nLab double limit

Redirected from "double limits".
Contents

Contents

Idea

A double limit is the appropriate notion of limit for a double category. Just as double categories generalise 2-categories, double limits generalise 2-limits.

Definition

For now, see Grandis and Paré (1999).

Examples

Properties

  • A double category has all small double limits if and only if it has small double products, double equalisers, and tabulators.

  • A 2-category admits all weighted 2-limits if and only if it admits all double limits. Double limits thus provide a “conical” approach to 2-limits that is an alternative to marked 2-limits.

  • A 2-category admits all flexible limits if and only if it admits all persistent double limits.

Variants

  • A pseudo double limit is analogous to a pseudo limit in 2-category theory.

  • A persistent double limit is a limit that is equivalent to a pseudo double limit. 𝕀\mathbb{I}-indexed double limits are persistent if and only if each connected component of the underlying 2-category of 𝕀\mathbb{I} has a weak initial object.

References

  • A. Bastiani, Charles Ehresmann, pages 272-273 of Multiple functors. I. Limits relative to double categories, Cah. Top. Géom. Différ. Catég. 15 (1974) 215–292

  • Robert Paré, Double limits, Talk given at Category Theory 1989, (pdf)

  • Dominic Verity, Enriched categories, internal categories and change of base Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 (TAC)

  • Marco Grandis and Robert Paré, Limits in double categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 40, No. 3, 1999, pp. 162-220. [cahiers]

  • Marco Grandis and Robert Paré, Adjoint for double categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 45, No. 3, 2004, pp. 193-240. [cahiers]

A notion of pseudo double limit is introduced in:

The connection between persistent double limits and flexible limits is proven in:

Last revised on December 9, 2023 at 22:28:19. See the history of this page for a list of all contributions to it.