nLab right-connected double category

Idea

A double category consists of objects, two classes of 1-morphisms (horizontal and vertical), and 2-morphisms between these.

The idea behind a right-connected double category is that each vertical morphism $f \colon A \nrightarrow B$ has an underlying horizontal morphism $U f \colon A \rightarrow B$. Therefore, the vertical morphisms should be understood as horizontal morphisms with certain properties or equipped with additional structure. Correspondingly, the main examples of right-connected double categories arise from orthogonal factorization systems or algebraic weak factorization systems, where the vertical morphisms are in the right class of the factorization system.

The concept of a right-connected double category is unrelated to the notion of a “connection on a double category”.

Definition

A double category $\mathbb{D} = (D_{0}, D_{1})$ is right-connected if its identity-assigning map $id \colon D_{0} \rightarrow D_{1}$ is right adjoint to its codomain-assigning map $cod \colon D_{1} \rightarrow D_{0}$.

Dually, a double category $\mathbb{D} = (D_{0}, D_{1})$ is left-connected if its identity-assigning map $id \colon D_{0} \rightarrow D_{1}$ is left adjoint to its domain-assigning map $dom \colon D_{1} \rightarrow D_{0}$.

Remark

This definition makes sense for any pseudo category object in a 2-category.

Let $\mathbf{RcDbl}$ denote the 2-category of right-connected double categories, unitary double functors (those which preserve vertical identities strictly), and horizontal transformations.

Unpacking the definition

We can unpack the definition of right-connectedness as follows. Let $\rho \colon 1_{D_{1}} \Rightarrow id \circ cod$ denote the unit of the adjunction $cod \dashv id$, and let the counit be the identity natural transformation by the axioms of an internal category. Given a vertical morphism $f \colon A \nrightarrow B$ in $\mathbb{D}$, the component of the unit may be depicted as follows: The triangle identities of an adjunction imply that:

1. the bottom boundary of $\rho_{f}$ is an identity morphism;

2. the component of $\rho$ at an identity vertical morphism $id_{A} \colon A \nrightarrow A$ is the identity cell on $A$.

Given a cell $\alpha$ in $\mathbb{D}$, naturality of $\rho$ states that the cells and are equal.

Examples

Example

For each category $C$, the horizontal double category $\mathbb{H}(C)$ —whose objects and horizontal morphisms come from $C$, and whose vertical morphisms and cells are identities — is right-connected.

Viewing double categories as internal categories in $\mathbf{Cat}$, we can see that $\mathbb{H}(C) = (C, C)$ is the discrete internal category on $C$, with identity-assigning map and codomain-assigning map are both the identity functor on $C$ and therefore adjoint to each other.

Example

For each category $C$, the double category of squares $\mathbb{S}q(C)$ is both right-connected and left-connected.

More generally, if $C$ is a category equipped with a wide subcategory $B$, the double category $\mathbb{S}q(C, B)$ – whose objects and horizontal morphisms come from $C$, whose vertical morphisms comes from $B$, and whose cells are commutative squares – is both right-connected and left-connected.

If we modify the above example to consider only pullback squares or pushout squares, then it is no longer right-connected or left-connected; we must also modify the classes of vertical morphisms.

Example

Consider a category $C$ equipped with a wide subcategory $M$ of monomorphisms stable under pullback along morphisms in $C$. The double category $\mathbb{P}b(C, M)$ — whose objects and horizontal morphisms come from $C$, whose vertical morphisms comes from $M$, and whose cells are pullback squares — is left-connected.

Dually, consider a category $C$ equipped with a wide subcategory $E$ of epimorphisms stable under pushout along morphisms in $C$. The double category $\mathbb{P}o(C, E)$ — whose objects and horizontal morphisms come from $C$, whose vertical morphisms comes from $E$, and whose cells are pushout squares — is right-connected.

Example

Given a category $C$, let $\mathbb{S}plEpi(C)$ denote the double category whose objects and horizontal morphisms come from $C$, whose vertical morphisms are split epimorphisms with a chosen section, and whose cells are squares such that $k \circ f = g \circ h$ and $h \circ s = t \circ k$. This double category is right-connected, since for each vertical morphism there is a cell: If $C$ has pullbacks, the codomain-assigning map of this double category is not only a left adjoint, but also a Grothendieck fibration, namely, the fibration of points.

Example

The double category $\mathbb{S}plFib$ — whose objects are categories, whose horizontal morphisms are functors, whose vertical morphisms are split Grothendieck fibrations, and whose 2-morphisms are commutative squares which preserve chosen cartesian lifts? — is right-connected.

