nLab companion pair

Companion pairs


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Companion pairs


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 f:ABf\colon A\to B be a vertical morphism and f:ABf'\colon A\to B 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:

A f B f ϕ id B id BandA id A id ψ f A f B \array{ A & \overset{f'}{\longrightarrow} & B \\ \mathllap{^f}\big\downarrow & \mathllap{^\phi}\swArrow & \big\downarrow\mathrlap{^id} \\ B & \underset{id}{\longrightarrow} & B } \qquad \qquad \text{and} \qquad \qquad \array{ A & \overset{id}{\longrightarrow} & A \\ \mathllap{^id}\big\downarrow & \mathllap{^\psi}\swArrow & \big\downarrow\mathrlap{^f} \\ A & \underset{f'}{\longrightarrow} & B }

such that ϕ hψ=id f\phi \circ_h \psi = id_{f'} and ϕ vψ=id f\phi \circ_v \psi = id_{f}, where h\circ_h and v\circ_v denote horizontal and vertical composition of 2-cells.

Given such a companion pair, we say that ff and ff' are companions of each other.



In the double category Sq(K)\mathbf{Sq}(K) of squares (quintets) in any 2-category KK, a companion pair is simply an invertible 2-cell between two parallel 1-morphisms of KK.


In the double category T T - Alg \mathbf{Alg} of algebras, lax morphisms, and colax morphisms for a 2-monad TT, an arrow (of either sort) has a companion precisely when it is a strong (= pseudo) TT-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 DD has a companion, then the functor fff\mapsto f' is a pseudofunctor VDHDV D\to H D 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 DD (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 DD into a proarrow equipment, or equivalently a framed bicategory.

  • Companion pairs and mate-pairs of 2-cells between them in any double category DD form a 2-category Comp(D)Comp(D). The functor Comp:DblCat2CatComp\colon DblCat \to 2Cat is right adjoint to the functor Sq:2CatDblCatSq\colon 2Cat \to DblCat sending a 2-category to its double category of squares.


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.