Theory and Applications of Categories, Vol. 26, No. 26, 2012, pp. 729–742.

SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES

SUSAN NIEFIELD

Abstract. Given a double category 𝔻 such that 𝔻_{0} has pushouts, we characterize oplax/lax adjunctions 𝔻 [figure] ℂospan(𝔻_{0}) for which the right adjoint is normal and restricts to the identity on 𝔻_{0}, where ℂospan(𝔻_{0}) is the double category on 𝔻_{0} whose vertical morphisms are cospans. We show that such a pair exists if and only if 𝔻 has companions, conjoints, and 1-cotabulators. The right adjoints are induced by the companions and conjoints, and the left adjoints by the 1-cotabulators. The notion of a 1-cotabulator is a common generalization of the symmetric algebra of a module and Artin-Wraith glueing of toposes, locales, and topological spaces.

1. Introduction

Double categories, first introduced by Ehresmann [2], provide a setting in which one can simultaneously consider two kinds of morphisms (called horizontal and vertical morphisms). Examples abound in many areas of mathematics. There are double categories whose objects are sets, rings, categories, posets, topological spaces, locales, toposes, quantales, and more.

There are general examples, as well. If D is a category with pullbacks, then there is a double category 𝕊pan(D) whose objects and horizontal morphisms are those of D, and vertical morphisms X_{0} [figure: •] X_{1} are spans, i.e., morphisms X_{0} [figure] X [figure] X_{1} of D, with vertical compositions via pullback. If D is a category with pushouts, then ℂospan(D) is defined dually, in the sense that vertical morphisms X_{0} [figure: •] X_{1} are cospans X_{0} [figure] X [figure] X_{1} with vertical compositions via pushout. Moreover, if D has both, then pushout of spans and pullback of cospans induce an oplax/lax adjunction (in the sense of Grandis-Paré [5, 12])

𝕊pan(D) [figure: F; G] ℂospan(D)

which restricts to the identity on the horizontal category D.

Now, suppose D is a category with pushouts, and we replace 𝕊pan(D) by a double category 𝔻 whose horizontal category is also D. Then several questions arise. Under what conditions on 𝔻 is there an oplax/lax adjunction 𝔻 [figure] ℂospan(D) which restricts to the identity on D? In particular, if 𝔻 has cotabulators, then there is an induced oplax functor F: 𝔻 [figure] ℂospan(D), and so one can ask when this functor F has a right adjoint. Similarly, if 𝔻 has companions and conjoints, there is a normal lax functor G: ℂospan(D) [figure] 𝔻 which

Received by the editors 2012-02-14 and, in revised form, 2012-11-30.

Transmitted by Robert Paré. Published on 2012-12-02.

2010 Mathematics Subject Classification: 18D05, 18A40, 18B25, 18B30, 06D22, 18D10, 15A78.

Key words and phrases: double category, lax functor, (co)span, (co)tabulator, companion, conjoint, symmetric algebra.

\begin{matrix}c \\ ◯ \end{matrix} Susan Niefield, 2012. Permission to copy for private use granted.

takes a cospan X_{0} [figure: c_{0}] X [figure: c_{1}] X_{1} to the composite X_{0} [figure: ^{c_{0}^{∗}}]_{•} X [figure: c_{1}^{∗}]_{•} X_{1}, and so one can ask when the induced functor G has a left adjoint.

We will see that these questions are related. In particular, there is an oplax/lax adjunction

𝔻 [figure: F; G] ℂospan(D)

such that G is normal and restricts to the identity on D precisely when 𝔻 has 1-cotabula-tors, conjoints, and companions (where for 1-cotabulators we drop the tetrahedron condition in the definition of cotabulator). Moreover, if D has pullbacks, then the result dualizes to show that there is oplax/lax adjunctions 𝕊pan(D) [figure] 𝔻 whose left adjoint is opnormal and restricts to the identity on D precisely when 𝔻 has 1-tabulators, conjoints, and companions.

The double categories mentioned above all have 1-cotabulators, companions, and conjoints, and the functors F and G are related to familiar constructions. In the double category of commutative rings (as well as, quantales), the functor F is given by the symmetric algebra on a bimodule and G is given by restriction of scalars. For categories (and posets), F is the collage construction. In the case of topological spaces, locales, and toposes, the functor F uses Artin-Wraith glueing.

