A survey on saturated and pre-saturated classes of colimits

Giacomo Tendas – 6 April 2022

Let Φ be a class of (indexing) categories; we say that Φ is saturated if for any small category C the existence and preservation of C-colimits is implied by that of Φ-colimits. We say Φ is pre-saturated if for any category C, every object X of ΦC (the free cocompletion of C under Φ-colimits) is a Φ-colimit of elements of C. When one replaces categories with weights (as it usually happens in the enriched context), it’s easy to see that every saturated class is pre-saturated; however the same result (to my knowledge) is not that easy to prove when Φ is a class of categories. The aim of this talk is to give some examples of non-pre-saturated, pre-saturated (but not saturated) and saturated classes of colimits, and to provide a proof of the fact that every saturated class of categories is pre-saturated. At the end we’ll see that a pre-saturated class is saturated if and only if it is closed under “codomains of final functors”.