Proposition [efr-0017]

Let \mathcal {C} be a category with finite limits. The internal nerve functor \mathsf {Cat}(\mathcal {C}) \to \mathsf {Simp}\mathcal {C} is fully faithful, and the essential image consists exactly of those simplicial objects X where the maps p_i: X[n] \to X[1] given by evaluating at p^i: [1] \to [n] : k \mapsto k+i, i = 0,\dots n-1 exhibits X[n] as the iterated pullback X[1] \times _{X[0]} \dots \times _{X[0]} X[1]