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:
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 $\kappa$.
Theorem. A functor $F \colon \mathsf{C} \to \mathsf{D}$
Set
Ab
R-Mod
Cat
Gpd
SSet
The definition is due to
The standard textbook is
Some further discussion is in proposition 3.4.16, page 220 of
and starting on page 150 of
See also section A.1.1 of
where locally presentable categories are called just presentable categories.
An enriched version of locally presentable can be found in
G. M. Kelly?, Structures defined by finite limits in the enriched context. I. Cahiers Topologie G´eom. Diff´erentielle, 23(1):3–42, 1982.
Michael Shulman, Set theory for category theory, 2008. arXiv:0810.1279