For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets whose unit is a natural weak equivalence (in the Quillen model structure) and whose algebras are Kan fibrations. A mere existence proof is not good enough, I need a construction. Does Kan's $\mathsf{Ex}^\infty$ construction do this? Are there other monads?

I am looking for (non trivial) model structures on simplicial objects in the effective topos. Unfortunately, the effective topos has no infinite colimits, so the small object argument is useless. The simplicial effective topos has enough injectives and enough projectives. These induce weak factorisation systems, which might be part of a model structure.