Let $X$ be a simplicial set. We say that $X$ is a $H$-space if it has a map $m:X\times X\to X$ and a point $e\in X$ which is a homotopy identity, that is, the map $m(e,-),m(-,e):X\to X$ are homotopic to the identity map.
How does an $H$-space compare with monoid in the category of simplicial sets? Is a monoid in the category of simplicial presheaves a topological monoid? One talks about group completion for a topological monoid $G$ as $\Omega BG$. Is there something same for a monoid?
Well, a simplicial monoid is associative and has a strict unit. Every $A_\infty$ Kan complex, which puts infinitely many more coherence constraints on the $H$-space structure than what an $H$-space requires, is homotopy equivalent to a simplicial monoid. Since a simplicial set isn't a topological space, a monoid in simplicial sets is not a topological monoid, although it induces one under geometric realization. Yes, there is a notion of group completion of a monoid, left adjoint to the forgetful functor.