The paper proceeds as follows. We begin in Section 2 with the double categories under consideration, followed by a review of companions and conjoints in Section 3. The notion of 1-tabulators (dually, 1-cotabulators) is then introduced in Section 4. After a brief discussion of oplax/lax adjunctions in Section 5, we present our characterization (Theorem 5.5) of those of the form 𝔻 [figure] ℂospan(D) such that right adjoint is normal and restricts to the identity on D. Along the way, we obtain a possibly new characterization (Proposition 5.3) of double categories with companions and conjoints, in the case where the horizontal category D has pushouts, as those for which the identity functor on D extends to a normal lax functor ℂospan(D) [figure] 𝔻. We conclude with the dual (Corollary 5.6) classification of oplax/lax adjunctions 𝕊pan(D) [figure] 𝔻 whose left adjoint is opnormal and restricts to the identity on D.

2. The Examples of Double Categories

Following Grandis-Paré [5, 12] and Shulman [13], we define a double category 𝔻 to be a weak internal category

[figure: 𝔻_{1} ×_{𝔻_{0}} 𝔻_{1}; 𝔻_{1}; c; 𝔻_{1}; 𝔻_{0}; d_{0}; 𝔻_{0}; 𝔻_{1}; Δ; 𝔻_{1}; 𝔻_{0}; d_{1}]

in CAT. It consists of objects (those of 𝔻_{0}), two types of morphisms: horizontal (those of 𝔻_{0}) and vertical (objects of 𝔻_{1} with domain and codomain given by d_{0} and d_{1}), and cells (morphisms of 𝔻_{1}) denoted by

[figure: X_{1}; Y_{1}; f_{1}; X_{0}; X_{1}; m; X_{0}; Y_{0}; f_{0}; Y_{0}; Y_{1}; n; •; •; φ]  (⋆)

Composition and identity morphisms are given horizontally in 𝔻_{0} and vertically via c and Δ, respectively.

The objects, horizontal morphisms, and special cells (i.e., ones in which the vertical morphisms are identities) form a 2-category called the horizontal 2-category of 𝔻. Since 𝔻 is a weak internal category in CAT, the associativity and identity axioms for vertical morphisms hold merely up to coherent isomorphism, and so we get an analogous vertical bicategory. When these isomorphisms are identities, we say that 𝔻 is a strict double category.

The following double categories are of interest in this paper.

2.1. Example. 𝕋op has topological spaces as objects and continuous maps as horizontal morphisms. Vertical morphisms X_{0} [figure: •] X_{1} are finite intersection-preserving maps O(X_{0}) [figure: •] O(X_{1}) on the open set lattices, and there is a cell of the form (⋆) if and only if f_{1}^{−1}n ⊆ mf_{0}^{−1}.

2.2. Example. 𝕃oc has locales as objects, locale morphisms (in the sense of [8]) as horizontal morphisms, and finite meet-preserving maps as vertical morphisms. There is a cell of the form (⋆) if and only if f_{1}^{∗}n ≤ mf_{0}^{∗}.

2.3. Example. 𝕋opos has Grothendieck toposes as objects, geometric morphisms (in the sense of [9]) as horizontal morphisms, finite limit-preserving functors as vertical morphisms, and natural transformations f_{1}^{∗}n [figure] mf_{0}^{∗} as cells of the form (⋆).

2.4. Example. ℂat has small categories as objects and functors as horizontal morphisms. Vertical morphisms m: X_{0} [figure: •] X_{1} are profunctors (also known as distributors and relators), i.e., functors m: X_{0}^{op} × X_{1} [figure] Sets, and natural transformations m [figure] n(f_{0}−, f_{1}−) are cells of the form (⋆).

2.5. Example. ℙos has partially-ordered sets as objects and order-preserving maps as horizontal morphisms. Vertical morphisms m: X_{0} [figure: •] X_{1} are order ideals m ⊆ X_{0}^{op} × X_{1}, and there is a cell of the form (⋆) if and only if (x_{0}, x_{1}) ∈ m ⇒ (f_{0}(x_{0}), f_{1}(x_{1})) ∈ n.

2.6. Example. For a category D with pullbacks, the span double category 𝕊pan(D) has objects and horizontal morphisms in D, and vertical morphisms which are spans in D, with

composition defined via pullback and the identities id^{•}: X [figure: •] X given by X [figure: ^{id^{X}}] X [figure: ^{id^{X}}] X.

The cells m [figure] n are commutative diagrams in D of the form

[figure: X; X_{1}; m_{1}; X_{0}; X; m_{0}; X_{0}; X_{1}; X; Y; f; X_{0}; Y_{0}; f_{0}; X_{1}; Y_{1}; f_{1}; Y; Y_{1}; n_{1}; Y_{0}; Y; n_{0}; Y_{0}; Y_{1}]

In particular, 𝕊pan(Sets) is the double category 𝕊et considered by Paré in [12], see also [1].

