kappa small colimit of kappa compact objects

Proposition

kappa-small colimits of kappa compact objects are again kappa compact

Proof

HTT: 5.3.4.15

Let \(I\) be \(kappa\)-small, \(J\) \(\kappa\)-filtered
Then we get

\begin{align*} \mathrm{map}_{\mathcal{C}}(\mathrm{colim}_I c_i, \mathrm{colim}_J c_j) \cong& \mathrm{lim}_I \mathrm{map}_{\mathcal{C}}( c_i, \mathrm{colim}_J c_j) \\ \cong& \mathrm{lim}_I \mathrm{colim}_J \mathrm{map}_{\mathcal{C}}(c_i, c_j) \\ \cong& \mathrm{colim}_J \mathrm{lim}_I \mathrm{map}_{\mathcal{C}}(c_i, c_j) \\ \cong& \mathrm{colim}_J \mathrm{map}_{\mathcal{C}}(\mathrm{colim}_I c_i, c_j) \\ \end{align*}

where

TODO

Date: nil

Author: Anton Zakrewski

Created: 2024-12-09 Mo 07:51