#
The Azimuth Project

Locally presentable category (Rev #1)

## Definitions

## Adjoint functor theorem for locally presentable categories

**Theorem.** A functor $F \colon \mathsf{C} \to \mathcal{D}$

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

## 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*.