2.7. Example. ℂospan(D) is defined dually, for a category D with pushouts. In particular, 𝕊pan(Top) is the double category used by Grandis [3, 4] in his study of 2-dimensional topological quantum field theory.

2.8. Example. For a symmetric monoidal category V with coequalizers, the double category 𝕄od(V) has commutative monoids in V as objects and monoid homomorphisms as horizontal morphisms. Vertical morphisms from X_{0} to X_{1} are (X_{0}, X_{1})-bimodules, with composition via tensor product, and cells are bimodule homomorphisms. Special cases include the double category ℝing of commutative rings with identity and the double category ℚuant of commutative unital quantales.

3. Companions and Conjoints

Recall [6] that companions and conjoints in a double category are defined as follows. Suppose f: X [figure] Y is a horizontal morphism. A companion for f is a vertical morphism f_{∗}: X [figure: •] Y together with cells

[figure: X; Y; f; X; X; ^{id^{•}^{X}}; X; X; ^{id^{X}}; X; Y; f_{∗}; •; •; η; Y; Y; _{id_{Y}}; X; Y; f_{∗}; X; Y; f; Y; Y; ^{id^{•}^{Y}}; •; •; ε]

whose horizontal and vertical compositions are identity cells. A conjoint for f is a vertical morphism morphism f^{∗}: Y [figure: •] X together with cells

[figure: X; X; _{id_{X}}; X; X; ^{id^{X}}; X; Y; f; Y; X; f^{∗}; •; •; α; X; Y; f; Y; X; f^{∗}; Y; Y; ^{id^{Y}}; Y; Y; ^{id^{•}^{Y}}; •; •; β]

whose horizontal and vertical compositions are identity cells. We say 𝔻 has companions and conjoints if every horizontal morphism has a companion and a conjoint. Such a double category is also known as a framed bicategory [13].

If f has a companion f_{∗}, then one can show that there is a bijection between cells of the following form

[figure: ·; ·; h; ·; ·; m; ·; Y; Y; ·; n; ·; X; g; X; Y; f; •; •; φ; ·; ·; h; ·; ·; m; ·; X; g; X; ·; X; Y; f_{∗}; Y; ·; n; •; •; •; ψ]

Similarly, if f has a conjoint f^{∗}, then there is a bijection between cells

[figure: ·; Y; ·; ·; m; ·; ·; g; ·; Y; n; ·; X; h; X; Y; f; •; •; φ; ·; X; h; ·; ·; m; ·; ·; g; ·; X; ·; Y; n; Y; X; f^{∗}; •; •; •; ψ]

There are two other cases of this process (called vertical flipping in [6]) which we do not recall here as they will not be used in the following.

All of the double categories mentioned in the previous section have well known companions and conjoints. In 𝕋op, 𝕃oc, and 𝕋opos, the companion and conjoint of f are the usual maps denoted by f_{∗} and f^{∗}. For ℂat, they are the profunctors defined by f_{∗}(x, y) = Y (fx, y) and f^{∗}(y, x) = Y (y, fx), and analogously, for ℙos. If V is a symmetric monoidal category and f: X [figure] Y is a monoid homomorphism, then Y becomes an (X, Y)-bimodule and a (Y, X)-bimodule via f, and so Y is both a companion and conjoint for f. Finally, for 𝕊pan(D) (respectively, ℂospan(D)) the companion and conjoint of f are the span (respectively, cospan) with f as one leg and the appropriate identity morphism as the other.

4. 1-Tabulators and 1-Cotabulators

Tabulators in double categories were defined as follows in [5] (see also [12]). Suppose 𝔻 is a double category and m: X_{0} [figure: •] X_{1} is a vertical morphism in 𝔻. A tabulator of m is an object T together with a cell

[figure: T; X_{1}; X_{0}; T; X_{0}; X_{1}; m; τ; •]

such that for any cell

[figure: Y; X_{1}; X_{0}; Y; X_{0}; X_{1}; m; φ; •]

there exists a unique morphism f: Y [figure] T such that τf = φ, and for any commutative tetrahedron of cells

[figure: Y_{1}; X_{1}; Y_{0}; Y_{1}; n; Y_{0}; X_{0}; X_{0}; X_{1}; m; •; •; Y_{0}; X_{1}; Y_{1}; X_{0}]

there is a unique cell ξ such that

[figure: T; Y_{1}; Y_{0}; T; Y_{0}; Y_{1}; n; T; X_{1}; X_{0}; T; X_{0}; X_{1}; m; τ; ξ; •; •]

gives the tetrahedron in the obvious way.

