Theory and Applications of Categories, Vol. 18, No. 20, 2007, pp. 631–653.

A UNIVERSAL PROPERTY OF THE MONOIDAL 2-CATEGORY OF COSPANS OF ORDINALS AND SURJECTIONS

M. MENNI AND N. SABADINI AND R. F. C. WALTERS

Abstract. We prove that the monoidal 2-category of cospans of ordinals and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition.

1. Introduction

Universal properties of cospan-like categories have been studied in geometry and computer science. For example, the category of 2-cobordisms has been shown to be the universal symmetric monoidal category with a symmetric Frobenius algebra (see [3] for an exposition and references). Further, Lack showed in [4] that the category of cospans of finite sets is the universal symmetric monoidal category with a symmetric separable algebra. Rosebrugh, Sabadini and Walters showed in [6] a similar property of the category of cospans of finite graphs.

The aim of this paper is to make a first step in extending these results to the 2dimensional structure of cospans. To concentrate attention we avoid symmetries and find that a very natural extension of Lack’s work characterizes the 2-category of cospans of monotone surjections between totally ordered sets, in the world of not-necessarily symmetric monoidal 2-categories.

Part of the work involves describing universal properties of bicategories of cospans. Work along these lines has been already done in [2] and [1] and it is possible that some of the results in this paper can be obtained as a byproduct of the work done in the papers just mentioned. On the other hand, our present concern allows us to make some simplifying assumptions and we have decided to prove the universal properties we need without appealing to more general work. We hope that the more concrete proofs presented here will make our work more accessible and, at the same time, allow one to see more clearly into the combinatorics of the structures involved.

Another relevant work is [7], which is however concerned with categories rather than 2-categories.

N. Sabadini and R. F. C. Walters are funded by Universitá dell’Insubria and the Italian Government PRIN project ART (Analisi di sistemi di Riduzione mediante sistemi di Transizione). M. Menni is funded by Conicet, Universitá dell’Insubria and ANPCyT.

Received by the editors 2007-07-17 and, in revised form, 2007-11-26.

Transmitted by R.J. Wood. Published on 2007-11-28.

2000 Mathematics Subject Classification: 18D5, 18D10, 18D35.

Key words and phrases: 2-categories, monoidal categories, structured objects.

\begin{matrix}c \\ ◯ \end{matrix} M. Menni and N. Sabadini and R. F. C. Walters, 2007. Permission to copy for private use granted.

2. The universal semigroup

Denote by Ord the category whose objects are ordinals n = {0 < … < n − 1} with n in ℕ and whose morphisms are monotone functions between these. (Recall that contravariant functors from Ord are sometimes called augmented simplicial objects [5].) Ordinal addition is a functor + : Ord × Ord → Ord which together with the initial object 0 induces a strict monoidal category (Ord, +, 0). This monoidal category is presented in detail in Section VII.5 of [5] where, in particular, it is proved that (1, ∇ : 1 + 1 → 1,! : 0 → 1) is the universal monoid in the sense that for any strict monoidal category (C, ⊗, I) together with a monoid (C, m : C ⊗ C → C, u : I → C) in C there exists a unique strict monoidal functor (Ord, +, 0) → (C, ⊗, I) which maps the monoid (1, ∇,!) to (C, m, u). (See Proposition 1 loc. cit.)

Now let sOrd be the subcategory of Ord determined by the surjective maps. The monoidal structure on Ord restricts to sOrd and exercise 3(b) of Section VII.5 of [5] states that (sOrd, +, 0) has the following universal property. A semigroup in (C, ⊗, I) is defined to be a pair (C, m : C ⊗ C → C) such that C is an object of C and m is associative. Then (1, ∇) is the universal semigroup.

The main results of this paper will also use the following results concerning pushouts in sOrd and their interaction with the tensor +. First let us say that a category has strict pushouts if every diagram y ← x → z in the category can be completed to a unique pushout square.

2.1. Lemma. sOrd has strict pushouts.

Proof. It is straightforward to see that the pushout (in the category of finite ordinals and all functions) of two surjections p ← m → n yields a pushout p → q ← n in the category of ordinals and surjective functions. Among these pushouts, there is a unique one making the function m → q order preserving.

The following simple fact will also be essential.

2.2. Lemma. The functor + : sOrd × sOrd → sOrd preserves pushouts.

That is, whenever the left and middle squares below are pushouts

\xymatrix{x \ar[d]_{f} \ar[r]^{g} & z \ar[d]^{p_{1}} & x \ar[d]_{f′} \ar[r]^{g′} & z \ar[d]^{p′_{1}} & x ⊗ x′ \ar[d]_{f⊗f′} \ar[r]^{g⊗g′} & z ⊗ z′ \ar[d]^{p_{1}⊗p′_{1}} \\ y \ar[r]_{p_{0}} & P & y \ar[r]_{p′_{0}} & P′ & y ⊗ y′ \ar[r]_{p_{0}⊗p′_{0}} & P ⊗ P′}

then so is the one on the right.

3. Cospans

THE MONOIDAL 2-CATEGORY OF COSPANS OF ORDINALS AND SURJECTIONS 633

In this section let C be a category with strict pushouts. Then cospan(C) has the structure of a 2-category and there are obvious y : C → cospan(C) and z : C^{op} → cospan(C) such that for every C in C, yC = zC. For every arrow f : C → C′ in C, yf is the cospan (f : C → C′ ← C′ : id) and zf is the cospan (id : C′ → C′ ← C : f).

Now let D be a 2-category. Each 2-functor cospan(C) → D induces by composition functors C → D and C^{op} → D which coincide at objects. In this section we describe what else is needed go the other way around.

3.1. Definition. A pair of functors F_{0} : C → D and F_{1} : C^{op} → D is called compatible if

1. they coincide at the level of objects (and so we write F_{0}X = F X = F_{1}X)

2. for every pushout square as on the left below,

\xymatrix{X \ar[d]_{α} \ar[r]^{β} & B \ar[d]^{p_{1}} & F X \ar[r]^{F_{0}β} & F B \\ A \ar[r]_{p_{0}} & P & F A \ar[u]^{F_{1}α} \ar[r]_{F_{0}p_{0}} & F P \ar[u]_{F_{1}p_{1}}}

the square on the right above commutes.

We write composition in ‘Pascal’ notation. So, for example, the commutative square above translates to the equation (F_{1}α); (F_{0}β) = (F_{0}p_{0}); (F_{1}p_{1}).

3.2. Lemma. The functors y and z are compatible.

Proof. Straightforward.

Another simple but important fact is the following.

3.3. Lemma. Let F_{0} : C → D and F_{1} : C^{op} → D be compatible functors and let G : D → E be a functor. Then F_{0}; G and F_{1}; G are also compatible.

Notice that in Definition 3.1 we are not requiring D to be a 2-category. For the next result let cospan_{0}(C) denote the underlying ordinary category of the 2-category cospan(C).

3.4. Lemma. Let D be a category and let F_{0} : C → D and F_{1} : C^{op} → D be functors. Then there exists a unique F : cospan_{0}(C) → D such that y; F = F_{0} and z; F = F_{1} if and only if F_{0} and F_{1} are compatible.

