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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.

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    $\begingroup$ Out of curiousity: what is a cloven fibered groupoid? $\endgroup$ Commented Feb 5, 2014 at 22:29
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    $\begingroup$ The fibrations of the canonical model structure on groupoids are Grothendieck fibrations. A cloven fibred groupoid is a cloven fibration between groupoids. $\endgroup$ Commented Feb 5, 2014 at 22:40
  • $\begingroup$ And what is a cloven fibration? $\endgroup$ Commented Feb 6, 2014 at 0:21
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    $\begingroup$ @TomGoodwillie A cloven (Grothendieck) fibration is one that is equipped with a choice of "pullbacks" (i.e. the data needed to determine the corresponding pseudofunctor uniquely). $\endgroup$
    – Zhen Lin
    Commented Feb 6, 2014 at 1:21
  • $\begingroup$ @WouterStekelenburg You may want modify your question somewhat. Garner's small object argument produces a monad on $[\mathbb{2}, \mathbf{sSet}]$ whose algebras are Kan fibrations with chosen liftings, and the unit is even a trivial cofibration. $\endgroup$
    – Zhen Lin
    Commented Feb 6, 2014 at 1:25

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