Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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 $u \colon A \nrightarrow B$ in a double category $\mathbb{D}$ consists of an object $Tu$ and a 2-morphism with the universal properties:
there is a unique horizontal morphism $x \colon X \rightarrow Tu$ such that $\tau_{u} x = \xi$.
If only the first universal property holds, we say that the vertical morphism has a $1$-tabulator.
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.
The definition of a (co)tabulator was first introduced in Section 5.3 of:
The $2$-dimensional universal property of a tabulator is stated in Section 3 of:
Other references on (co)tabulators include:
Susan Niefield, Span, cospan, and other double categories Theory and Applications of Categories_, 26, 2012. (url)
Marco Grandis, Higher Dimensional Categories. From Double to Multiple Categories, 2019. (doi:10.1142/11406)
Last revised on December 9, 2023 at 20:41:01. See the history of this page for a list of all contributions to it.