nLab L-finite category

-finite categories


Category theory

Limits and colimits

LL-finite categories



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

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


(relation to K-finite sets)
The notion of L-finite category (Def. ) is a sort of categorification of the notion of K-finite set:

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



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


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

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