# nLab L-finite category

-finite categories

# $L$-finite categories

## Definition

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

## 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 $1\in Set^C$ 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

1. There is a typo in (Pare) in the statement of this equivalence: it says “final” instead of “initial”.

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