Presentable $(\infty, n)$-categories

Abstract

We define for each $n \geq 1$ a symmetric monoidal $(\infty, n+1)$-category $n\mathrm{Pr}^L$ whose objects we call presentable $(\infty,n)$-categories, generalizing the usual theory of presentable $(\infty,1)$-categories. We show that each object $\mathcal{C}$ in $n\mathrm{Pr}^L$ has an underlying $(\infty,n)$-category $\psi_n(\mathcal{C})$ which admits all conical colimits, and that conical colimits of right adjointable diagrams in $\psi_n(\mathcal{C})$ can be computed in terms of conical limits after passage to right adjoints.