Suppose we are given a category enriched over [**semi-simplicial**](http://ncatlab.org/nlab/show/semi-simplicial+set) sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this category all **inner** Kan conditions are satisfied, i.e. we have $\mathrm{Kan}(n,k)$ for all $n$ and $1\le k \le n-1$, but not necessarily $\mathrm{Kan}(n,0)$ and $\mathrm{Kan}(n,n)$. 


Is there any "canonical" way to make this category become enriched over **simplicial** sets, i.e. to find "canonical up to equivalence" degeneracy maps, so that the category will become an $(\infty,1)$-category?


There are several papers on making a Kan-simplicial set out of Kan-semi-simplicial one, including the classical paper by [Rouke and Sanderson](http://www.maths.ed.ac.uk/~aar/papers/deltars.pdf) and a more recent by [McClure](http://arxiv.org/abs/1210.5650), but I haven't found anything concerning quasi-categories.



Defining degeneracies for semi-simplicial sets with inner Kan conditions
The short answer is no. If you take the semi-simplicial set $X$ with one $0$-simplex and no higher simplices, then it has fillers for all inner horns (because it has no inner horns) but clearly does not admit degeneracies.


The question then becomes: what in addition to inner horn fillers could we assume to make this true (without restricting attention to Kan complexes)? One possible guess would be: add lifts along the inclusion of $X$ into $E[1]$, i.e. (the underlying semi-simplicial set of) the nerve of the groupoid freely generated by one isomorphism. I don't know whether this is sufficient, but this is a notable lifting property that follows from fillers for inner horns in the simplicial setting, but doesn't in the semi-simplicial one.


