# nLab companion pair

Companion pairs

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

#### Higher category theory

higher category theory

# Companion pairs

## Idea

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.

## Definition

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

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

such that $\phi \circ_h \psi = id_{f'}$ and $\phi \circ_v \psi = id_{f}$, where $\circ_h$ and $\circ_v$ denote horizontal and vertical composition of 2-cells.

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

## Examples

###### Example

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

###### Example

In the double category $T$-$\mathbf{Alg}$ of algebras, lax morphisms, and colax morphisms for a 2-monad $T$, an arrow (of either sort) has a companion precisely when it is a strong (= pseudo) $T$-morphism. This is important in the theory of doctrinal adjunction.

## Properties

• 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 $D$ has a companion, then the functor $f\mapsto f'$ is a pseudofunctor $V D\to H D$ 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 $D$ (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 $D$ into a proarrow equipment, or equivalently a framed bicategory.

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

## References

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.