Let \(\mathcal D\) be a (possibly large) \(\infty\)-category.
Let \(\mathcal K\) be a collection of \(\infty\)-categories and \(S\) a small set of objects of \(\mathcal D\). Then \(\mathcal D\) is generated by \(S\) under \(\mathcal K\)-indexed colimits if \(\mathcal D\) has all colimits indexed by categories in \(\mathcal K\) and is the smallest full subcategory of \(\mathcal D\) which contains the objects in \(S\) and is closed under \(\mathcal K\)-indexed colimits.
Let \(\kappa\) be an infinite regular cardinal and assume \(\mathcal D\) admits \(\kappa\)-filtered colimits. Then an object \(d\in \mathcal D\) is called \(\kappa\)-compact if the functor \(\mathrm{Hom}_{\mathcal D}(d,-)\colon \mathcal D\rightarrow\mathcal S\) preserves \(\kappa\)-filtered colimits.
The \(\infty\)-category \(\mathcal D\) is called accessible if it is locally small and there exists a regular cardinal \(\kappa\) and a small set \(S\) of \(\kappa\)-compact objects in \(\mathcal C\) that generates \(\mathcal C\) under \(\kappa\)-filtered colimits.
The \(\infty\)-category \(\mathcal D\) is called presentable if it has all small colimits and is accessible.