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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 $F \colon \mathsf{C} \to \mathsf{D}$

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