Proof. One direction is trivial by Lemmas 3.2 and 3.3. On the other hand, assume that F_{0} and F_{1} are compatible. The conditions y; F = F_{0} and z; F = F_{1} determine the definition of F on objects. In order to define F on 1-cells notice that every cospan p = (p_{0} : A → P ← B : p_{1}) is the result of the composition (yp_{0}); (zp_{1}). So, as F must preserve composition, we have F p = (F_{0}p_{0}); (F_{1}p_{1}) : F A → F B. So the definition of F is forced, but it remains to check that F is a functor. Identities are preserved because F_{0} and F_{1} preserve them. Concerning composition, let f = (f_{0} : X → A ← Y : f_{1}) and g = (g_{0} : Y → B ← Z : g_{1}) be a pair of composable cospans. If we let the following square be the pushout of f_{1} and g_{0}

\xymatrix{Y \ar[d]_{f_{1}} \ar[r]^{g_{0}} & B \ar[d]^{p_{1}} \\ A \ar[r]_{p_{0}} & P}

then f; g is the cospan (f_{0}; p_{0}) : X → P ← Z : (g_{1}; p_{1}). Now calculate using compatibility (and recall that F_{1} is contravariant):

F (f; g) = F_{0}(f_{0}; p_{0}); F_{1}(g_{1}; p_{1}) = (F_{0}f_{0}); (F_{0}p_{0}); (F_{1}p_{1}); (F_{1}g_{1}) =
= (F_{0}f_{0}); (F_{1}f_{1}); (F_{0}g_{0}); (F_{1}g_{1}) = (F f); (F g)

so the result is proved.

An analogous, more general result is dealt with in Example 5.3 of [4]. We prefer to be somewhat more specific as it will allow us to see more clearly how to extend the results one dimension up.

4. The extension to 2-cells

When considering 2-categories, ι_{( )} denotes the operation providing identities for horizontal and vertical composition. That is, 2-cells of the form ι_{f} act as units for vertical composition and those of the form ι_{id_{A}} act as units for horizontal composition. Also, vertical composition of 2-cells is denoted by · and the horizontal one by ∗. In all cases we write composition in ‘Pascal’ order.

4.1. Definition. Let D be a 2-category and F_{0} : C → D and F_{1} : C^{op} → D be functors. A compatible selection of 2-cells is a function τ_{( )} that assigns to each map f : X → Y in C a two cell τ_{f} : id_{F X} ⇒ (F_{0}f); (F_{1}f) in D such that:

1. τ_{id_{X}} = ι_{(id_{F X})}
2. τ_{α;β} = τ_{α} · ((F_{0}α) ∗ τ_{β} ∗ (F_{1}α))

Graphically, this can be stated as follows: if α : X → Y and β : Y → Z then the two 2-cells id_{F X} ⇒ (F_{0}(α; β)); (F_{1}(α; β)) below

are the same.

[figure: F X; F_{0}α; id; τ_{α;β}; F X; F X; F_{0}α; id; τ_{α}; F X; F Y; F_{0}β; F Z; F_{1}β; F Y; F_{1}α; F Y; F_{0}β; τ_{β}; F Y; F_{1}α; F Z; F_{1}β]

3. for every pushout in C as below

\xymatrix{X \ar[d]_{α} \ar[r]^{β} & B \ar[d]^{p_{1}} \\ A \ar[r]_{p_{0}} & P}

the following identities hold:

τ_{α} ∗ (F_{0}β) = (F_{0}β) ∗ τ_{p_{1}} (F_{1}α) ∗ τ_{β} = τ_{p_{0}} ∗ (F_{1}α).

The idea is, of course, that a compatible selection of 2-cells is exactly what is needed to extend Lemma 3.4 to two dimensions. But before we prove the result let us prove a couple of technical lemmas.

First notice that each α : A → B in C induces a 2-cell \overline{α} : id_{A} ⇒ (yα); (zα).

4.2. Lemma. The assignment α ↦ \overline{α} is compatible with y and z.

Proof. Straightforward.

Now we need a result analogous to Lemma 3.3.

4.3. Lemma. Let C be a category and let D and E be 2-categories. Also, let F_{0} : C → D and F_{1} : C^{op} → D be compatible functors and let G : D → E be a 2-functor. If τ_{( )} is a selection of 2-cells compatible with F_{0} and F_{1} then Gτ_{( )} is compatible with F_{0}; G and F_{0}; G.

Proof. Easy.

4.4. Lemma. Let p_{0} and p_{1} be the pushout of α and β as in Definition 4.1 and let γ = α; p_{0} = β; p_{1}. Then τ_{α} ∗ τ_{β} = τ_{γ}.

Proof. Calculate:

τ_{α;p_{0}} = τ_{α} · ((F_{0}α) ∗ τ_{p_{0}} ∗ (F_{1}α)) = τ_{α} · ((F_{0}α) ∗ (F_{1}α) ∗ τ_{β}) =
= (τ_{α} ∗ ι_{X}) · ([(F_{0}α) ∗ ι_{A} ∗ (F_{1}α)] ∗ τ_{β}) = (τ_{α} · [(F_{0}α) ∗ τ_{id_{A}} ∗ (F_{1}α)]) ∗ (ι_{X} · τ_{β}) =
= τ_{(α;id_{A})} ∗ τ_{β} = τ_{α} ∗ τ_{β}

We can now prove the following.

4.5. Proposition. Let D be a 2-category, let F_{0} : C → D and F_{1} : C^{op} → D be functors and let τ_{( )} be a function assigning a 2-cell of D to each map in C. Then the following are equivalent:

1. there exists a unique 2-functor F : cospan(C) → D such that y; F = F_{0}, z; F = F_{1} and τ_{( )} = F \overline{( )} hold; (here and for the rest of the paper F \overline{( )} is denoting the operation that to each 1-cell f in C assigns the 2-cell F (\overline{f}))

2. F_{0} and F_{1} are compatible and τ_{( )} is a selection of 2-cells that is compatible with them.

Proof. Assume that the first item holds. Lemma 3.4 shows that y; F = F_{0} and z; F = F_{1} are compatible. Lemmas 4.2 and 4.3 show that τ_{( )} is a selection of 2-cells that is compatible with them.

To prove the converse notice that we can apply Lemma 3.4 again to conclude that there exists a unique ordinary functor F : cospan_{0}(C) → D such that y; F = F_{0} and z; F = F_{1}. So we are left to show that this F extends uniquely to a 2-functor in such a way that τ_{( )} = F \overline{( )} holds.

First assume that the functor F does extend to a 2-functor and consider an arbitrary 2-cell α as below.

\xymatrix{ & A \ar[dd]^{α} \\ X \ar@{-->}[ur]^{f_{0}} \ar[dr]_{g_{0}} &  & Y \ar@{-->}[ul]_{f_{1}} \ar[dl]^{g_{1}} \\  & B}

Consider cospans f = (f_{0} : X → A ← Y : f_{1}) and g = (g_{0} : X → A ← Y : g_{1}). Then it is not difficult to see that, for α considered as a 2-cell f ⇒ g, α = (yf_{0}) ∗ \overline{α} ∗ (zf_{1}). So F (α : f ⇒ g) = F (yf_{0}) ∗ F \overline{α} ∗ F (zf_{1}) and hence the definition of F at the level of 2-cells is completely determined by F (α : f ⇒ g) = (F_{0}f_{0}) ∗ τ_{α} ∗ (F_{1}f_{1}).

