nLab
lifted limit

Contents

Contents

Definition

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

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.

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.

Last revised on March 10, 2021 at 01:33:15. See the history of this page for a list of all contributions to it.