Pointwise inductive limits in accessible categories (Q2723231)

A sketch is called normal (resp. positive) if its distinguished projective cones have non empty connected (resp. non empty) indexation. Normally (resp. positively) -- sketchable categories are characterized as follows, in this paper.NEWLINENEWLINENEWLINELet \(\alpha\) be a regular cardinal. A category is \(\alpha\)-pseudo-filtered if its \(\alpha\)-connected component are \(\alpha\)-filtered. A category \({\mathcal A}\) is called normally accessible if there exists a regular cardinal \(\alpha\) such that:NEWLINENEWLINENEWLINE(a) \({\mathcal A}\) has \(\alpha\)-pseudo-filtered colimits.NEWLINENEWLINENEWLINE(b) \({\mathcal A}\) has a small full subcategory \({\mathcal A}'\) whose objects are \(\alpha\)-presentable and connected, and such that any object of \({\mathcal A}\) is a \(\alpha\)-pseudo-colimit of objects of \({\mathcal A}'\).NEWLINENEWLINENEWLINEIt is proved that normally accessible categories are precisely normally sketchable categories. On the other hand, positively sketchable categories are precisely accessible categories having a strict initial object.