Finally, we prove that if we define F (α : f ⇒ g) to be the 2-cell (F_{0}f_{0}) ∗ τ_{α} ∗ (F_{1}f_{1}) as above then we do obtain a 2-functor.

1. F ι_{f} = ι_{F f}
F ι_{f} = (F_{0}f_{0}) ∗ τ_{id_{A}} ∗ (F_{1}f_{1}) = (F_{0}f_{0}) ∗ ι_{id_{A}} ∗ (F_{1}f_{1}) =
= (F_{0}f_{0}) ∗ (F_{1}f_{1}) = ι_{(F_{0}f_{0});(F_{1}f_{1})} = ι_{(F f)}

2. Consider maps α : A → B and β : B → C inducing 2-cells in unique possible way (starting from f). Then calculate

F (α · β) = (F_{0}f_{0}) ∗ τ_{(α;β)} ∗ (F_{1}f_{1}) =

= (F_{0}f_{0}) ∗ (τ_{α} · ((F_{0}α) ∗ τ_{β} ∗ (F_{1}α))) ∗ (F_{1}f_{1}) =
= (ι_{(F_{0}f_{0})} · ι_{(F_{0}f_{0})}) ∗ (τ_{α} · ((F_{0}α) ∗ τ_{β} ∗ (F_{1}α))) ∗ (F_{1}f_{1}) =
= [(ι_{(F_{0}f_{0})} ∗ τ_{α}) · (ι_{(F_{0}f_{0})} ∗ ((F_{0}α) ∗ τ_{β} ∗ (F_{1}α)))] ∗ (F_{1}f_{1}) =
= [(F_{0}f_{0}) ∗ τ_{α}) · ((F_{0}f_{0}) ∗ (F_{0}α) ∗ τ_{β} ∗ (F_{1}α))] ∗ (ι_{(F_{1}f_{1})} · ι_{(F_{1}f_{1})}) =

= [(F_{0}f_{0}) ∗ τ_{α} ∗ (F_{1}f_{1})] · [(F_{0}f_{0}) ∗ (F_{0}α) ∗ τ_{β} ∗ (F_{1}α)) ∗ (F_{1}f_{1})] = (F α) · (F β)

3. Preservation of horizontal composition. Suppose we have 2-cells α : f ⇒ f′ and β : g ⇒ g′ as in the diagram below.

\xymatrix{ & A \ar[dd]^{α} &  &  & B \ar[dd]^{β} \\ X \ar[ur]^{f_{0}} \ar[dr]_{f′_{0}} &  & Y \ar[ul]_{f_{1}} \ar[dl]^{f′_{1}} & Y \ar[ur]^{g_{0}} \ar[dr]_{g′_{0}} &  & Z \ar[ul]_{g_{1}} \ar[dl]^{g′_{1}} \\  & A′ &  &  & B′}

In order to calculate α ∗ β calculate the following pushout and resulting map (every small square is a push out).

[figure: Y; f_{1}; g_{0}; B; p_{1}; β; B′; r; A; α; p_{0}; P; α∗β; α′; β′; R; p′_{1}; A′; q; Q; p′_{0}; P′]

Now use Lemma 4.4 to calculate:

F (α ∗ β) = F_{0}(f_{0}; p_{0}) ∗ τ_{α∗β} ∗ F_{1}(g_{1}; p_{1}) =
= (F_{0}f_{0}) ∗ (F_{0}p_{0}) ∗ τ_{α′} ∗ τ_{β′} ∗ (F_{1}p_{1}) ∗ (F_{1}g_{1}) =
= (F_{0}f_{0}) ∗ τ_{α} ∗ (F_{0}p_{0}) ∗ (F_{1}p_{1}) ∗ τ_{β} ∗ (F_{1}g_{1}) =
= (F_{0}f_{0}) ∗ τ_{α} ∗ (F_{1}f_{1}) ∗ (F_{0}g_{0}) ∗ τ_{β} ∗ (F_{1}g_{1}) = (F α) ∗ (F β)

5. Adding the monoidal structure

In this section let (C, ⊕, 0) be a strict monoidal category with strict pushouts. We have already seen that cospan(C) is a 2-category. We want to ‘extend’ the tensor ⊕ on C to one on cospan(C).

5.1. Lemma. There exists a 2-iso cospan(C) × cospan(C) → cospan(C × C) such that the following diagram commutes.

\xymatrix{C × C \ar[dr]_{y} \ar[r]^{y×y} & cospan(C) × cospan(C) \ar[d]^{≅} & C^{op} × C^{op} \ar[l]_{z×z} \ar[d]^{≅} \\  & cospan(C × C) & _{(C × C)_{op}} \ar[l]_{z}}

Proof. The obvious one.

So we need only build a 2-functor ⊕ : cospan(C × C) → cospan(C) with the right properties.

5.2. Lemma. The functors

\xymatrix{C × C \ar[r]^{⊕} & C \ar[r]^{y} & cospan(C) & (C × C)^{op} \ar[r]^{^{⊕^{op}}} & C^{op} \ar[r]^{z} & cospan(C)}

are compatible if and only if ⊕ preserves pushouts. Moreover, in this case (α, β) ↦ \overline{α ⊕ β} is a compatible selection of 2-cells.

Proof. The functors coincide at the level of objects. Now, a pushout in C × C is a pair of pushouts α; p_{0} = β; p_{1} and α′; p′_{0} = β′; p′_{1} in C. The compatibility condition reduces, in this case, to (α ⊕ α′); (p_{0} ⊕ p′_{0}) = (β ⊕ β′); (p_{1} ⊕ p′_{1}) being a pushout. So the first part of the result follows.

For the second part denote let σ_{(α,β)} = \overline{α ⊕ β} and recall that \overline{( )} is a compatible selection of 2-cells (Lemma 4.2). It is easy to show that σ_{(id_{X},id_{Y} )} = ι_{id_{X⊕Y}}. In order to check the second condition calculate:

σ_{(α,α′);(β,β′)} = σ_{((α;β),(α′;β′))} = \overline{(α; β) ⊕ (α′; β′)} = \overline{(α ⊕ α′); (β ⊕ β′)} =

= \overline{(α ⊕ α′)} · (y(α ⊕ α′) ∗ \overline{(β ⊕ β′)} ∗ z(α ⊕ α′)) = σ_{(α,α′)} · (y(α ⊕ α′) ∗ σ_{(β,β′)} ∗ z(α ⊕ α′))

In order to check the final condition assume that we have a pushout in C × C as on the left below

[figure: (α,α′); (β,β′); (p_{1},p′_{1}); α⊕α′; β⊕β′; p_{1}⊕p′_{1}; (p_{0},p_{1}); p_{0}⊕p_{1}]

then the square on the right above is a pushout because ⊕ preserves them. Then calculate:

σ_{(α,α′)} ∗ y(β ⊕ β′) = \overline{α ⊕ α′} ∗ y(β ⊕ β′) = y(β ⊕ β′) ∗ \overline{p_{1} ⊕ p′_{1}} = y(β ⊕ β′) ∗ σ_{(p_{1},p′_{1})}

The equation z(α ⊕ α′) ∗ σ_{(β,β′)} = σ_{(p_{0},p′_{0})}z(α ⊕ α′) is dealt with similarly.

