Inductive and nearly inductive sketches (Q2762307)

A sketch is a small category \({\mathcal E}\) equipped with a set \(\Gamma\) of projective or inductive cônes. It is called projective, inductive, nearly inductive if \(\Gamma\) is a set of projective cônes, a set of inductive cônes, or a set of cônes whose projective ones have empty base, respectively. The category \({\mathcal M}od({\mathcal E})\) of models of \({\mathcal E}\) in \({\mathcal S}et\) is the full subcategory of \({\mathcal S}et^{\mathcal E}\) whose objects send the distinguished projective (resp. inductive) cônes on limits (resp. colimits), and is well known to be an accessible category.NEWLINENEWLINENEWLINEIt is proved that the category \({\mathcal M}od({\mathcal E})\) of models of a nearly inductive sketch is multicomplete, i.e., has multilimits. Let us recall that a multilimit of a small diagram \(\delta:{\mathcal D}\to{\mathcal A}\) in a category \({\mathcal A}\), is a small family of projective cones \((\lambda_i:L_i \to\delta)_{i\in I}\) of \({\mathcal A}\) based on \(\delta\), such that, for any object \(A\) of \({\mathcal A}\), we have \(\text{Hom}_{\mathcal A} (A,\delta) \simeq \coprod_{i\in I}\text{Hom}_{\mathcal A}(A,L_i)\) in a natural way. As a consequence, any nearly inductive sketch is equivalent to a special projective sketch. Specific examples are given.