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Locally presentable category

Definitions

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

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

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

Remark. If κλ\kappa \le \lambda, then LocPres κLocPres λ\mathsf{LocPres}_\kappa \subset \mathsf{LocPres}_\lambda.

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

Adjoint functor theorem for locally presentable categories

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

Examples

References

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