nLab L-finite category

Definition

Proposition

(characterizations of L-finite limits)
A category $C$ is $L$-finite if the following equivalent conditions hold, which are all equivalent:

• The terminal object of the functor category $[C,Set]$ to 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.¹

• $C$-limits lie in the saturation of the class of finite limits.

(Paré 1990, p. 740 (10 of 16), around Prop. 7)

Remark

(relation to K-finite sets)
The notion of L-finite category (Def. ) 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 $1\in Set^C$ belongs to the closure of the representables under finite colimits.

In Paré 1990, p. 741 (11 0f 16) this observation is attributed to Richard Wood.

Examples

Example

(categories with initial objects are L-finite)
Any category $\mathcal{C}$ with an initial object $\varnothing \,\in\, \mathcal{C}$ is L-finite, with the inclusion of the terminal category mapping to this initial object $\{\varnothing\} \xhookrightarrow{\;} \mathcal{C}$ being an initial functor (by this exp.) as required by Def. .

Related concepts

• finite category

• filtered colimit

• sound doctrine

References

• Robert Paré, Section 3 of: Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746 (doi:10.4153/CJM-1990-038-6)

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1. There is a typo in Paré Prop. 7 in the statement of this equivalence: it says “final” instead of “initial”. ↩