conservative functor preserving limits and colimits and commuting
1. Proposition
Let \(\mathcal{C}, \mathcal{D}\) be categories, \(\mathcal{I}, \mathcal{J}\) be diagrams over \(\mathcal{C}\) and \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\) be a conservative functor which
- preserves colimits of the shape \(J\)
- preserves limits of the shape \(I\)
TFAE:
- \(\mathrm{colim}_J \mathrm{lim}_I = \mathrm{lim}_I \mathrm{colim}_J\) in \(\mathcal{C}\)
- \(\mathrm{colim}_J \mathrm{lim}_I \mathcal{F} = \mathrm{lim}_I \mathrm{colim}_J \mathcal{F}\)