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Locally presentable category (Rev #1)


Adjoint functor theorem for locally presentable categories

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

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


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.