nLab lifted limit

Definition

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is said to lift limits of a particular shape $I$ if for any diagram $J:I\to \mathcal{C}$, any limiting cone for $F \circ J$ in $\mathcal{D}$ is the image of a limiting cone for $J$ in $\mathcal{C}$.

The above definition is not invariant under equivalences of categories. It can be made invariant if we demand instead that any limiting cone for $F\circ J$ is isomorphic to the image of a limiting cone for $J$. Alternatively, this says that, if $F \circ J$ has a limit, then $J$ also has a limit and that limit is preserved by $F$.

Terminological remarks

Lifting limits is closely related to creating them. The relationships between these notions were the subject of a post by Aleks Kissinger at the categories mailing list, here, but there is some dispute about its correctness.

Related concepts

• preserved limit

• reflected limit

• created limit

• topological concrete category

• lifted limit

References

See Definition 13.17 in Adamek, Herrlich?, Strecker?: Abstract and Concrete Categories. Remark 13.38 provides a useful diagram of relations between reflected/created/lifted limits.