Example

The double category $\mathbb{L}ens$ — whose objects are categories, whose horizontal morphisms are functors, and whose vertical morphisms are delta lenses — is right-connected (see Clarke, Section 3.3).

The previous three examples (although Example only if $C$ has coproducts) are all instances of the following.

Example

For each algebraic weak factorization system $(R, L)$ on a category $C$, the double category $R$-$\mathbb{A}lg$ of $R$-algebras is right-connected.

Example

Consider an orthogonal factorization system $(R, L)$ on a category $C$, where $R$ denotes the right class and $L$ denotes the left class. Then the double category $\mathbb{S}q(C, R)$ is right-connected. This is a special case of both Example and Example .

Properties

Proposition

If a double category is right-connected, then it has all 1-cotabulators.

Proposition

If $\mathbb{D} = (D_{0}, D_{1})$ is a right-connected double category, then there is a double functor $U \colon \mathbb{D} \rightarrow \mathbb{S}q(D_{0})$ with assignment given below.

A proof can be found in Bourke & Garner, Section 3.5.

Let $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ denote the functor underlying the double functor $U \colon \mathbb{D} \rightarrow \mathbb{S}q(D_{0})$ above.

Proposition

A right-connected double category is thin if and only if the functor $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ is faithful.

In the terminology of Bourke & Garner, Section 2.8, a thin right-connected double category $\mathbb{D} = (D_0, D_1)$ is an example of a concrete double category over $D_{0}$.

Let $Obj \colon \mathbf{RcDbl} \rightarrow \mathbf{Cat}$ denote the 2-functor which assigns a double category $\mathbb{D} = (D_{0}, D_{1})$ to its category $D_{0}$ of objects and horizontal morphisms. Let $\mathbb{H} \colon \mathbf{Cat} \rightarrow \mathbf{RcDbl}$ be the $2$-functor which assigns each category to its horizontal double category (see Example ), and let $\mathbb{S}q \colon \mathbf{Cat} \rightarrow \mathbf{RcDbl}$ be the $2$-functor which assigns each category to its double category of squares (see Example ).

Proposition

There is an adjoint triple of $2$-functors:

Proof

(Idea) The component of the counit of the adjunction $\mathbb{H} \dashv \mathrm{Obj}$ at a right-connected double category $\mathbb{D} = (D_{0}, D_{1})$ is given by the double functor $\mathbb{H}(D_{0}) \rightarrow \mathbb{D}$ determined by the pair of functors $(1 \colon D_{0} \rightarrow D_{0}, id \colon D_{0} \rightarrow D_{1})$. The component of the unit of the adjunction $\mathrm{Obj} \dashv \mathbb{S}q$ is given in Proposition .

Let $\mathbf{AWFS}_{lax}$ denote the $2$-category of algebraic weak factorization systems. There is a $2$-functor $\mathbf{AWFS}_{lax} \rightarrow \mathbf{RcDbl}$ which assigns each algebraic weak factorisation system to its double category of $R$-algebras (see Example ). The following theorem characterises the essential image of the this $2$-functor.

Theorem

The $2$-functor $\mathbf{AWFS}_{lax} \rightarrow \mathbf{RcDbl}$ has in its essential image exactly those right-connected double categories $\mathbb{D} = (D_0, D_1)$ for which the functor $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ is strictly monadic.

The proof combines the results in Theorem 6 and Proposition 11 of Bourke & Garner.

Corollary

An orthogonal factorization system correspond exactly those right-connected double categories $\mathbb{D} = (D_0, D_1)$ for which the functor $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ is fully faithful and strictly monadic.

This is essentially Bourke & Garner, Proposition 3.

References

The notion was first defined in Section 3.5 of:

• John Bourke, Richard Garner, Algebraic weak factorisation systems I: Accessible AWFS, Journal of Pure and Applied Algebra, 220, 2016 (arXiv:1412.6559, doi:10.1016/j.jpaa.2015.06.002)

Further discussion of right-connected double categories appears in Section 2.5 of:

• John Bourke, Richard Garner, Algebraic weak factorisation systems II: Categories of weak maps, Journal of Pure and Applied Algebra, 220, 2016 (arXiv:1412.6560, doi:10.1016/j.jpaa.2015.06.003)

The example of the double category of delta lenses first appeared in:

• Bryce Clarke, The double category of lenses, PhD thesis, Macquarie University, 2022. (doi:10.25949/22045073.v1)