Tabulators (and their duals cotabulators) arise in the next section, but we do not use the tetrahedron property in any of our proofs or constructions. Thus, we drop this condition in favor of a weaker notion which we call a 1-tabulator (and dually, 1-cotabulator) of a vertical morphism. When the tetrahedron condition holds, we call these tabulators (respectively, cotabulators) strong.

It is easy to show that the following proposition gives an alternative definition in terms of adjoint functors.

4.1. Proposition. A double category 𝔻 has 1-tabulators (respectively, 1-cotabulators) if and only if Δ: 𝔻_{0} [figure] 𝔻_{1} has a right (respectively, left) adjoint which we denote by Σ (respectively, Γ).

4.2. Corollary. 𝕄od(V) does not have 1-tabulators.

Proof. Suppose (V, ⊗, I) is a symmetric monoidal category. Then I is an initial object 𝕄od(V)_{0}, which is the category of commutative monoids in V. Since ΔI is not an initial object of 𝕄od(V)_{1}, we know Δ does not have a right adjoint, and so the result follows from Proposition 4.1.

The eight examples under consideration have 1-cotabulators, and we know that 1tabulators exist in all but one (namely, 𝕄od(V)). In fact, we will prove a general proposition that gives the existence of 1-tabulators from a property of 1-cotabulators (called 𝟚-glueing in [11]) shared by 𝕋op, 𝕃oc, 𝕋opos, ℂat, and ℙos. We will also show that the 1-cotabulators in ℝing are not strong, and so consideration of only strong cotabulators would eliminate this example from consideration.

The cotabulator of m: X_{0} [figure: •] X_{1} in ℂat (and similarly, ℙos), also known as the collage, is the the category X over 𝟚 whose fibers over 0 and 1 are X_{0} and X_{1}, respectively, and morphism from objects of X_{0} to those of X_{1} are given by m: X_{0}^{op} × X_{1} [figure] Sets, with the obvious cell m [figure] idX^{•}.

Cotabulators in 𝕋opos, 𝕃oc, and 𝕋op are constructed using Artin-Wraith glueing (see [9], [10], [11]). In particular, given m: X_{0} [figure: •] X_{1} in 𝕋op, the points of Γm are given by the disjoint union of X_{0} and X_{1} with U open in Γm if and only if U_{0} is open in X_{0}, U_{1} is open in X_{1}, and U_{1} ⊆ m(U_{0}), where U_{i} = U ∩ X_{i}.

For ℂospan(D), the cotabulator of X_{0} [figure: c_{0}] X [figure: c_{1}] X_{1} is given by X with cell (c_{0}, id_{X}, c_{1}). If D has pullbacks and pushouts, then the cotabulator of X_{0} [figure: s_{0}] X [figure: s_{1}] X_{1} in 𝕊pan(D) is the pushout of s_{0} and s_{1}.

The situation in 𝕄od(V) is more complicated. The following example (which generalizes to any V with a “symmetric algebra” functor) shows that ℝing has 1-cotabulators which do not satisfy the tetrahedron condition.

4.3. Example. Suppose M: X_{0} [figure: •] X_{1} is a vertical morphism in ℝing, i.e., M is an (X_{0}, X_{1})-bimodule. Then M is an X_{0} ⊗X_{1}-module, and so we can consider the symmetric X_{0} ⊗ X_{1}-algebra

SM = ⨁_{n≥0} M^{⊗n}

together with the canonical inclusion ι: M [figure] SM, where M^{⊗0} = X_{0} ⊗ X_{1} and M^{⊗n} is the tensor product over X_{0} ⊗ X_{1} of n copies of M. We claim that the corresponding cell

[figure: SM; X_{1}; X_{0}; SM; X_{0}; X_{1}; M; ι; •]

is a 1-cotabulator of M: X_{0} [figure: •] X_{1}. Since X_{0} ⊗ X_{1} is the coproduct of X_{0} and X_{1} as commutative rings and S is left adjoint to the forgetful functor from X_{0} ⊗ X_{1}-algebras to X_{0} ⊗ X_{1}-modules, we know that any cell

[figure: Y; X_{1}; X_{0}; Y; X_{0}; X_{1}; M; φ; •]

corresponds to an X_{0} ⊗ X_{1}-module homomorphism φ: M [figure] Y, and hence, a unique X_{0} ⊗ X_{1}-algebra homomorphism f: SM [figure] Y such that fι = φ. Thus, SM has the desired structure of a 1-cotabulator in ℝing.

