Category theory; homological algebra Definition

Locally presentable category

Builds on Accessible category Colimit Regular cardinal

Definition

Let $\lambda$ be a regular cardinal and let $\mathcal{C}$ be a locally small category. The category $\mathcal{C}$ is locally $\lambda$ -presentable if both conditions hold.

$\mathcal{C}$ is $\lambda$ -accessible.
$\mathcal{C}$ has all small colimits.

The category $\mathcal{C}$ is locally presentable if it is locally $\lambda$ -presentable for some regular cardinal $\lambda$ .

Remarks

Equivalently, $\mathcal{C}$ is cocomplete and has a set $S$ of $\lambda$ -presentable objects such that every object is a $\lambda$ -filtered colimit of objects of $S$ . For $\lambda = \aleph_0$ one says locally finitely presentable.

Example: the category of sets is locally finitely presentable. It is cocomplete, and it is finitely accessible because every set is the filtered colimit of its finite subsets, which are finitely presentable.

Non-example: the category of fields and field homomorphisms is accessible but not locally presentable. It has no initial object, which is the colimit of the empty diagram. An initial field would map to both $\Q$ and $\F_2$ , but a field homomorphism preserves the characteristic. So clause 2 fails.

A locally presentable category is also complete, well-powered and co-well-powered.

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Tag	`QZFTEZKGNX`
URI	`m2://QZFTEZKGNX`
Kind	definition
Writing	W0 stub
MSC	18C35
Defines	Locally presentable category
Uses	Accessible category, Colimit, Regular cardinal

Locally presentable category

Definition

Remarks

Reading path

Further reading

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