Presentable $(\infty, n)$-categories

TL;DR

This paper introduces a new framework for presentable $( , n)$-categories using symmetric monoidal $( , n+1)$-categories, extending the classical theory of presentable $( , 1)$-categories.

Contribution

It defines a symmetric monoidal $( , n+1)$-category $n\mathrm{Pr}^L$ for each $n\geq 1$, generalizing the concept of presentable $(\n, 1)$-categories, and studies their properties.

Findings

Objects in $n\mathrm{Pr}^L$ have underlying $(\infty,n)$-categories with all conical colimits.

Conical colimits of right adjointable diagrams can be computed via conical limits after taking right adjoints.

The framework generalizes classical presentability to higher categorical contexts.

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.