The Azimuth Project
Locally presentable category

Definitions

Definition. Let κ\kappa be a regular cardinal. A category JJ is κ\kappa-filtered if there is a cone under any diagram with fewer than κ\kappa morphisms.

Definition. Let κ\kappa be a regular cardinal. A locally small category CC is locally κ\kappa-presentable if it is cocomplete and if it has a set of objects SS such that:

  • Every object in CC can be written as a colimit of a diagram valued in the subcategory spanned by the objects in SS.
  • For each object sSs \in S, the functor C(s,):CSetC(s, -) \colon C \to \mathsf{Set} preserves κ\kappa-filtered colimits.

Definition. A functor between locally κ\kappa-presentable categories is accessible if it preserves κ\kappa-filtered colimits.

Remark. If κλ\kappa \le \lambda, then LocPres κLocPres λ\mathsf{LocPres}_\kappa \subset \mathsf{LocPres}_\lambda.

Definition. A category is locally presentable if it is κ\kappa-locally presentable for some κ\kappa.

Adjoint functor theorem for locally presentable categories

Theorem. A functor F:CDF \colon \mathsf{C} \to \mathsf{D}

  • admits a right adjoint iff it is cocontinuous.
  • admits a left adjoint iff it is continuous and accessible.

Examples

  • Set

  • Ab

  • R-Mod

  • Cat

  • Gpd

  • SSet

References

The definition is due to

  • Pierre Gabriel?, Friedrich Ulmer?, Lokal präsentierbare Kategorien, Springer LNM 221, 1971

The standard textbook is

  • Jiří Adámek?, Jiří Rosický?, Locally presentable and accessible categories?, Cambridge University Press, (1994)

Some further discussion is in proposition 3.4.16, page 220 of

  • Francis Borceux?, Handbook of Categorical Algebra: III Categories of Sheaves.

and starting on page 150 of

  • Emily Riehl?, Category Theory in Context, Dover Publications (2017). (pdf), book website

See also section A.1.1 of

  • Jacob Lurie, Higher Topos Theory. pdf

where locally presentable categories are called just presentable categories.

An enriched version of locally presentable can be found in