Sur les catégories accessibles multicomplètes. (On multicomplete accessible categories) (Q1602665)

A category \({\mathcal A}\) is accessible if there exists a regular cardinal \(\beta\) such that: 1) \({\mathcal A}\) has all \(\beta\)-filtered colimits, 2) its full subcategory \({\mathcal A}_\beta\) of \(\beta\)-presentable objects is small and dense in \({\mathcal A}\), 3) and, for any object \(A\) in \({\mathcal A}\), the category \({\mathcal A}_\beta/A\) is \(\beta\)-filtered. A category \({\mathcal A}\) is multicomplete if any small diagram \(\delta:{\mathcal D}\to {\mathcal A}\) of \({\mathcal A}\) has a multilimit, i.e., a small family \((\lambda_i: L_i\to \delta)_{i\in I}\) of projective cônes of \({\mathcal A}\) based on \(\delta\), such that for any object \(A\) in \({\mathcal A}\), we have \(\text{Hom}_{\mathcal A}(A,\delta) \simeq\coprod_{i\in I} \text{Hom}_{\mathcal A} (A,L_i)\) in a natural way. A connected colimit in a category \({\mathcal A}\) is a colimit of a small nonempty connected diagram in \({\mathcal A}\). It is proved that an accessible category \({\mathcal A}\) is multicomplete if and only if it has all connected colimits.