I have two semi-simplicial complexes $X_\bullet$ and $Y_\bullet$, along with a simplicial map $f_\bullet: X_\bullet \to Y_\bullet$. Now $f_\bullet$ is not a Kan fibration: for an $n$-simplex in $Y_\bullet$ and a lift of any of its horns to $X_\bullet$ there will not, in general, be a filler for the lifted horn. However, there will be a collection of $n$-simplices in $X_\bullet$ whose boundary contains the lifted horn. Think of subdividing the face not included in the lifted horn into a union of $(n-1)$-simplicies, and taking the cone on that; its boundary will be the original lifted horn, along with a subdivision of the opposite face.
In the situation that I'm studying, it is possible to fill the lifted horn in $X_\bullet$ with such a subdivided $n$-simplex. Of course, I have somewhat lost control of how the opposite face descends to the opposite face of the original horn in $Y_\bullet$, since there is a single $(n-1)$-simplex in the opposite face of the horn in $Y_\bullet$, but many in the ``filler" in $X_\bullet$.
My question is this: does this weakening of the notion of a Kan fibration have any good homotopical properties? In particular, can I conclude anything about the connectivity of the map from the fibre of $f_\bullet$ to the homotopy fibre?
If it makes life easier in answering this question, feel free to assume that $X_\bullet$ and $Y_\bullet$ have degeneracies, and are in fact simplicial sets.
