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A companion pair in a double category is a way of saying that a horizontal morphism and a vertical morphism are “isomorphic”, even though they do not live in the same 1-category/2-category.
A connection pair in a double category is a strictly 2-functorial choice of companion pairs for every vertical morphism.
Let be a vertical morphism and a horizontal morphism in a double category. These are said to be a companion pair if they come equipped with 2-morphisms of the form:
such that and , where and denote horizontal and vertical composition of 2-cells.
Given such a companion pair, we say that and are companions of each other.
In the double category of squares (quintets) in any 2-category , a companion pair is simply an invertible 2-cell between two parallel 1-morphisms of .
In the double category - of algebras, lax morphisms, and colax morphisms for a 2-monad , an arrow (of either sort) has a companion precisely when it is a strong (= pseudo) -morphism. This is important in the theory of doctrinal adjunction.
The horizontal (or vertical) dual of a companion pair is a conjunction.
Companion pairs (and conjunctions) have a mate correspondence generalizing the calculus of mates in 2-categories.
If every vertical arrow in some double category has a companion, then the functor is a pseudofunctor from the vertical 2-category to the horizontal one, which is the identity on objects, and locally fully faithful by the mate correspondence. A choice of companions that make this a strict 2-functor is called a connection on (an arbitrary choice of companions may be called a “pseudo-connection”). A double category with a connection is thereby equivalent to an F-category. If every vertical arrow also has a conjoint, then this makes into a proarrow equipment, or equivalently a framed bicategory.
Companion pairs and mate-pairs of 2-cells between them in any double category form a 2-category . The functor is right adjoint to the functor sending a 2-category to its double category of squares.
Marco Grandis and Robert Pare, Adjoints for double categories, NUMDAM
Robert Dawson and Robert Pare and Dorette Pronk, The Span construction, TAC.
Michael Shulman, Framed bicategories and monoidal fibrations, TAC
This latter reference explains the relationship between companions to connection pairs and foldings:
Ronnie Brown and C.B. Spencer, Double groupoids and crossed modules, Cahiers de Topologie et Géométrie Différentielle Catégoriques 17 (1976), 343–362.
Ronald Brown and Ghafar H. Mosa, Double categories, 2-categories, thin structures and connections, Theory and Application of Categories 5.7 (1999): 163-1757.
Thomas M. Fiore, Pseudo Algebras and Pseudo Double Categories, Journal of Homotopy and Related Structures, Volume 2, Number 2, pages 119-170, 2007. 51 pages.
Last revised on January 2, 2024 at 21:41:51. See the history of this page for a list of all contributions to it.