nLab
L-finite category

Definition

A category $C$ is $L$-finite if the following equivalent conditions hold:

• The terminal object of $[C,\mathrm{Set}]$ is ($\omega$-)compact.
• $C$-limits commute with filtered colimits in Set.
• $C$ has an initial finitely generated? subcategory.
• $C$ admits an initial functor from a finite category.
• $C$-limits lie in the saturation of the class of finite limits.

Remarks

The notion of L-finite category is a sort of categorification of the notion of K-finite set:

• A set $X$ is $K$-finite if the top element $1\in {\Omega }^X$ belongs to the closure of the singletons under finite unions.

• A category $C$ is $L$-finite if the terminal object $1\in {\mathrm{Set}}^C$ belongs to the closure of the representables under finite colimits.

References

• Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS