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

  1. preserves colimits of the shape \(J\)
  2. preserves limits of the shape \(I\)

TFAE:

  1. \(\mathrm{colim}_J \mathrm{lim}_I = \mathrm{lim}_I \mathrm{colim}_J\) in \(\mathcal{C}\)
  2. \(\mathrm{colim}_J \mathrm{lim}_I \mathcal{F} = \mathrm{lim}_I \mathrm{colim}_J \mathcal{F}\)

2. Proof

2.1. 1) \(\implies\) 2)

2.2. 2) \(\implies\) 1)

Date: nil

Author: Anton Zakrewski

Created: 2024-10-14 Mo 08:55