# The Azimuth Project Locally presentable category

## Definitions

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

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

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

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

Remark. If $\kappa \le \lambda$, then $\mathsf{LocPres}_\kappa \subset \mathsf{LocPres}_\lambda$.

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

## Adjoint functor theorem for locally presentable categories

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

• 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