5.3. Proposition. Let (C, ⊕, 0) be a strict monoidal category with strict pushouts then the following are equivalent:

1. there exists a unique 2-functor

⊕ : cospan(C) × cospan(C) → cospan(C)

such that the following diagrams commute

\xymatrix{C × C \ar[d]_{y×y} \ar[r]^{⊕} & C \ar[d]^{y} \\ cospan(C) × cospan(C) \ar[r]_{⊕} & cospan(C)}

\xymatrix{C^{op} × C^{op} \ar[d]_{z×z} \ar[r]^{^{⊕^{op}}} & C^{op} \ar[d]^{z} \\ cospan(C) × cospan(C) \ar[r]_{⊕} & cospan(C)}

and such that \overline{α} ⊕ \overline{β} = \overline{α ⊕ β}.

2. The functor ⊕ : C × C → C preserves pushouts.

In this case, the structure (cospan(C), ⊕, 0) is a monoidal 2-category and the functors y and z extend to strict monoidal (C, ⊕, 0) → cospan(C) and (C^{op}, ⊕, 0) → cospan(C) respectively.

Proof. Lemma 5.2 together with Proposition 4.5 show that if ⊕ : C × C → C preserves pushouts then there exists a 2-functor ⊕ : cospan(C × C) → cospan(C) satisfying a number of properties which, after precomposing with the isomorphism of Lemma 5.1, turn out to be exactly the ones in the statement of the present result. The rest is trivial by strictness.

In particular:

5.4. Corollary. (cospan(sOrd), +, 0) is a strict monoidal 2-category.

5.5. Monoidal 2-functors from cospan(C). In this section let (C, ⊕, 0) be a strict monoidal category such that C has strict pushouts and ⊕ preserves these. By Proposition 5.3 we have the strict monoidal 2-category (cospan(C), ⊕, 0).

5.6. Lemma. Let (D, ⊗, I) be a strict monoidal 2-category, consider compatible functors F_{0} : C → D and F_{1} : C^{op} → D and let τ_{( )} be a compatible selection of 2-cells. Then the induced 2-functor F : cospan(C) → D is strict monoidal (cospan(C), ⊕, 0) → (D, ⊗, I) if and only if F_{0} and F_{1} are strict monoidal and τ_{α⊕β} = τ_{α} ⊗ τ_{β}.

Proof. Assume that F is a strict monoidal 2-functor. Then clearly F_{0} = y; F is a strict monoidal functor and similarly for F_{1}. To check the condition on the selection of 2-cells just calculate:

τ_{α⊕β} = F (\overline{α ⊕ β}) = F (\overline{α} ⊕ \overline{β}) = F \overline{α} ⊗ F \overline{β} = τ_{α} ⊗ τ_{β}

Conversely, assume that the conditions stated for F_{0}, F_{1} and τ_{( )} hold. Clearly F (X ⊕ Y) = F X ⊗ F Y because F and F_{0} coincide at the level of objects. Now let f = (f_{0} : X → A ← Y : f_{1}) and f′ = (f′_{0} : X′ → A′ ← Y′ : f′_{1}) be 1-cells. Then calculate

F (f ⊕ f′) = F (f_{0} ⊕ f′_{0} : X ⊕ X′ → A ⊕ A′ ← Y ⊕ Y′ : f_{1} ⊕ f′_{1}) =
= F_{0}(f_{0} ⊕ f′_{0}); F_{1}(f_{1} ⊕ f′_{1}) = ((F_{0}f_{0}) ⊗ (F_{0}f′_{0})); ((F_{1}f_{1}) ⊗ (F_{1}f′_{1})) =
= ((F_{0}f_{0}); (F_{1}f_{1})) ⊗ ((F_{0}f′_{0}); (F_{1}f′_{1})) = (F f) ⊗ (F f′)

Finally, consider a 2-cells α from f and β from f′ and calculate using that ⊗ is a 2-functor:

F (α ⊕ β) = F_{0}(f_{0} ⊕ f′_{0}) ∗ τ_{α⊕β} ∗ F_{1}(f_{1} ⊕ f′_{1}) =
= ((F_{0}f_{0}) ⊗ (F_{0}f′_{0})) ∗ (τ_{α} ⊗ τ_{β}) ∗ ((F_{1}f_{1}) ⊗ (F_{1}f′_{1})) =
= ((F_{0}f_{0}) ∗ τ_{α} ∗ (F_{1}f_{1})) ⊗ ((F_{0}f′_{0}) ∗ τ_{β} ∗ (F_{1}f′_{1})) = (F α) ⊗ (F β)

6. Separable pre-algebras

In this section we introduce the fundamental 1-dimensional structure to be studied in the paper.

6.1. Definition. Let (D, ⊗, I) be a strict monoidal category. A bi-semigroup (X, ∇, Δ) is an object X in D together with morphisms ∇ : X ⊗ X → X and Δ : X → X ⊗ X such that (X, ∇) is a semigroup and (X, Δ) is a ‘co-semigroup’ in the sense that Δ is coassociative.

It is useful to have a graphical notation for expressions involving ∇ and Δ. A couple of examples will suffice to introduce it. Consider a bi-semigroup (X, ∇, Δ). The identity on X will be denoted by a straight line. On the other hand, id_{X} ⊗ id_{X} will be denoted by two parallel horizontal lines. More importantly, ∇ will be denoted as in the left diagram below and Δ will be denoted as on the right above.

So that, for example, the diagram

[figure]

represents the expression (id ⊗ ∇); ∇; Δ.

6.2. Definition. Let (D, ⊗, I) be a strict monoidal category. A separable pre-algebra is a bi-semigroup (D, ∇, Δ) in D such that:

1. (Separable) Δ; ∇ = id : D → D

\xymatrix{D \ar[d]_{Δ} \ar[dr]^{id} \\ D ⊗ D \ar[r]_{∇} & D}

2. (Frobenius) (Δ ⊗ id_{D}); (id_{D} ⊗ ∇) = ∇; Δ = (id_{D} ⊗ Δ); (∇ ⊗ id_{D})

\xymatrix{D ⊗ D \ar[d]_{Δ⊗id} \ar[r]^{∇} & D \ar[d]^{Δ} & D ⊗ D \ar[d]_{id⊗Δ} \ar[r]^{∇} & D \ar[d]^{Δ} \\ D ⊗ D ⊗ D \ar[r]_{id⊗∇} & D ⊗ D & D ⊗ D ⊗ D \ar[r]_{∇⊗id} & D ⊗ D}

Graphically, separability can be expressed as saying that the following two diagrams

[figure]

are equal. The authors have found it useful to think of separability as allowing to pop the ‘bubble’ on the left.

On the other hand, Frobenius says that the two diagrams below

[figure]

which represent (Δ ⊗ id_{D}); (id_{D} ⊗ ∇) and (id_{D} ⊗ Δ); (∇ ⊗ id_{D}) respectively, are equal to

[figure]

which represents ∇; Δ.

