I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here.

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 (perfect) group $G$?

In other words, can we make a plus construction in simplicial sets, so that it commutes with the geometric realization functor?

homotopy groupsare the homology groups of the original simplicial set. This is very different from the plus construction, which for example leaves simply connected spaces unchanged. $\endgroup$