Notes on commutation of limits and colimits
  We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all (small) colimits. On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is contained in that of pseudo-filtered colimits. 
Marie Bjerrum, Peter Johnstone, Tom Leinster, William F. Sawin
math.CT math.GR