Theory and Applications of Categories, Vol. 20, No. 3, 2008, pp. 25–47.

FROBENIUS OBJECTS IN CARTESIAN BICATEGORIES

R.F.C WALTERS AND R.J. WOOD

Abstract. Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, Map(B)(T, A) is a groupoid.

1. Introduction

The notion of locally ordered cartesian bicategory was introduced by Carboni and Walters [C&W] for the axiomatization of the bicategory of relations of a regular category. The notion has since been extended by Carboni, Kelly, Walters, and Wood [CKWW] to the case of a general bicategory, to include examples such as bicategories of spans, cospans, and profunctors.

A crucial further axiom introduced by Carboni and Walters in that paper was the so-called discreteness axiom, now known as the Frobenius axiom, since it was recognized to be equivalent to Lawvere’s equational version [LAW] of Frobenius algebra. With this axiom one can define the notion of Frobenius object in a monoidal category, the Frobenius axiom being an equation satisfied by monoid and comonoid structures on the object.

The Frobenius axiom has found a large variety of uses. For example, the 2-dimensional cobordism category has been shown to be the symmetric monoidal category with a generic commutative Frobenius object. (For a presentation of this result see J. Kock [Ko].) Related results are the characterization of the symmetric monoidal category of cospans of finite sets in [LACK] and the characterization of the symmetric monoidal category of cospans of finite graphs in [RSW]. Another example is that, in the algebra of quantum measurement [Co&P], classical data types are Frobenius objects. In [G&H] the Frobenius equation is a crucial equation in an algebraic presentation of double pushout graph rewriting, and in [KaSW] the equation is one of the main equations in a compositional theory of automata. The 2-dimensional version of Frobenius algebra has also been introduced in the characterization of a certain monoidal 2-category in [MSW].

There is a rather obvious way of extending the notion of Frobenius object to the context of a monoidal bicategory: instead of requiring equations between operations, certain canonical 2-cells are required to be invertible. This paper develops properties of such 2-dimensional Frobenius objects, for the canonical monoid and comonoid structure on each object which is part of the cartesian bicategory structure. The two principal results are (i) that maps (left adjoint arrows) between Frobenius objects are the same as comonoid homomorphisms, and (ii) that if A is a Frobenius object then, for any object T in the cartesian bicategory B, Map(B)(T, A) is a groupoid. This second result was noticed for the special case of Profunctors at the time of the Carboni-Walters paper by Carboni and Wood, independently, but has never been published. We develop in this paper techniques in a general cartesian bicategory which enable us to lift the profunctor proof.

The authors gratefully acknowledge financial support from the Italian Government PRIN project ART (Analisi di sistemi di Riduzione mediante sistemi di Transizione) and the Canadian NSERC. Diagrams typeset using M. Barr’s diagram package, diagxy.tex.

Received by the editors 2007-09-04 and, in revised form, 2008-01-25.

Transmitted by R. Paré. Published on 2008-01-27.

2000 Mathematics Subject Classification: 18A25.

Key words and phrases: cartesian bicategory, Frobenius object, dual object, groupoid.

\begin{matrix}c \\ ◯ \end{matrix} R.F.C Walters and R.J. Wood, 2008. Permission to copy for private use granted.

The results of this paper will be used in a following paper [W&W] characterizing bicategories of spans.

We thank Bob Paré for his helpful comments when these results were presented at the ATCAT seminar in Halifax, Canada.

2. Preliminaries

2.1. We recall from [CKWW] that a bicategory B is cartesian if the subbicategory of maps (by which we mean left adjoint arrows), M = MapB, has finite products (− × −, 1) with projections denoted p:X [figure] X × Y [figure] Y :r; each hom-category B(X, A) has finite products (−∧−, ⊤) with projections denoted π:R [figure] R∧S [figure] S:ρ; and an evident derived tensor product on B, (− ⊗ −, I) extending the product structure of M, is functorial. It was shown that the derived tensor product of a cartesian bicategory underlies a symmetric monoidal bicategory structure. Throughout this paper, B is assumed to be a cartesian bicategory and, as in [CKWW], we assume, for ease of notation, that B is normal, meaning that the identity compositional constraints of B are identity 2-cells.

