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 Tu 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.



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:

Last revised on December 9, 2023 at 20:41:01. See the history of this page for a list of all contributions to it.