nLab double limit

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

• tabulator

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. $\mathbb{I}$-indexed double limits are persistent if and only if each connected component of the underlying 2-category of $\mathbb{I}$ has a weak initial object.

Related concepts

• 2-limit
• marked 2-limit

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:

• Marco Grandis and Robert Paré, Persistent double limits, Cahiers de topologie et géométrie différentielle catégoriques, Vol. 60, No. 3, 2019, pp. 255-297. [cahiers]

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

• Marco Grandis and Robert Paré, Persistent double limits and flexible weighted limits, 2019, pdf