kan, Is an ss complex a css complex? Advances in Mathematics Volume 4, Issue 2, April 1970, Pages 170–171.
(as explained in the comments "ss" refers to semi-simplicial set -- without degeneracies -- and "css" to what is known as simplicial set nowadays).
The proposition is: "An ss complex $X$ which satisfies the extension condition can be completed (although, in general, in many diferent ways)." and he states afterwards that "It is clear that any two such completions will have the same homotopy type."
The proof proceeds by an inductive construction of a simplicial set $WX$ such that there is an isomorphism $FWX \to X$ where $F$ is the forgetful functor from simplicial sets to semi-simplicial sets. As a crucial ingredient he uses the geometric realization from Rourke and Sanderson (he gives a reference to [1,1.3], which should probably be Proposition 2.1. on p.325 in the published version of Rourke and Sanderson's On $\Delta$-sets, I).