6.3. Remark. Notice that the notion of separable pre-algebra can be described as that of commutative separable algebra (in the sense of Definition 2.7 in [6]) but where the underlying (co)monoid is not required to have (co)unit.

6.4. Lemma. If we denote the cospan (id : 1 → 1 ← 1 + 1 : ∇) by Δ : 1 → 1 + 1 then (1, ∇, Δ) is a separable pre-algebra in cospan_{0}(sOrd).

Proof. This is a simple exercise left for the reader. But it is important to mention now that this separable pre-algebra plays an important role in everything that follows.

7. A universal property of (1, ∇, Δ)

The universal property we discuss in this section was independently observed by Lack on the one hand [4] and by Rosebrugh, Sabadini and Walters on the other [6].

It is important to recall (see Lemma in Chapter VII.5 of [5]) the fact that every surjection in sOrd can be factored in a unique way as a composition (satisfying certain conditions) of maps (id+∇+id). (The conditions ensuring uniqueness will not be relevant for us here.)

Let F_{0} : C → D and F_{1} : C^{op} → D be functors agreeing on objects. We say that F_{0} and F_{1} indulge a commutative square

[figure: α; β; p_{1}; p_{0}]

if, as in Definition 3.1, (F_{1}α); (F_{0}β) = (F_{0}p_{0}); (F_{1}p_{1}). (Notice that there is a handedness in this notion. The fact that the functors indulge the square above does not seem to imply that it indulges the square obtained by flipping the same square along its diagonal. That is (F_{1}α); (F_{0}β) = (F_{0}p_{0}); (F_{1}p_{1}) does not seem to imply (F_{1}β); (F_{0}α) = (F_{0}p_{1}); (F_{1}p_{0}).)

7.1. Lemma. If F_{0} and F_{1} indulge the two squares below separately

[figure: α; β; α′; β′; p′_{1}; p_{0}; p′_{0}]

then they indulge the rectangle.

Proof. Trivial.

There is also a ‘vertical’ version which we shall use when necessary.

7.2. Lemma. [See [4]] Let F_{0} : sOrd → D and F_{1} : sOrd^{op} → D be monoidal functors agreeing on objects. Then they are compatible if and only if they indulge the following pushout squares

\xymatrix{1 + 1 \ar[d]_{∇} \ar[r]^{∇} & 1 \ar[d]^{id} & 1 + 1 + 1 \ar[d]_{∇+1} \ar[r]^{1+∇} & 1 + 1 \ar[d]^{∇} & 1 + 1 + 1 \ar[r]^{∇+1} \ar[d]_{1+∇} & 1 + 1 \ar[d]^{∇} \\ 1 \ar[r]_{id} & 1 & 1 + 1 \ar[r]_{∇} & 1 & 1 + 1 \ar[r]_{∇} & 1}

Proof. One direction is trivial. Consider a pushout of the form below

\xymatrix{k \ar[d]_{f} \ar[r]^{g} & n \ar[d]^{p_{1}} \\ m \ar[r]_{p_{0}} & t}

If f or g are identities then the square is trivially indulged. So we can assume that f and g are non-trivial compositions of the type recalled above. Say, f = (l_{0} + ∇ + l_{1}); f′ and g = (l′_{0} + ∇ + l′_{1}); g′. The idea of the proof is to split the pushout into four smaller pushouts as below.

[figure: k; l_{0}+∇+l_{1}; l′_{0}+∇+l′_{1}; l′_{0} + 1 + l′_{1}; g′; n; l_{0} + 1 + l_{1}; f′; m; t]

The inductive hypothesis can deal with the two bottom squares and the top right one. If we can prove that the top left one is indulged then Lemma 7.1 implies that the big pushout is indulged.

So, concerning the top left pushout, the following things can happen:

1. l_{0} + 2 ≤ l′_{0}, that is, f’s first ∇ is strictly to the left of g’s,

2. l_{0} + 1 = l′_{0}, that is, f’s first ∇ “touches” g’s but f and g do not start in the same way,

3. l_{0} = l′_{0}, that is, f and g start in the same way,

4. l_{0} = l′_{0} + 1, analogous to the first item but to the right,

5. l_{0} ≥ l′_{0} + 2, analogous to the second item.

Consider the first case. We let k = k_{0} + 2 + k_{1} + 2 + k_{2}, f = (k_{0} + ∇ + k′_{1}); f′ and g = (k′_{0} + ∇ + k_{2}); g′ where k′_{0} = k_{0} + 2 + k_{1} and k′_{1} = k_{1} + 2 + k_{2}. Then the pushout is calculated as below:

\xymatrix{k_{0} + 2 + k_{1} + 2 + k_{2} \ar[d]_{k_{0}+∇+k′_{1}} \ar[r]^{k′_{0}+∇+k_{2}} & k_{0} + 2 + k_{1} + 1 + k_{2} \ar[d]^{k_{0}+∇+k_{1}+1+k_{2}} \\ k_{0} + 1 + k_{1} + 2 + k_{2} \ar[r]_{k_{0}+1+k_{1}+∇+k_{2}} & k_{0} + 2 + k_{1} + 1 + k_{2}}

and it is indulged because it is the sum of trivial pushouts (that are indulged) and moreover F_{0} and F_{1} are monoidal so the tensor of indulged squares is indulged.

Consider now the second case. Let k = k_{0} + 1 + 1 + 1 + k_{1}, f = (k_{0} + ∇ + 1 + k_{1}); f′ and g = (k_{0} + 1 + ∇ + k_{1}); g′. In this case the pushout in question is calculated as follows

\xymatrix{k_{0} + 1 + 1 + 1 + k_{1} \ar[d]_{k_{0}+∇+1+k_{1}} \ar[r]^{k_{0}+1+∇+k_{1}} & k_{0} + 1 + 1 + k_{1} \ar[d]^{k_{0}+∇+k_{1}} \\ k_{0} + 1 + 1 + k_{1} \ar[r]_{k_{0}+∇+k_{1}} & k_{0} + 1 + k_{1}}

Again, the pushout is a sum of two squares that are trivially indulged and one that is indulged by assumption.

In order to deal with the third case let k = k_{0} + 1 + 1 + k_{1}, f = (k_{0} + ∇ + k_{1}); f′ and g = (k_{0} + ∇ + k_{1}); g′. In this case the pushout in question is calculated as follows

\xymatrix{k_{0} + 1 + 1 + k_{1} \ar[d]_{k_{0}+∇+k_{1}} \ar[r]^{k_{0}+∇+k_{1}} & k_{0} + 1 + k_{1} \ar[d]^{id} \\ k_{0} + 1 + k_{1} \ar[r]_{id} & k_{0} + 1 + k_{1}}

Again, the pushout is a sum of two squares that are trivially indulged and one that is indulged by assumption. The remaining two cases are analogous.

7.3. Corollary. [See [4] and [6]] Let (D, ⊗, I) be a strict monoidal category with a separable pre-algebra (D, ∇, Δ). Then there exists a unique strict monoidal functor

(cospan_{0}(sOrd), +, 0) → (D, ⊗, I)

mapping (1 + 1, ∇, Δ) to (D, ∇, Δ).

