Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that resembles the plus construction in $\mathsf{Top}$?
More precisely, let $G$ be a (perfect) group and $|BG|$ the classifying space of $G$, by the geometric realization of the nerve construction (so $BG$ is the simplicial set with $B_{n}G$ = set of all tuples $(g_1, \cdots ,g_n)$, with the natural face and degeneracy maps. And, $|BG| = $ geometric realization of the simplicial set $BG$. I'm using the construction given in Weibel's "An introduction to Homological Algebra" book, Page 257).
The question is as follows :
Does there exist a functor (denoted with slight abuse of notation) $(-)^{+}\colon\mathsf{sSet}\to\mathsf{sSet}$, such that $|(BG)^{+}| \cong (|BG|)^{+}$ for any group $G$?
In other words, can we make a plus construction in simplicial sets, so that it commutes with the geometric realization functor?