nLab right-connected double category

Contents

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:A↛Bf \colon A \nrightarrow B has an underlying horizontal morphism Uf:A→BU 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 𝔻=(D 0,D 1)\mathbb{D} = (D_{0}, D_{1}) is right-connected if its identity-assigning map id:D 0→D 1id \colon D_{0} \rightarrow D_{1} is right adjoint to its codomain-assigning map cod:D 1→D 0cod \colon D_{1} \rightarrow D_{0}.

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

Remark

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

Let RcDbl\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 ρ:1 D 1⇒id∘cod\rho \colon 1_{D_{1}} \Rightarrow id \circ cod denote the unit of the adjunction cod⊣idcod \dashv id, and let the counit be the identity natural transformation by the axioms of an internal category. Given a vertical morphism f:A↛Bf \colon A \nrightarrow B in 𝔻\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 ρ f\rho_{f} is an identity morphism;

  2. the component of ρ\rho at an identity vertical morphism id A:A↛Aid_{A} \colon A \nrightarrow A is the identity cell on AA.

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

Examples

Example

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

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

Example

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

More generally, if CC is a category equipped with a wide subcategory BB, the double category 𝕊q(C,B)\mathbb{S}q(C, B) – whose objects and horizontal morphisms come from CC, whose vertical morphisms comes from BB, 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 CC equipped with a wide subcategory MM of monomorphisms stable under pullback along morphisms in CC. The double category ℙb(C,M)\mathbb{P}b(C, M) — whose objects and horizontal morphisms come from CC, whose vertical morphisms comes from MM, and whose cells are pullback squares — is left-connected.

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

Example

Given a category CC, let 𝕊plEpi(C)\mathbb{S}plEpi(C) denote the double category whose objects and horizontal morphisms come from CC, whose vertical morphisms are split epimorphisms with a chosen section, and whose cells are squares such that k∘f=g∘hk \circ f = g \circ h and h∘s=t∘kh \circ s = t \circ k. This double category is right-connected, since for each vertical morphism there is a cell: If CC 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 𝕊plFib\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 𝕃ens\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 CC has coproducts) are all instances of the following.

Example

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

Example

Consider an orthogonal factorization system (R,L)(R, L) on a category CC, where RR denotes the right class and LL denotes the left class. Then the double category 𝕊q(C,R)\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 𝔻=(D 0,D 1)\mathbb{D} = (D_{0}, D_{1}) is a right-connected double category, then there is a double functor U:𝔻→𝕊q(D 0)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:D 1→Sq(D 0)U_{1} \colon D_{1} \rightarrow Sq(D_{0}) denote the functor underlying the double functor U:𝔻→𝕊q(D 0)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:D 1→Sq(D 0)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 𝔻=(D 0,D 1)\mathbb{D} = (D_0, D_1) is an example of a concrete double category over D 0D_{0}.

Let Obj:RcDbl→CatObj \colon \mathbf{RcDbl} \rightarrow \mathbf{Cat} denote the 2-functor which assigns a double category 𝔻=(D 0,D 1)\mathbb{D} = (D_{0}, D_{1}) to its category D 0D_{0} of objects and horizontal morphisms. Let ℍ:Cat→RcDbl\mathbb{H} \colon \mathbf{Cat} \rightarrow \mathbf{RcDbl} be the 22-functor which assigns each category to its horizontal double category (see Example ), and let 𝕊q:Cat→RcDbl\mathbb{S}q \colon \mathbf{Cat} \rightarrow \mathbf{RcDbl} be the 22-functor which assigns each category to its double category of squares (see Example ).

Proposition

There is an adjoint triple of 22-functors:

Proof

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

Let AWFS lax\mathbf{AWFS}_{lax} denote the 22-category of algebraic weak factorization systems. There is a 22-functor AWFS lax→RcDbl\mathbf{AWFS}_{lax} \rightarrow \mathbf{RcDbl} which assigns each algebraic weak factorisation system to its double category of RR-algebras (see Example ). The following theorem characterises the essential image of the this 22-functor.

Theorem

The 22-functor AWFS lax→RcDbl\mathbf{AWFS}_{lax} \rightarrow \mathbf{RcDbl} has in its essential image exactly those right-connected double categories 𝔻=(D 0,D 1)\mathbb{D} = (D_0, D_1) for which the functor U 1:D 1→Sq(D 0)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 𝔻=(D 0,D 1)\mathbb{D} = (D_0, D_1) for which the functor U 1:D 1→Sq(D 0)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:

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

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

Last revised on February 10, 2023 at 16:39:27. See the history of this page for a list of all contributions to it.