nLab L-finite category

-finite categories

category theory

Applications

Limits and colimits

limits and colimits

$L$-finite categories

Definition

Proposition

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

(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. .

References

1. 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.