Proof. The semigroup (D, Δ) is essentially the same thing as a strict monoidal functor F_{0} : sOrd → D (mapping ∇ to ∇) while the co-semigroup (D, Δ) is essentially the same thing as a strict monoidal F_{1} : sOrd^{op} → D (mapping ∇ to Δ). As F_{0} and F_{1} are strict monoidal and coincide on 1, they agree on objects. So we are left to prove that F_{0} and F_{1} are compatible. By Lemma 7.2 it is enough to check that F_{0} and F_{1} indulge three pushout squares. But notice that indulgence of these squares is equivalent to Separability and Frobenius.

7.4. An alternative proof of Corollary 7.3. Corollary 7.3 can be interpreted as saying that the free monoidal category with a separable pre-algebra is cospan_{0}(sOrd). In this short section we sketch a ‘graphical’ proof which makes a lot more evident the relation between the result and the calculation of colimits.

What is the free monoidal category generated by ∇ and Δ subject to the equations in Definition 6.2? First, given only ∇ we can build diagrams of the form

\overline{}

⋮  ⋮  ⋮

\overline{}

The associative law says that the order of applying ∇s does not matter so with only ∇s we can can build exactly surjective monotone functions. Similarly, using only Δ we can produce exactly the reverses of monotone surjections. So using both we can produce cospans of monotone surjections. But perhaps we can produce more? The answer is no. If in an expression of ∇s and Δs a Δ occurs to the left of a ∇ then only 4 cases can occur. The first one is when the Δ and the ∇ do not interact:

⋮  ⋮  ⋮

\overline{}

The second case is given by the bubble as drawn after Definition 6.2. The third and fourth cases are given by the two diagrams representing the expressions in the Frobenius condition and drawn below the bubble after Definition 6.2. In all four cases, the Δs can be moved to the right of the ∇s. In the first case trivially, in the second by popping the bubble (separability) and in the third and fourth cases by Frobenius. So the free monoidal category with a separable pre-algebra is cospan_{0}(sOrd).

7.5. Remark. It is important to notice that the process of moving Δs to the right of ∇s is really calculating the pushout involved in the composition of cospans.

In Section 8.10 we add 2-dimensional data so that the free monoidal 2-category on this data is cospan(sOrd). But first let us extract some more information from Lemma 7.2.

7.6. Monoidal 2-functors from cospan(sOrd). Here we characterize when two functors from sOrd to a 2-category are compatible. Let us say that a selection of 2-cells τ_{( )} indulges a square α; p_{0} = β; p_{1} if the two equations in Definition 4.1 relating the square and τ_{( )} hold.

7.7. Lemma. Let F_{0} : sOrd → D and F_{1} : sOrd^{op} → D be compatible monoidal functors. Let τ_{( )} be a selection of 2-cells satisfying the first two conditions of Definition 4.1. Then τ_{( )} is a compatible selection of 2-cells if and only if it indulges the squares in the statement of Lemma 7.2.

Proof. Analogous to that of Lemma 7.2.

8. Adjoint bi-semigroups

In this section we introduce what we believe are the right liftings to 2-dimensions of the Frobenius and separability conditions.

8.1. Definition. Let (D, ⊗, I) be a strict monoidal 2-category. An adjoint bi-semigroup (X, ∇, Δ, η, ϵ) is a bi-semigroup (X, ∇, Δ) in D_{0} together with 2-cells η : id_{X⊗X} ⇒ ∇; Δ and ϵ : Δ; ∇ ⇒ id_{X} witnessing that ∇ ⊣ Δ.

We now lift the conditions of separability and Frobenius to the level of adjoint bisemigroups. We first deal with Frobenius.

8.2. Frobenius adjoint bi-semigroups. In order to justify the definition consider first the following result.

8.3. Lemma. Let X = (X, ∇, Δ, η, ϵ) be an adjoint bi-semigroup. If the (1-dimensional) structure (X, ∇, Δ) satisfies Frobenius as a bi-semigroup then the following two items are equivalent:

1. the mates of the associative laws

\xymatrix{X ⊗ X ⊗ X \ar[d]_{∇⊗X} \ar[r]^{X⊗∇} & X ⊗ X \ar[d]^{∇} & X ⊗ X ⊗ X \ar[d]_{X⊗∇} \ar[r]^{∇⊗X} & X ⊗ X \ar[d]^{∇} \\ X ⊗ X \ar[r]_{∇} & X & X ⊗ X \ar[r]_{∇} & X}

are identity 2-cells

2. (η ⊗ X) ∗ (X ⊗ ∇) = (X ⊗ ∇) ∗ η and (X ⊗ η) ∗ (∇ ⊗ X) = (∇ ⊗ X) ∗ η.

Proof. Consider the mate of one of the associative laws

\xymatrix{X ⊗ X \ar[dr]_{id} \ar[r]^{Δ⊗X} & X ⊗ X ⊗ X \ar[d]_{⟸}^{∇⊗X} \ar[r]^{X⊗∇} & X ⊗ X \ar[d]^{∇} \ar[dr]^{id} \\  & X ⊗ X \ar[r]_{∇} & X \ar[r]^{⟸}_{Δ} & X ⊗ X}

where inside the triangles we have the 2-cells given by ϵ ⊗ X : (Δ ⊗ X); (∇ ⊗ X) ⇒ id and η : id ⇒ ∇; Δ. Notice that the outside of this diagram is one of the Frobenius laws. Now assume that (η ⊗ D) ∗ (D ⊗ ∇) = (D ⊗ ∇) ∗ η and calculate:

[(Δ ⊗ X) ∗ (X ⊗ ∇) ∗ η] · [(ϵ ⊗ X) ∗ ∇ ∗ Δ] =
= [(Δ ⊗ X) ∗ (η ⊗ X) ∗ (X ⊗ ∇)] · [(ϵ ⊗ X) ∗ (Δ ⊗ X) ∗ (X ⊗ ∇)] =
= [((Δ ∗ η) ⊗ X) ∗ (X ⊗ ∇)] · [((ϵ ∗ Δ) ⊗ X) ∗ (X ⊗ ∇)] =
= [((Δ ∗ η) ⊗ X) · ((ϵ ∗ Δ) ⊗ X)] ∗ (X ⊗ ∇) =
= [((Δ ∗ η) · (ϵ ∗ Δ)) ⊗ X] ∗ (X ⊗ ∇) = (Δ ⊗ X) ∗ (X ⊗ ∇)

which shows that the mate is the identity 2-cell. Similarly if one assumes that the other equation holds then the corresponding mate is the identity.

