Introduction

Presentability is the right size condition for category theory: it is large enough to contain “all the categories that come up in nature” — modules over a ring spectrum, sheaves on a site, parametrised spectra, anima, etc. — yet small enough that the adjoint functor theorem applies and that $\mathrm{Hom}$-functors are controlled by an essentially small skeleton.

The intuition is that a presentable category is generated by a small piece (its $\kappa$-compact objects) under filtered colimits. Everything else is recovered as a limit of these atoms. This is what makes the category “tame”: universal constructions exist, adjoint functors exist as soon as one side preserves the right kind of (co)limit, and almost all the categories of homotopy theory live here.

Body

A category $\mathcal{C}$ is presentable iff:

  1. $\mathcal{C}$ admits all small colimits.
  2. There is a regular cardinal $\kappa$ and an essentially small set of $\kappa$-compact objects that generate $\mathcal{C}$ under $\kappa$-filtered colimits.

Equivalently (Lurie / Adámek–Rosický in the classical case), $\mathcal{C}$ is presentable iff it is an accessible localization of a presheaf category $\mathsf{Fun}(\mathcal{D}^{\mathrm{op}}, \mathsf{An})$ for some small $\mathcal{D}$.

The category of presentable categories with colimit-preserving functors is $\mathsf{Pr}^L$. It carries the Lurie tensor product, and is itself complete and cocomplete.

Two important refinements live inside $\mathsf{Pr}^L$:

  • A Compactly generated category is a presentable category whose $\kappa$-compact generators can be taken with $\kappa = \aleph_0$.
  • A Compactly assembled category is a presentable category that is a retract in $\mathsf{Pr}^L$ of a compactly generated one — equivalently, a dualizable object of $\mathsf{Pr}^L_{\mathrm{st}}$ in the stable case.

For the 1-categorical (i.e. classical) version of these notions, see Adámek–Rosický.

Pointers

  • Lurie, Higher Topos Theory, §5.5
  • Adámek–Rosický, Locally Presentable and Accessible Categories

See also: Compactly generated category, Compactly assembled category, Stable category.