2.2. If f is a map of B, an arrow of M, we will write η_{f}, ϵ_{f} : f ⊣ f^{∗} for a chosen adjunction in B that makes it so. As in [CKWW], we write

[figure: M; M; G; M; ∂_{0}; G; M; ∂_{1}]

for the Grothendieck span corresponding to

M^{op} × M [figure: i^{op}×i] B^{op} × B [figure: B(−,−)] CAT

where i:M [figure] B is the inclusion. A typical arrow of G, (f, α, u):(X, R, A) [figure] (Y, S, B) can be depicted by a square in B

[figure: A; B; u; X; A; R; X; Y; f; Y; B; S; α]  (1)

in which f and u are maps, and such arrows are composed by pasting. A 2-cell (ϕ, ψ): (f, α, u) [figure] (g, β, v) in G is a pair of 2-cells ϕ: f [figure] g, ψ: u [figure] v in M which satisfy the obvious equation.

2.3. In part of this and subsequent work it will be useful to revisit certain of the arrows of G from another point of view. Consider

[figure: X; Y; R; T; X; x; T; T; 1_{T}; T; Y; y; ρ]

On the one hand it is just an arrow from 1_{T} to R in G but each of the three reformulations of ρ that result from taking mates have their uses.

[figure: X; Y; R; T; X; x; T; T; 1_{T}; T; Y; y^{∗}; ρ̂; X; Y; R; T; X; x^{∗}; T; T; 1_{T}; T; Y; y^{∗}; ρ^{∗}; X; Y; R; T; X; x^{∗}; T; T; 1_{T}; T; Y; y; ρ̃]

In the first of these, ρ̂: 1_{T} [figure] y^{∗}Rx, it is sometimes convenient to write R(y, x) = y^{∗}Rx and regard ρ̂ as a 1_{T} -element of R(y, x). In the special case where R is 1_{X}:X [figure] X we write X(y, x) = y^{∗}x (invoking normality of B). (This hom-notation is similar to that employed first in [S&W]. It was adapted for this compositional context in [Wd].) The second we will use without further comment except to say that, for R = 1_{X}, ρ^{∗} is the usual way of making the process of taking right adjoints functorial. The third will appear in our discussion of tabulations in the forthcoming [W&W]. Note that the R(y, x) notation extends to 2-cells so that, for η:y′ [figure] y and ξ:x [figure] x′, we have R(η, ξ):R(y, x) [figure] R(y′, x′).

The chief purpose of the notation R(y, x) is to guide intuition so that constructions in such cartesian bicategories as that of categories, profunctors, and equivariant 2-cells (which we call prof) can be usefully generalized. Observe that if τ : R [figure] S is a 2cell in B and ξ: x [figure] x′ then we have automatically such identities as τ(y, x′).R(y, ξ) = S(y, ξ).τ(y, x), both providing the horizontal composite τξ whiskered with y^{∗} as below.

[figure: T; X; x; T; X; x′; X; Y; R; X; Y; S; ξ; τ; Y; T; y^{∗}]

For the most part, we will use such calculations with little comment.