Conversely, assume that the mates of associativity are identity 2-cells and contemplate the following diagram:

\xymatrix{X ⊗ X ⊗ X \ar[d]_{X⊗∇} \ar[dr]^{id} \\ X ⊗ X \ar[dr]^{⟸}_{id} \ar[r]^{X⊗Δ} & X ⊗ X ⊗ X \ar[d]_{⟸}^{X⊗∇} \ar[r]^{∇⊗X} & X ⊗ X \ar[d]^{∇} \ar[dr]^{id} \\  & X ⊗ X \ar[r]_{∇} & X \ar[r]^{⟸}_{Δ} & X ⊗ X}

where the triangles are filled with the 2-cells X ⊗ η, X ⊗ ϵ and η. Pasting 2-cells one obtains that (X ⊗ η) ∗ (∇ ⊗ X) = (∇ ⊗ X) ∗ η. Indeed, one can calculate:

(X ⊗ η) ∗ (∇ ⊗ X) =

= [(X ⊗ η) ∗ (∇ ⊗ X)] · [(X ⊗ ∇) ∗ (X ⊗ Δ) ∗ (∇ ⊗ X) ∗ η] · [(X ⊗ ∇) ∗ (X ⊗ ϵ) ∗ ∇ ∗ Δ] =

= [(∇ ⊗ X) ∗ η] · [(X ⊗ η) ∗ (∇ ⊗ X) ∗ ∇ ∗ Δ] · [(X ⊗ ∇) ∗ (X ⊗ ϵ) ∗ ∇ ∗ Δ] =
= [(∇ ⊗ X) ∗ η] · [(X ⊗ η) ∗ (X ⊗ ∇) ∗ ∇ ∗ Δ] · [(X ⊗ (∇ ∗ ϵ)) ∗ ∇ ∗ Δ] =
= [(∇ ⊗ X) ∗ η] · [(X ⊗ (η ∗ ∇)) ∗ ∇ ∗ Δ] · [(X ⊗ (∇ ∗ ϵ)) ∗ ∇ ∗ Δ] =
= [(∇ ⊗ X) ∗ η] · [[(X ⊗ (η ∗ ∇)) · (X ⊗ (∇ ∗ ϵ))] ∗ (∇ ∗ Δ)] =
= (∇ ⊗ X) ∗ η

The proof of the other equation is analogous.

Because of this, we find it natural to introduce the following definition.

8.4. Definition. Let X = (X, ∇, Δ, η, ϵ) be an adjoint bi-semigroup such that the 1dimensional structure (X, ∇, Δ) satisfies Frobenius as a bi-semigroup. We say that X satisfies ∇-Frobenius if the equivalent conditions of Lemma 8.3 hold.

It is interesting and useful to notice that the equalities in the second item of Lemma 8.3 can be thought of as rewrite rules. Indeed, notice that in the notation we have used for expressions with Δs and ∇s, the left 2-cell of the first equation of item 2 has domain the left hand diagram below

[figure]

and codomain the right hand diagram below. In other words, the 2-cell pinches the first two strings. The reader is invited to draw the other 2-cells and exercise in applying the pinching and popping rules.

Back to the lifting of the Frobenius condition, it must be mentioned that one can prove the following in a way analogous to Lemma 8.3.

8.5. Lemma. Let X = (X, ∇, Δ, η, ϵ) be an adjoint bi-semigroup. If the (1-dimensional) structure (X, ∇, Δ) satisfies Frobenius as a bi-semigroup then the following two items are equivalent:

1. the mates of

\xymatrix{X \ar[d]_{Δ} \ar[r]^{Δ} & X ⊗ X \ar[d]^{Δ⊗X} & X \ar[d]_{Δ} \ar[r]^{Δ} & X ⊗ X \ar[d]^{X⊗Δ} \\ X ⊗ X \ar[r]_{X⊗Δ} & X ⊗ X ⊗ X & X ⊗ X \ar[r]_{Δ⊗X} & X ⊗ X ⊗ X}

are identity 2-cells

2. (Δ ⊗ D) ∗ (D ⊗ η) = η ∗ (Δ ⊗ D) and (D ⊗ Δ) ∗ (η ⊗ D) = η ∗ (D ⊗ Δ).

So, just as in Definition 8.4 we say that X satisfies Δ-Frobenius if the equivalent conditions of Lemma 8.5 hold.

8.6. Definition. Let X = (X, ∇, Δ, η, ϵ) be an adjoint bi-semigroup such that (X, ∇, Δ) satisfies Frobenius as a bi-semigroup. We say that X satisfies Frobenius if it satisfies both ∇-Frobenius and Δ-Frobenius.

8.7. Separable adjoint bi-semigroups. In this section we introduce the notion of separable adjoint bi-semigroup and show that for, these semigroups, 1-dimensional Frobenius implies 2-dimensional Frobenius.

8.8. Definition. We say that an adjoint bi-semigroup (X, ∇, Δ, η, ϵ) is separable if (X, ∇, Δ) is separable as a bi-semigroup and moreover ϵ = ι_{id_{X}}.

Notice that in a separable bi-semigroup, η ∗ ∇ = ι_{∇} and Δ ∗ η = ι_{Δ}.

8.9. Lemma. Let X = (X, ∇, Δ, η, ϵ) be an adjoint bi-semigroup. If the 1-dimensional structure (X, ∇, Δ) satisfies Frobenius and X is separable then X satisfies Frobenius.

Proof. Because of separability, the triangular identities witnessing that ∇ ⊣ Δ become η ∗ ∇ = ι_{∇} and Δ ∗ η = ι_{Δ}. To prove ∇-Frobenius we need to show that the mates of associativity are identity 2-cells. In particular, we need to show that

[(Δ ⊗ X) ∗ (X ⊗ ∇) ∗ η] · [(ϵ ⊗ X) ∗ ∇ ∗ Δ]

is ι_{∇;Δ}. Under separability, we need only prove that

[(Δ ⊗ X) ∗ (X ⊗ ∇) ∗ η] · ι_{∇;Δ} = (Δ ⊗ X) ∗ (X ⊗ ∇) ∗ η

is the identity 2-cell ι_{∇;Δ}. As (X, ∇, Δ) satisfies Frobenius and Δ ∗ η = ι_{Δ}, we can calculate:

(Δ ⊗ X) ∗ (X ⊗ ∇) ∗ η = ∇ ∗ Δ ∗ η = ∇ ∗ Δ

so, indeed, the mate of associativity is the identity. The other condition is dealt with in an analogous way so X satisfies ∇-Frobenius. To prove Δ-Frobenius one uses the same idea. For example, one of the conditions is proved as follows:

[η ∗ (X ⊗ Δ) ∗ (∇ ⊗ X)] · [∇ ∗ Δ ∗ (ϵ ⊗ X)] = η ∗ ∇ ∗ Δ = ∇ ∗ Δ

so, altogether, X satisfies Frobenius.

Since the counit is the identity, separable adjoint bi-semigroups will usually be denoted by (X, ∇, Δ, η).

8.10. A universal property of cospan(sOrd). In this section we prove a universal property of cospan(sOrd) as a monoidal 2-category. For brevity let us introduce the following definition.

8.11. Definition. Let (D, ⊗, I) be a strict monoidal 2-category. A Como-algebra is a separable adjoint bi-semigroup (D, ∇, Δ, η) such that (D, ∇, Δ) satisfies Frobenius.

Alternatively, one can say that a Como-algebra is a structure (D, ∇, Δ, η) such that (D, ∇, Δ) is a separable pre-algebra in D_{0} and η : id_{D⊗D} ⇒ ∇; Δ is a 2-cell satisfying

η ∗ ∇ = ι_{∇} and Δ ∗ η = ι_{Δ}

(essentially saying ∇ ⊣ Δ with trivial counit).

