Limits and colimits of topoi

The categories \(\Topos\) and \(\Logos\) admit small limits and colimits. Furthermore, the following properties are satisfied:

1. The inclusion \(\Logos \hookrightarrow \Cat\), \(\phi \mapsto \phi^*\), preserves limits.

2. The inclusion \(\Topos \hookrightarrow \Cat\), \(\phi \mapsto \phi_*\), preserves filtered limits.

3. The inclusion \(\Logos \hookrightarrow \CAlg(\Cat^{\colim})\) preserves finite coproducts. In particular:

   • \(\An\) is the terminal topos.

   • The product \(T \times S\) of two topoi is computed as the Lurie tensor product \(T \otimes S\) of cocomplete categories.