If

[figure: X; Y; R; T; X; x; T; T; 1_{T}; T; Y; y^{∗}; ρ̂; and; Y; Z; S; T; Y; y; T; T; 1_{T}; T; Z; z^{∗}; σ̂]

are 1_{T} -elements of R(y, x) and S(z, y) respectively then it is easy to see that ρ̂□σ, where ρ□σ is the paste composite of ρ and σ, is a 1_{T} -element of (SR)(z, x). The 1_{T} -element ρ̂□σ can be given in several ways. We will have occasion to give it via the pasting composite

[figure: X; T; X; x; T; T; 1_{T}; T; X; Y; R; ρ; T; Y; y; T; T; 1_{T}; T; Y; y^{∗}; η_{y}; Z; T; T; T; 1_{T}; T; Z; z^{∗}; Y; Z; S; σ^{∗}]

We note that a paste composite such as ρ□σ as below

[figure: X; Y; R; T; X; x; T; T; 1_{T}; T; Y; y; ρ; Y; Z; S; T; Y; y; T; T; 1_{T}; T; Z; z; σ]

may result from several different y:T [figure] Y. For example, in

[figure: X; Y; R; T; X; x; T; T; 1_{T}; T; Y; y; ρ; Y; Z; S; T; Y; y′; T; T; 1_{T}; T; Z; z; η; σ]

we have (ρ□η)□σ = ρ□(η□σ) suggesting that some of the 1_{T} -elements of (SR)(z, x) are given by an obvious coend over y in the category M(T, Y).

However, our prof-like notation has its limitations. For fixed T we can associate to X the category X̃ = M(T, X) and to R: X [figure] Y the profunctor R̃: X̃ [figure] Ỹ where R̃(y, x) = B(T, T)(1_{T} , y^{∗}Rx) but we see no reason why a general 1_{T} -element of (SR)(z, x) in a general cartesian bicategory should arise from pasting a 1_{T} -element of S(z, y) to a 1_{T} -element of R(y, x) for some y: T [figure] Y. In short, while there is a 2-cell S̃R̃ [figure] S̃R in prof there seems to be no reason why it should have surjective components. That said, S̃R̃ [figure] S̃R is an isomorphism in case B = SpanE, for any category E with finite limits, and for any cartesian B we have isomorphisms 1X̃ [figure] ˜1_{X} in prof, for any X in B. So there is always a normal lax functor

˜(−):B [figure] prof

which in some cases is a pseudofunctor. Fortunately, we have no need for invertibility of the S̃R̃ [figure] S̃R.

2.4. Quite generally, an arrow of G as given by the square (1) will be called a commutative square if α is invertible. The arrow (1) of G will be said to satisfy the Beck-Chevalley condition if the mate of α under the adjunctions f ⊣ f^{∗} and u ⊣ u^{∗}, as given in the square below (no longer an arrow of G), is invertible.

[figure: A; B; u^{∗}; X; A; R; X; Y; f^{∗}; Y; B; S; α^{∗}]

Thus Proposition 4.8 of [CKWW] says that projection squares of the form p̃_{R,1_{Y}} and r̃_{1_{X},S} satisfy the Beck-Chevalley condition. (Also, Proposition 4.7 of [CKWW] says that the same projection squares are commutative. In general, neither commutative nor Beck-Chevalley implies the other.) If R and S are also maps and α is invertible then α^{−1} gives rise to another arrow of G which may or may not satisfy the Beck-Chevalley condition. The point here is that a commutative square of maps gives rise to two, generally distinct, Beck-Chevalley conditions. It is well known that, for bicategories of the form SpanE and RelE all pullback squares of maps satisfy both Beck-Chevalley conditions. A [bi]category with finite products has automatically a number of pullbacks which we might call product-absolute pullbacks because they are preserved by all [pseudo]functors which preserve products.

3. Frobenius Objects in Cartesian Bicategories

For any object A in B, we have the following two G arrows:

[figure: A ⊗ A; A ⊗ (A ⊗ A); 1⊗d; A; A ⊗ A; d; A; A ⊗ A; d; A ⊗ A; A ⊗ (A ⊗ A); A ⊗ A; (A ⊗ A) ⊗ A; d⊗1; (A ⊗ A) ⊗ A; A ⊗ (A ⊗ A); a; A ⊗ A; A ⊗ (A ⊗ A); A; A ⊗ A; d; A; A ⊗ A; d; A ⊗ A; A ⊗ (A ⊗ A); 1⊗d; A ⊗ A; (A ⊗ A) ⊗ A; d⊗1; (A ⊗ A) ⊗ A; A ⊗ (A ⊗ A); a]

obtained from the same equality of arrows in MapB. (With a suitable choice of conventions we have equality rather than a mere isomorphism.) For each square, observe that the data regarded as a square in M provide an example of a product-absolute pullback.

