The homotopy groups of a Kan complex are defined combinatorially, and they coincide with the homotopy groups of the geometric realization. For a general complex $X$, homotopy isn't an equivalence relation, so there are no combinatorial homotopy groups, and, as far as I know, one must define the homotopy groups as those of the geometric realization $|X|$, equivalently, as those of the (Kan) complex $S_{\bullet}(|X|)$. For my purpose this is impractical.
Is there a combinatorial way to define the homotopy groups of a quasi category (i.e. a weak Kan complex) $X\;$?
The following surely is wishful thinking.
Is there a (functorial) Kan complex $X'$ and a (natural) weak equivalence $X\xrightarrow{\,\sim\,} X'$ such that if $X$ is finite (in the sense that it only has a finite amount of nondegenerate simplices) then so is $X'\;$?