nLab tabulator

Idea

A tabulator is a kind of double limit, i.e. a limit in a double category, generalizing the notion of the graph of a profunctor (also called the cocollage). Dually, the notion of a cotabulator generalizes that of the cograph of a profunctor (also called the collage).

Definition

The tabulator of a vertical morphism $u \colon A \nrightarrow B$ in a double category $\mathbb{D}$ consists of an object $Tu$ and a 2-morphism with the universal properties:

1. For every 2-morphism

there is a unique horizontal morphism $x \colon X \rightarrow T u$ such that $\tau_{u} x = \xi$.

1. The tetrahedron condition holds (see Paré, page 478).

If only the first universal property holds, we say that the vertical morphism has a $1$-tabulator.

Properties

• A double category $\mathbb{D} = (D_{0}, D_{1})$ has all $1$-tabulators (resp. $1$-cotabulators) if and only if the identity-assigning map $id \colon D_{0} \rightarrow D_{1}$ has a right adjoint (resp. left adjoint).

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

• A tabulator in a 2-category, viewed as a double category, is a power by the interval category.

References

An early reference for what is essentially the universal property of an (effective) tabulator is the definition of a tabulation, based on a similar definition in an allegory:

• Aurelio Carboni, Stefano Kasangian? and Ross Street, Bicategories of spans and relations, Journal of Pure and Applied Algebra, Vol. 33, No. 3, 1984, pp. 259-267. [doi:10.1016/0022-4049(84)90061-6]

The definition of a (co)tabulator was first introduced in Section 5.3 of:

• Marco Grandis, Robert Pare, Limits in double categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 40, 1999. (url)

The $2$-dimensional universal property of a tabulator is stated in Section 3 of:

• Robert Pare, Yoneda theory for double categories, Theory and Applications of Categories_, 25, 2011. (url)

Other references on (co)tabulators include:

• Susan Niefield, Span, cospan, and other double categories Theory and Applications of Categories_, 26, 2012. (url)

• Marco Grandis and Bob Paré, Span and cospan representations of weak double categories, Categories and General Algebraic Structures with Applications 6.Speical Issue on the Occasion of Banaschewski’s 90th Birthday (I) (2017): 85-105.

• Marco Grandis, Higher Dimensional Categories. From Double to Multiple Categories, 2019. (doi:10.1142/11406)