Definition [efr-003A]

Let X: \Delta ^\mathrm {op} \to \mathcal {C} be a simplicial object in a category, and suppose \mathcal {C} has all limits. Then the right Kan extension along the coYoneda embedding \mathrm {Ran}_yX : \mathsf {sSet}^\mathrm {op} = [\Delta ^\mathrm {op},\mathsf {Set}]^\mathrm {op} \to \mathcal {C} exists. If S is a simplicial set, we denote by a minor abuse of notation X[S] the value of this Kan extension at this set. Thus X[n] = X[\Delta ^n].

We will also use this notation when \mathcal {C} does not have all limits, as long as this Kan extension exists at the given point. For example, if \mathcal {C} has merely finite limits, it exists for all simplicial sets with finitely many nondegenerate simplices.