nLab companion pair

Idea

A companion pair in a double category is a way of saying that a loose arrow and a tight 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 tight morphism.

Definition

Let $f\colon A\to B$ be a tight morphism and $g\colon A \nrightarrow B$ a loose arrow in a double category. These are said to be a companion pair if they come equipped with 2-morphisms of the form: such that $\phi \circ_\tau \psi = id_{g}$ and $\phi \circ_\lambda \psi = id_{f}$, where $\circ_\tau$ and $\circ_\lambda$ denote, respectively, loose and tight composition of 2-cells.

Given such a companion pair, we say that $f$ and $g$ are companions of each other. A double category for which every tight morphism admits a companion may be called companionable.

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 loose (or tight) 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 tight arrow in some double category $D$ has a companion, then the functor $f\mapsto g$ is a pseudofunctor $T D \to L D$ from the tight 2-category to the loose 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 tight 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.

• A double category $\mathbb{C}$, carried by the span of categories $C_0 \xleftarrow{s} C_1 \xrightarrow{t} C_0$ has all companions iff such span is a two-sided fibration in the sense of Street. Indeed, consider a tight map $h:A \to B$ in $\mathbb{C}$, thus a morphism in $C_0$, and the unit loose arrow $U_A$ of $A$: since $s$ is an opfibration, we obtain a square $\eta:=\mathrm{cocart}_h: U_A \Rightarrow h_* U_A$, with source (top boundary) $h$ and target (bottom boundary) necessarily $1_B$ since $s$-cartesian maps are $t$-vertical in a two-sided fibration. Dually, we get a cartesian square $\varepsilon:=\mathrm{cart}_h : h^*U_B \Rightarrow U_B$. We claim $\mathrm{cocart}_h$ and $\mathrm{cart}_h$ are, respectively, the unit and counit of a companionship between $h$ and $h':=h^*U_B = h_*U_A$. First, observe this latter equation holds since the identiy square of $h$ factors as $\varepsilon \eta$, using either universal property. This proves also the first companionship equation. As for the fact $\eta \odot \varepsilon = 1_{h'}$, it follows again from the aforementioned factorization: $\eta \odot \varepsilon$ is an $(s,t)$-vertical morphism which is thus isomorphic to the identity of $h'$ by uniqueness of the $s$-vertical, $(s,t)$-vertical, $t$-vertical factorization of a morphism in $C_1$. Vice versa, companions can be used to construct the above co/cartesian lifts, as shown in (Shulman ‘08).

Related concepts

• retrocell
• conjunction

References

• 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.

• Robert Pare, Seeing double, Talk given at FMCS 2018, (pdf)