To see that the tetrahedron condition does not hold, consider the 1-cotabulator of 0: ℤ [figure: •] ℤ, where S0 ≅ ℤ and ι is the unique homomorphism 0 [figure] ℤ. Taking ι_{1}(n) = (n, 0) and ι_{2}(n) = (0, n), the diagram

[figure: ℤ; ℤ; ℤ; ℤ; 0; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ⊕ℤ; 0; ι_{1}; =; ℤ; ℤ; ℤ; ℤ; 0; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ⊕ℤ; 0; ι_{2}]

defines a commutative tetrahedron which does not factor as

[figure: ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; 0; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ; ℤ⊕ℤ; ξ; 0; •; •]

for any homomorphism ξ: ℤ [figure] ℤ ⊕ ℤ. Thus, the 1-cotabulators are not strong in ℝing.

Now, suppose 𝔻 has 1-cotabulators and a terminal object 1, and let 𝟚 denote the image under Γ of the vertical identity morphism on 1, where Γ: 𝔻_{1} [figure] 𝔻_{0} is left adjoint to Δ (see Proposition 4.1). Then Γ induces a functor 𝔻_{1} [figure] 𝔻_{0}/𝟚, which we also denote by Γ. If this functor is an equivalence of categories, then we say 𝔻 has 𝟚-glueing.

In ℂat (and similarly, ℙos), 𝟚 is the category with two objects and one non-identity morphism. It is the Sierpinski space 𝟚 in 𝕋op, the Sierpinski locale O(𝟚) in 𝕃oc, and the Sierpinski topos S^{𝟚} in 𝕋opos. That ℂat has 𝟚-glueing is Bénabou’s equivalence cited in [14]. For 𝕋op, 𝕃oc, and 𝕋opos, the equivalence follows from the glueing construction (see [9], [10], [11]). Note that in each of these cases, 𝟚 is exponentiable in 𝔻_{0} (see [7, 9]), and so the functor 𝟚^{∗}: 𝔻_{0} [figure] 𝔻_{0}/𝟚 has a right adjoint, usually denoted by Π_{𝟚}.

4.4. Proposition. If 𝔻 has 𝟚-glueing and 𝟚 is exponentiable in 𝔻_{0}, then 𝔻 has 1-tabulators.

Proof. Consider the composite F: 𝔻_{0} [figure: 𝟚^{∗}] 𝔻_{0}/𝟚 ≃ 𝔻_{1}. Since it is not difficulty to show that Γ takes the vertical identity on X to the projection X ×𝟚 [figure] 𝟚, it follows that F = Δ, and so Δ has a right adjoint Σ, since 𝟚^{∗} does. Thus, 𝔻 has 1-tabulators by Proposition 4.1.

Applying Proposition 4.4, we see that ℂat, ℙos, 𝕋op, 𝕃oc, and 𝕋opos have 1-tabulators (which can be shown to be strong). Unraveling the construction of Σ given in the proof above, one gets the following descriptions of 1-tabulators in ℂat and 𝕋op which can be shown to be strong.

Given m: X_{0} [figure: •] X_{1} in ℂat (and similarly ℙos), the tabulator is the category of elements of m, i.e., the objects of Σm are of the form (x_{0}, x_{1}, α), where x_{0} is an object of X_{0}, x_{1} is an object of X_{1}, and α ∈ m(x_{0}, x_{1}). Morphisms from (x_{0}, x_{1}, α) to (x′_{0}, x′_{1}, α′)

in Σm are pairs (x_{0}  [figure] x′_{0}, x_{1}  [figure] x′_{1}) of morphisms, compatible with α and α′.

The tabulator of m: X_{0} [figure: •] X_{1} in 𝕋op is the set

Σm = {(x_{0}, x_{1}) | ∀U_{0} ∈ O(X_{0}), x_{1} ∈ m(U_{0}) ⇒ x_{0} ∈ U_{0}} ⊆ X_{0} × X_{1}

with the subspace topology. Note that one can directly see that this is the tabulator of m by showing that (f_{0}, f_{1}): Y [figure] X_{0} × X_{1} factors through Σm if and only if f_{1}^{−1}m ⊆ f_{0}^{−1}.

Although 𝕊pan(D) and ℂospan(D) do not have 𝟚-glueing (since Γ(id1^{•}) = 1), and so Proposition [11] does not apply, the construction of their tabulators is dual to that of their cotabulators. Finally, ℝing and ℚuant do not have 1-tabulators, as they are special cases of Corollary 4.2.

5. The Adjunction

Recall from [5] that a lax functor F: 𝔻 [figure] 𝔼 consists of functors F_{i}: 𝔻_{i} [figure] 𝔼_{i}, for i = 0, 1, compatible with d_{0} and d_{1}; together with identity and composition comparison cells

ρ_{X}: idF X^{•} [figure] F (idX^{•}) and ρ_{m,m′}: F m′^{•}F m [figure] F (m′^{•}m)

for every object X and every vertical composite m′^{•}m of 𝔻, respectively; satisfying naturality and coherence conditions. If ρ_{X} is an isomorphism, for all X, we say that F is a normal lax functor. An oplax functor is defined dually with comparison cells in the opposite direction.