3.1. Definition. An object A is said to be Frobenius if both of the G arrows above satisfy the Beck-Chevalley condition. This is to demand invertibility both of δ_{0}:d.d^{∗} [figure] 1_{A}⊗ d^{∗}.a.d ⊗ 1_{A}, the mate of the first equality above, and of δ_{1}:d.d^{∗} [figure] d^{∗} ⊗ 1_{A}.a^{∗}.1_{A} ⊗ d, the mate of the second equality above.

3.2. Lemma. The Beck-Chevalley condition for either square implies the condition for the other.

Proof. Explicitly, in notation suppressing ⊗, δ_{0} and δ_{1} are given by

[figure: δ_{0} =; AA; A; d^{∗}; AA; AA; 1; AA; A; d; AA; (AA)A; dA; (AA)A; A(AA); a; A; AA; d; AA; AA; 1; A(AA); AA; Ad; A(AA); AA; Ad^{∗}; ϵ; |; Aη]

and

Assume that δ_{0} is invertible and paste at its top and right edges the following pasting composite at its bottom edge.

[figure: δ_{1} =; AA; A; d^{∗}; AA; AA; 1; AA; A; d; AA; A(AA); Ad; A(AA); (AA)A; a^{∗}; A; AA; d; AA; AA; 1; (AA)A; AA; dA; (AA)A; AA; d^{∗}A; ϵ; |; ηA] [figure: AA; AA; 1; AA; (AA)A; dA; (AA)A; A(AA); a; AA; AA; s; AA; AA; s; (AA)A; A(AA); s; A(AA); (AA)A; s; AA; AA; 1; AA; A(AA); Ad; A(AA); A(AA); As; A(AA); (AA)A; a^{∗}; (AA)A; (AA)A; sA; ≅; ≅; ≅; A(AA); AA; Ad^{∗}; (AA)A; AA; d^{∗}A; AA; AA; s; ≅]

The squares are pseudonaturality squares for symmetry as in 4.5 of [CKWW] and the hexagon bounds an invertible modification constructed from those relating the associativity equivalence a and the symmetry equivalence s. Next, observe that we have sd ≅ d and, since s is an equivalence with s_{A,B}^{∗} ≅ s_{B,A}, d^{∗}s ≅ d^{∗}. By functoriality of ⊗ we have also (As)(Ad) ≅ Ad and (d^{∗}A)(sA) ≅ d^{∗}A. Noting the compatibility of the pseudonatural transformation s with the 2-cell ηA, the large pasting composite is seen to be δ_{1}. The derivation of invertibility of δ_{0} from that of δ_{1} is effected in a similar way.

3.3. Axiom. Frobenius A cartesian bicategory B is said to satisfy the Frobenius axiom if, for each A in B, A is Frobenius.

3.4. Proposition. In a cartesian bicategory B, the Frobenius objects are closed under finite products.

Proof. Consider a Frobenius object A so that we have invertible δ_{0} = δ_{0}(A) in

[figure: A ⊗ A; A ⊗ (A ⊗ A); 1⊗d^{∗}; A; A ⊗ A; d; A; A ⊗ A; d^{∗}; A ⊗ A; A ⊗ (A ⊗ A); A ⊗ A; (A ⊗ A) ⊗ A; d⊗1; (A ⊗ A) ⊗ A; A ⊗ (A ⊗ A); a; δ_{0}]

