universal property of Ind kappa

Proposition

Let \(\kappa\) be a regular cardinal, \(\mathcal{C}\) a small infinity category and \(\mathrm{Ind}_{\kappa}(\mathcal{C})\) the infinity category Ind kappa.
Then for each \(\infty\)-category \(\mathcal{D}\) which admits \(\kappa\)-filtered colimits, there exists an joyal equivalence between

\begin{align*} \mathrm{Fun}^{\kappa}(\mathrm{Ind}_{\kappa}(\mathcal{C}),\mathcal{D}) \cong \mathrm{Fun}(\mathcal{C}, \mathcal{D}) \end{align*}

where \(\mathrm{Fun}^{\kappa}(-,-)\) is the full infinity subcategory containing infinity functors which preserve kappa filtered colimits

Proof

FW: 6.61 c)

Date: nil

Author: Anton Zakrewski

Created: 2024-12-10 Di 11:11