(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.
(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.
(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. .
There is a typo in Paré Prop. 7 in the statement of this equivalence: it says “final” instead of “initial”. ↩
Last revised on August 25, 2021 at 11:12:03. See the history of this page for a list of all contributions to it.