nLab tabulator


2-category theory

Limits and colimits



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).


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

  1. For every 2-morphism

there is a unique horizontal morphism x:XTux \colon X \rightarrow T u such that τ ux=ξ\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 11-tabulator.



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:

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 22-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)

Last revised on April 6, 2024 at 21:02:45. See the history of this page for a list of all contributions to it.