Category theory; homological algebra Definition

Preservation of limits

Definition

Let $G \colon \mathcal{C} \to \mathcal{D}$ be a functor, and let $F \colon J \to \mathcal{C}$ be a diagram. For a cone $(L, (\lambda_j)_{j \in J})$ over $F$ , the image $(G L, (G \lambda_j)_{j \in J})$ is a cone over $G \circ F$ . $G$ preserves the limit of $F$ if for every limit cone $(L, \lambda)$ of $F$ , the cone $(G L, G \lambda)$ is a limit cone of $G \circ F$ . $G$ preserves limits if it preserves the limit of every diagram $F \colon J \to \mathcal{C}$ with $J$ a small category.

Remarks

A functor that preserves limits is also called continuous. Dually, $G$ preserves colimits if it sends colimit cones to colimit cones. Since limits are unique up to isomorphism, it suffices to check one limit cone of each diagram.

Example: the forgetful functor $U$ from groups to sets preserves products. The product of groups $H \times K$ has underlying set $U H \times U K$ , and the projections are the projections of sets.

Non-example: the functor $X \mapsto X \sqcup \set{*}$ from sets to sets does not preserve the terminal object. The terminal object is a one-point set, which is the limit of the empty diagram, and its image has two points.

A related notion goes the other way. $G$ reflects limits if every cone over $F$ whose image under $G$ is a limit cone is itself a limit cone.

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Cone concept
Diagram concept
Functor concept
Limit concept
Preservation of limits definition

Tag	`QD9H1WHHSW`
URI	`m2://QD9H1WHHSW`
Kind	definition
Writing	W0 stub
MSC	18A35
Defines	Preservation of limits
Uses	Functor, Diagram, Cone, Limit
Changed	2 November 2026

Preservation of limits

Definition

Remarks

Reading path

Further reading

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