Definition Limit sketch [efr-001M]

A limit sketch consists of a (small) category \mathcal {I} equipped with a collection of cones (J_i \in \mathsf {Cat}, D_i: J_i^\triangleleft \to \mathcal {I}) (recall that J^\triangleleft is the "cone on J", the result of freely adjoining an initial object to J).

If \mathcal {C} is a category, a model of the limit sketch is a functor M: \mathcal {I} \to \mathcal {C} so that each composite J_i^\triangleleft \to \mathcal {I} \to \mathcal {C} is a limit cone. A morphism of models is any natural transformation. In other words, the category of models is the full subcategory of the functor category spanned by the models. It is denoted \mathsf {Mod}_\mathcal {I}(\mathcal {C}) (the collection of diagrams is left implicit).

Given a bicategory or other higher category \mathcal {C}, a model is a pseudofunctor \mathcal {I} \to \mathcal {C} which carries every diagram to a homotopy limit cone. \mathsf {Mod}_\mathcal {I}(\mathcal {C}) \subseteq \mathsf {Fun}(\mathcal {I},\mathcal {C}) is again the full sub-bicategory of the bicategory of pseudofunctors spanned by the models.