An oplax/lax adjunction consists of an oplax functor F: 𝔻 [figure] 𝔼 and a lax double functor G: 𝔼 [figure] 𝔻 together with double cells

[figure: X_{1}; GF X_{1}; η_{X_{1}}; X_{0}; X_{1}; m; X_{0}; GF X_{0}; η_{X_{0}}; GF X_{0}; GF X_{1}; GF m; •; •; η_{m}; F GY_{1}; Y_{1}; ε_{Y_{1}}; F GY_{0}; F GY_{1}; F Gn; F GY_{0}; Y_{0}; ε_{Y_{0}}; Y_{0}; Y_{1}; n; •; •; ε_{n}]

satisfying naturality and coherence conditions, as well as the usual adjunction identities (see [6]).

5.1. Example. Suppose 𝔻 is a double category with 1-cotabulators and 𝔻_{0} has pushouts. Then, by Proposition 4.1, the functor Δ: 𝔻_{0} [figure] 𝔻_{1} has a left adjoint (denoted by Γ), and so there is an oplax functor F: 𝔻 [figure] ℂospan(𝔻_{0}) which is the identity on objects and horizontal morphisms, and defined on vertical morphisms and cells by

[figure: X_{1}; X′_{1}; f_{1}; X_{0}; X_{1}; m; X_{0}; X′_{0}; f_{0}; X′_{0}; X′_{1}; m′; •; •; φ; ⟼; Γm; X_{1}; i_{1}; X_{0}; Γm; i_{0}; X_{0}; X_{1}; Γm′; X′_{1}; i′_{1}; X′_{0}; Γm′; i′_{0}; X′_{0}; X′_{1}; X_{0}; X′_{0}; f_{0}; X_{1}; X′_{1}; f_{1}; Γm; Γm′; f]

where f is induced by the universal property of the 1-cotabulator. The comparison cells F (idX^{•}) [figure] idF X^{•} and F (m′^{•}m) [figure] F m′^{•}F m also arise via the universal property, with the latter given by the horizontal morphism Γ(m′^{•}m) [figure] P corresponding to the diagram

[figure: Γm′; X_{2}; X_{1}; Γm′; X_{1}; X_{2}; Γm; X_{1}; X_{0}; Γm; X_{0}; X_{1}; X_{0}; X_{0}; ^{id^{X_{0}}}; X_{2}; X_{2}; _{id_{X_{2}}}; X_{0}; X_{2}; m′^{•}m; X_{0}; X_{1}; m; X_{1}; X_{2}; m′; •; •; •; ι_{m}; ι_{m′}; Γm; P; Γm′; P]

where P is a pushout and the large rectangle is the canonical cell.

Dually, we get:

5.2. Example. Suppose 𝔻 is a double category with 1-tabulators and 𝔻_{0} has pullbacks. Then there is a lax functor F: 𝔻 [figure] 𝕊pan(𝔻_{0}) which is the identity on objects and horizontal morphisms and takes m: X_{0} [figure: •] X_{1} to the span X_{0} [figure] Σm [figure] X_{1}, where Σ is the right adjoint to Δ.

5.3. Proposition. Suppose 𝔻 is a double category and 𝔻_{0} has pushouts. Then 𝔻 has companions and conjoints if and only the identity functor on 𝔻_{0} extends to a normal lax functor G: ℂospan(𝔻_{0}) [figure] 𝔻.

Proof. Suppose 𝔻 has companions and conjoints. Then it is not difficult to show that there is a normal lax functor G: ℂospan(𝔻_{0}) [figure] 𝔻 which is the identity on objects and horizontal morphisms, and is defined on cells by

[figure: Y; Y_{1}; c_{1}; Y_{0}; Y; c_{0}; Y_{0}; Y_{1}; Y; Y′; g; Y_{0}; Y′_{0}; g_{0}; Y_{1}; Y′_{1}; g_{1}; Y′; Y′_{1}; c′_{1}; Y′_{0}; Y′; c′_{0}; Y′_{0}; Y′_{1}; ⟼; Y; Y′; g; Y_{0}; Y; ^{c_{0}^{∗}}; Y_{0}; Y′_{0}; g_{0}; Y′_{0}; Y′; ^{c′_{0}^{∗}}; Y_{1}; Y′_{1}; g_{1}; Y; Y_{1}; c_{1}^{∗}; Y; Y′; Y′; Y′_{1}; ^{c′_{1}^{∗}}; •; •; •; •; ψ_{0}; ψ_{1}]

where ψ_{0} and ψ_{1} arise from the commutativity of the squares in the cospan cell, and the definitions of companion and conjoint.