By Lemma 8.9 every Como-algebra satisfies Frobenius. Notice also that 1 + 1 has an obvious Como-algebra structure: just take η = \overline{∇}.

8.12. Proposition. Let (D, ⊗, I) be a strict monoidal 2-category with a Como-algebra (D, ∇, Δ, η). Then there exists a unique strict monoidal 2-functor

(cospan(sOrd), +, 0) → (D, ⊗, I)

mapping (1 + 1, ∇, Δ, η) to (D, ∇, Δ, η).

Proof. By Corollary 7.3 be have a strict monoidal functor cospan_{0}(sOrd) → D mapping the universal separable pre-algebra to the one in D. In order to extend this functor to a strict monoidal 2-functor we need a compatible selection of 2-cells satisfying the conditions of Lemma 5.6. That is, a compatible selection τ_{( )} satisfying τ_{f+g} = τ_{f} ⊗ τ_{g}. Now, Definition 4.1 forces τ_{( )} on identities and composition. As every map in sOrd is built from ∇ and using tensor and composition, a selection of 2-cells as the one we need is determined by its value τ_{∇} : id_{D⊗D} ⇒ ∇; Δ. Let us call this selection η. When does the selection of such a 2-cell induces a compatible selection? The answer is given by Lemma 7.7. But indulgence of the three distinguished pushouts is equivalent to the validity of the following equations:

1. η ∗ ∇ = ι_{∇} and Δ ∗ η = ι_{Δ}

2. (η ⊗ D) ∗ (D ⊗ ∇) = (D ⊗ ∇) ∗ η and (Δ ⊗ D) ∗ (D ⊗ η) = η ∗ (Δ ⊗ D)

3. (D ⊗ η) ∗ (∇ ⊗ D) = (∇ ⊗ D) ∗ η and (D ⊗ Δ) ∗ (η ⊗ D) = η ∗ (D ⊗ Δ)

The first item is exactly separability while the other two items are exactly Frobenius (Definition 8.6). But a Como-algebra is separable by definition and it always satisfies Frobenius by Lemma 8.9. So the result follows.

9. Como-units

Let iOrd be the full subcategory of Ord determined by injective monotone functions. The monoidal structure (Ord, +, 0) restricts to iOrd and the inclusion iOrd → Ord is strict monoidal. By results in [5], all maps in iOrd are built out of ! : 0 → 1.

9.1. Definition. A unit in a monoidal category (D, ⊗, I) is an object X in D equipped with a map u : I → X.

The object 1 in iOrd together with ! : 0 → 1 is the universal object with unit.

9.2. Lemma. The category iOrd has strict pullbacks and + preserves them.

9.3. Lemma. Every pullback in iOrd is a pasting of trivial pullbacks and pullbacks of the form

\xymatrix{0 \ar[d]_{id} \ar[r]^{id} & 0 \ar[d]^{!} \\ 0 \ar[r]_{!} & 1}

Proof. Similar to the proof of Lemma 7.2.

9.4. Definition. Let (D, ⊗, I) be a monoidal category. A split-unit is a structure (X, s : I → X, r : X → I) such that (X, s : I → X) is a unit and r : X → I is such that s; r = id_{I}.

The object 1 has a unique split-unit structure (1,! : 0 → 1,? : 1 → 0) in the monoidal category (cospan_{0}(iOrd^{op}), +, 0).

9.5. Corollary. If (D, ⊗, I) is a strict monoidal category and (X, s : I → X, r : X → I) is a split-unit in it then there exists a unique strict monoidal cospan_{0}(iOrd^{op}) → D mapping (1,! : 0 → 1,? : 1 → 0) to (X, s, r).

Proof. The map r : X → I induces a strict monoidal functor F_{0} : iOrd^{op} → D while the map s : I → X induces a strict monoidal F_{1} : iOrd → D. The functors clearly agree on objects. By Lemma 9.3, the functors are compatible if and only if they indulge the pushout

\xymatrix{1 \ar[d]_{_{!_{op}}} \ar[r]^{^{!^{op}}} & 0 \ar[d]^{_{id_{op}}} \\ 0 \ar[r]_{_{id_{op}}} & 0}

in iOrd^{op}. This means exactly that s; r = id.

9.6. Definition. Let (D, ⊗, I) be a monoidal 2-category. A Como-unit is a split-unit (X, s : I → X, l : X → I) together with a 2-cell η : ι_{X} ⇒ l; s such that l ⊣ s with unit η and counit ι_{id_{I}}.

The split-unit (1,! : 0 → 1,? : 1 → 0) is a Como-unit when considered as an object in (cospan(iOrd^{op}), +, 0). We denote the unit of the adjunction ? ⊣! by η.

In a way analogous to Proposition 8.12 we obtain the following corollary.

9.7. Corollary. For every strict monoidal 2-category (D, ⊗, I) and Como-unit object (X, s, l, η) in it, there exists a unique strict monoidal 2-functor

(cospan(iOrd^{op}), +, 0) → (D, ⊗, I)

mapping (1,! : 0 → 1,? : 1 → 0, η) to (X, s, l, η).

Proof. By Corollary 9.5 we have a unique strict monoidal functor

(cospan_{0}(iOrd^{op}), +, 0) → (D_{0}, ⊗, I)

mapping the split-unit (1,! : 0 → 1,? : 1 → 0) to (X, s, l). In order to extend this functor to a strict monoidal 2-functor we need a selection τ_{( )} of 2-cells satisfying τ_{f+g} = τ_{f} ⊗ τ_{g}. Such a selection of 2-cells is determined by its value τ_{(!^{op}:1→0)} : ι_{X} ⇒ l; s. Naturally, we define τ_{(!^{op}:1→0)} = η. Is the resulting selection compatible? We need to check that τ_{( )} indulges all pushout squares in iOrd^{op}. By Lemma 9.3 we need only check that it indulges the square in the statement of that lemma. But this says exactly that η ∗ l = l and s ∗ η = s. Which means that l ⊣ s with unit η and counit ι_{I}.

Acknowledgments

We would like to thank the anonymous referee for his/her comments and careful reading of the paper.

References

[1] R. J. M. Dawson, R. Paré, and D. A. Pronk. Universal properties of span. Theory and Applications of Categories, 13:61–85, 2004.

[2] Claudio Hermida. Representable multicategories. Advances in Mathematics, 151:164– 225, 2000.

[3] Joachim Kock. Frobenius algebras and 2D topological quantum field theories, volume 59 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2004.

[4] S. Lack. Composing Props. Theory and Applications of Categories, 13:147–163, 2004.

[5] S. Mac Lane. Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer Verlag, 1971.

[6] R. Rosebrugh, N. Sabadini, and R. F. C. Walters. Generic commutative separable algebras and cospans of graphs. Theory and Applications of Categories, 15:164–177, 2005.

[7] R. Rosebrugh and R. J. Wood. Distributive laws and factorization. Journal of Pure and Applied Algebra, 175:327–353, 2002.

N. Sabadini and R. F. C. Walters Department of Science of Culture, Politics and Information, via Valleggio 11, Como, Italy.

M. Menni C.C. 11 (1900) La Plata Argentina

Email: matias.menni@gmail.com
      nicoletta.sabadini@uninsubria.it
      robert.walters@uninsubria.it

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