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:
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Where can I find this construction written down?
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Is the choice of degeneracies unique in some "up to homotopy" sense? Ideally, there would be a space of choices which is always contractible.
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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.)
Best Answer
The answer to (1) is to be found in
Rourke, C. P.; Sanderson, B. J.$\Delta$-sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. (2) 22 (1971), 321–338.
It is shown there that a Kan "semi-simplicial" set admits a compatible system of degeneracies.
By the way, the term "semi-simplicial" set is not the usual name for this term; it is usually called a "$\Delta$-set."
Added: Given Mike's comment below, I realize now that the following sketch doesn't do the job.
I haven't looked at this paper recently, but I would imagine the way it goes is as follows: Let $$ \Delta^{\text{inj}} = \text{category of finite ordered sets and order preserving injections} $$
$$ \Delta = \text{category of finite ordered sets and order preserving maps} $$ The we have an inclusion functor $j: \Delta^{\text{inj}} \to \Delta$. Given a "semi-simplicial" set $X$ (i.e., a functor $X: \Delta^{\text{inj}} \to \text{Sets}$) we can form the left Kan extension $j_*X$ and then the restriction $j^*j_*X$ to get a natural map $X \to j^*j_*X$. One can ask whether this is a weak equivalence of simplicial sets. Perhaps what Rourke and Sanderson are doing is showing this map to be a weak homotopy equivalence, or maybe just so when $X$ is Kan? (I don't have the paper at hand, so this is speculation on my part.) One might argue as follows: Step a): show that $j^\ast j_\ast$ preserves colimits, Step b) show that the map $X \to j^\ast j_\ast X$ is a weak equivalence when $X$ is a standard "semi-simplicial" $n$-simplex, Step c) infer the general case by induction on simplices.
At any rate, if this is how it goes, then the outcome also provides an answer to (3).