Conversely, suppose there is a normal lax functor G: ℂospan(𝔻_{0}) [figure] 𝔻 which is the identity on objects and horizontal morphisms. Then the companion and conjoint of f: X [figure] Y are defined as follows. Consider

f_{∗} = G(X [figure: f] Y [figure: ^{id^{Y}}] Y): X [figure: •] Y f^{∗} = G(Y [figure: ^{id^{Y}}]Y [figure: f] X): Y [figure: •] X

Applying G to the cospan diagrams

[figure: X; X; ^{id^{X}}; X; X; ^{id^{X}}; X; X; X; X; ^{id^{X}}; X; Y; f; X; Y; f; Y; Y; ^{id^{Y}}; X; Y; f; X; Y; Y; Y; ^{id^{Y}}; X; Y; f; X; Y; X; Y; f; Y; Y; ^{id^{Y}}; Y; Y; _{id_{Y}}; Y; Y; ^{id^{Y}}; Y; Y; ^{id^{Y}}; Y; Y]

and composing with ρ_{X} and ρ_{Y}^{−1}, we get cells

[figure: X; X; _{id_{X}}; X; X; ^{id^{•}^{X}}; X; X; ^{id^{X}}; X; X; •; •; ρ_{X}; X; Y; f; X; X; ^{id^{X}}^{∗}; X; X; ^{id^{X}}; X; Y; f_{∗}; •; G(id_{X} ,f,f); Y; Y; _{id_{Y}}; X; Y; f_{∗}; X; Y; f; Y; Y; ^{id^{Y}}^{∗}; •; •; G(f,idy,id_{Y} ); Y; Y; _{id_{Y}}; Y; Y; Y; Y; ^{id^{Y}}; Y; Y; ^{id^{•}^{Y}}; •; ρ_{Y}^{−1}]

which serve as η and ε, respectively, making f_{∗} the companion of f. Note that the horizontal and vertical identities for η and ε follow from the normality and coherence axioms of G, respectively.

Similarly, the cells α and β for f^{∗} arise from the cospan diagrams

[figure: X; X; ^{id^{X}}; X; X; ^{id^{X}}; X; X; X; Y; f; X; Y; f; X; X; _{id_{X}}; Y; X; f; Y; Y; ^{id^{Y}}; Y; X; Y; X; f; Y; Y; ^{id^{Y}}; Y; X; Y; Y; ^{id^{Y}}; Y; Y; ^{id^{Y}}; X; Y; f; Y; Y; ^{id^{Y}}; Y; Y; ^{id^{Y}}; Y; Y]

and it follows that 𝔻 has companions and conjoints.

5.4. Corollary. Suppose 𝔻 is a double category and 𝔻_{0} has pullbacks. Then 𝔻 has companions and conjoints if and only the identity functor on 𝔻_{0} extends to an opnormal oplax functor 𝕊pan(𝔻_{0}) [figure] 𝔻.

Proof. Apply Proposition 5.3 to 𝔻^{op}.

5.5. Theorem. The following are equivalent for a double category 𝔻 such that 𝔻_{0} has pushouts:

(a) there is an oplax/lax adjunction 𝔻[figure: F; G] ℂospan(𝔻_{0}) such that G is normal and restricts

to the identity on 𝔻_{0};

(b) 𝔻 has companions, conjoints, and 1-cotabulators;

(c) 𝔻 has companions and conjoints, and the induced normal lax functor ℂospan(𝔻_{0}) [figure: G] 𝔻 has an oplax left adjoint; (d) 𝔻 has companions, conjoints, and 1-cotabulators, and the induced oplax functor

𝔻 [figure: F] ℂospan(𝔻_{0}) is left adjoint to the induced normal lax functor ℂospan(𝔻_{0}) [figure: G] 𝔻;

(e) 𝔻 has 1-cotabulators and the induced oplax functor 𝔻 [figure: F] ℂospan(𝔻_{0}) has a normal lax right adjoint.

Proof. (a)⇒(b) Given F ⊣ G such that G is normal and restricts to the identity on 𝔻_{0}, we know 𝔻 has companions and conjoints by Proposition 5.3. To see that 𝔻 has 1-cotabulators, by Proposition 4.1, it suffices to show that Δ: 𝔻_{0} [figure] 𝔻_{1} has a left adjoint.

