Introduction

Compactly generated categories are the tame end of the presentable spectrum. Every object is a filtered colimit of compact ones, and most homological invariants — K-theory, Hochschild homology, cyclic homology — are determined by the small subcategory of compact objects. The idea: a “big” category is recovered, up to filtered colimits, from a small skeleton of finite/perfect/dualizable atoms.

Examples are everywhere in algebra and topology: spectra (compact = finite spectra), modules over a ring spectrum (compact = perfect), the derived category of a ring (compact = perfect complexes). The fact that all of homotopy theory’s “large” categories of interest are compactly generated is why the theory of K-theory, traces and dimensions is as workable as it is.

Body

A presentable category $\mathcal{C}$ is compactly generated (or $\aleph_0$-presentable) if it is generated under filtered colimits by its compact objects $\mathcal{C}^{\omega} \subset \mathcal{C}$.

Equivalently:

\[ \mathcal{C} \;\simeq\; \mathsf{Ind}(\mathcal{C}^{\omega}), \]

so a compactly generated $\mathcal{C}$ is recovered as the Ind-completion of its small subcategory of compact objects. In the classical case, this recovers the notion of a locally finitely presentable category in the sense of Adámek–Rosický.

Examples include:

  • The derived category $\mathsf{D}(R)$ of a ring $R$ (compact = perfect complexes).
  • The stable category of spectra $\mathsf{Sp}$ (compact = finite spectra).
  • $\mathsf{Mod}_R$ over a ring spectrum (compact = perfect modules).

Many naturally occurring categories — sheaves on a locally compact Hausdorff space, for instance — are not compactly generated, but the slightly weaker notion of Compactly assembled category still applies.

Relation to other notions

  • A compactly generated category is automatically a Presentable category (with $\kappa = \aleph_0$).
  • A Compactly assembled category is, by Lurie–Clausen, a retract in $\mathsf{Pr}^L$ of a compactly generated one.
  • In the stable setting, a Stable category that is compactly generated is in particular dualizable in $\mathsf{Pr}^L_{\mathrm{st}}$, with dual $\mathsf{Ind}((\mathcal{C}^{\omega})^{\mathrm{op}})$.

Pointers

  • Lurie, Higher Topos Theory, §5.5.7
  • Krause–Nikolaus–Pützstück, Sheaves on Manifolds, §2

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