Since Δ factors as 𝔻_{0} [figure: Δ] Cospan(𝔻_{0}) [figure: G_{1}] 𝔻_{1}, by normality of G, and both these functors have left adjoints, the desired result follows. (b)⇒(c) Suppose 𝔻 has companions, conjoints, and 1-cotabulators, and consider the induced functor F: 𝔻 [figure] ℂospan(𝔻_{0}). Given m: X_{0} [figure: •] X_{1} and Y_{0} [figure: c_{0}] Y [figure: c_{1}] Y_{1}, applying the definition of 1-cotabulator, we know that every cell in ℂospan(𝔻_{0}) of the form

[figure: Γm; X_{1}; X_{0}; Γm; X_{0}; X_{1}; X_{0}; Y_{0}; f_{0}; Γm; Y; X_{1}; Y_{1}; f_{1}; Y; Y_{1}; c_{1}; Y_{0}; Y; c_{0}; Y_{0}; Y_{1}]

corresponds to a unique cell

[figure: X_{1}; Y; c_{1}f_{1}; X_{0}; X_{1}; m; X_{0}; Y; c_{0}f_{c}; Y; Y; ^{id^{Y}}; φ; •; •]

and hence, a unique cell

[figure: X_{1}; Y_{1}; f_{1}; X_{0}; X_{1}; m; X_{0}; Y_{0}; f_{0}; Y_{0}; Y_{1}; Y_{0}; Y; ^{c_{0}^{∗}}; Y; Y_{1}; c_{1}^{∗}; •; •; •; ψ]

by vertical flipping, and it follows that F ⊣ G. (c)⇒(d) Suppose 𝔻 has companions and conjoints, and the induced normal lax functor G has an oplax left adjoint F. As in the proof of (a)⇒(b), we know that 𝔻 has 1cotabulators, and so, it suffices to show that F is the induced functor. We know F takes m: X_{0} [figure: •] X_{1} to a cospan of the form X_{0} [figure: c_{0}] Γm [figure: c_{1}] X_{1}, since F is the identity on objects and the left adjoint of Δ: 𝔻_{0} [figure] Cospan(𝔻_{0}) takes X_{0} [figure: c_{0}] Γm[figure: c_{1}] X_{1} to X, and so the desired result easily follows.

(d)⇒(e) is clear. (e)⇒(a) Suppose 𝔻 has 1-cotabulators and the induced oplax functor F has a normal lax right adjoint G. Since F restricts to the identity on 𝔻_{0}, then so does G, and the proof is complete.

Note that this proof shows that if there is an oplax/lax adjunction

𝔻 [figure: F; G] ℂospan(𝔻_{0})

such that G is normal and restricts to the identity on 𝔻_{0}, then F is the induced by 1-cotabulators and G by companions and conjoints. Since the double categories in Examples 2.1–2.8 all have companions, conjoints, and 1-cotabulators, it follows that they each admits a unique (up to equivalence) oplax/lax adjunction of this form and it is induced in this manner.

Applying Theorem 5.5 to 𝔻^{op}, we get:

5.6. Corollary. The following are equivalent for a double category 𝔻 such that 𝔻_{0} has pullbacks:

(a) there is an oplax/lax adjunction 𝕊pan(𝔻_{0})[figure: G; F] 𝔻 such that G is opnormal and restricts

to the identity on 𝔻_{0};

(b) 𝔻 has companions, conjoints, and 1-tabulators;

(c) 𝔻 has companions and conjoints, and the induced opnormal oplax functor 𝕊pan(𝔻_{0}) [figure: G] 𝔻 has a lax right adjoint;

(d) 𝔻 has companions, conjoints, and 1-tabulators, and the induced lax functor 𝔻 [figure: F] 𝕊pan(𝔻_{0}) is right adjoint to the induced opnormal oplax functor 𝕊pan(𝔻_{0}) [figure: G] 𝔻;

(e) 𝔻 has 1-tabulators and the induced lax functor 𝔻 [figure: F] 𝕊pan(𝔻_{0}) has an opnormal oplax left adjoint.

As in the cospan case, if there is an oplax/lax adjunction

𝕊pan(𝔻_{0}) [figure: G; F] 𝔻

such that G is opnormal and restricts to the identity on 𝔻_{0}, then F is the induced by 1-tabulators and G is by companions and conjoints. Since the double categories in Examples 2.1–2.7 (i.e., all but 𝕄od(V) ) have companions, conjoints, and 1-tabulators, it follows they each admits is a unique (up to equivalence) oplax/lax adjunction of this form, and it is induced by companions, conjoints, and 1-tabulators.

742  SUSAN NIEFIELD

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Union College Department of Mathematics Schenectady, NY 12308

Email: niefiels@union.edu

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