For B also Frobenius, form the tensor product of the diagrams for δ_{0}(A) and δ_{0}(B), noting that δ_{0}(A)⊗δ_{0}(B) is also invertible. The diagram for δ_{0}(A⊗B) is easily formed from that of δ_{0}(A)⊗δ_{0}(B) by pasting to its exterior the requisite permutations of the A and B and using such isomorphisms as m(d_{A}⊗d_{B}) ≅ d_{A⊗B}, where m:(A⊗A)⊗(B⊗B) [figure] (A⊗B)⊗(A⊗B) is the middle-four interchange equivalence. Thus A⊗B is Frobenius when A and B are so. Invertibility of δ_{0}(I) follows easily since d_{I} is an equivalence, showing that I is Frobenius.

Write FrobB for the full subbicategory of B determined by the Frobenius objects. It follows immediately from Proposition 3.4 that

3.5. Proposition. For a cartesian bicategory B, the full subbicategory FrobB is a cartesian bicategory which satisfies the Frobenius axiom.

In any (pre)cartesian bicategory we have, for each object X, the following arrows:

N_{X} = I [figure: t_{X}^{∗}] X [figure: d_{X}] X ⊗ X and E_{X} = X ⊗ X [figure: d_{X}^{∗}] X [figure: t_{X}] I

Since the cartesian bicategory B is a (symmetric) monoidal bicategory it can be seen as a one-object tricategory, so that pseudo adjunctions N, E: X ⊣ A, where X and A are objects of B (and N and E are arrows of B), are well defined. (We note that, especially since B is symmetric, it is customary to speak of such X and A as duals.)

3.6. Proposition. For a Frobenius object X in a cartesian bicategory, N_{X} and E_{X} provide the unit and counit for a pseudo-adjunction X ⊣ X.

Proof. (Sketch) We are to exhibit isomorphisms

(E_{X} ⊗ X)a^{∗}(X ⊗ N_{X}) ≅ s_{X,I} and (X ⊗ E_{X})a(N_{X} ⊗ X) ≅ s_{I,X}

subject to two coherence equations. Consider:

[figure: X ⊗ I; X ⊗ X; X⊗t_{X}^{∗}; X ⊗ X; X ⊗ (X ⊗ X); X⊗d_{X}; X ⊗ (X ⊗ X); (X ⊗ X) ⊗ X; a^{∗}; (X ⊗ X) ⊗ X; X ⊗ X; d_{X}^{∗}⊗X; X ⊗ X; X; d_{X}^{∗}; X; X ⊗ X; d_{X}; X ⊗ I; X; r; X ⊗ X; I ⊗ X; t_{X}⊗X; X; I ⊗ X; l; ≃^{δ_{1}}; ≅; ≅]

[figure: I ⊗ X; X ⊗ X; t_{X}^{∗}⊗X; X ⊗ X; (X ⊗ X) ⊗ X; d_{X}⊗X; (X ⊗ X) ⊗ X; X ⊗ (X ⊗ X); a; X ⊗ (X ⊗ X); X ⊗ X; X⊗d_{X}^{∗}; X ⊗ X; X; d_{X}^{∗}; X; X ⊗ X; d_{X}; I ⊗ X; X; l; X ⊗ X; X ⊗ I; X⊗t_{X}; X; X ⊗ I; r; ≃^{δ_{0}^{−1}}; ≅; ≅]

For the coherence requirements let us abbreviate ⊗ by juxtaposition, as we have before, but now work as if the bicategory constraints of B and those of the monoidal structure (B, ⊗, I) are strict. (In general, this is not acceptable because a monoidal bicategory is not tri-equivalent to a one-object 3-category. However, our monoidal structure, being given by universal properties, is less problematical.) Temporarily, write N : I [figure] X^{∘}X and E : XX^{∘} [figure] I, just to mark the role of the X’s. Write α : 1_{X} [figure] (EX)(XN) and β:(X^{∘}E)(NX^{∘}) [figure] 1_{X^{∘}} for the isomorphisms built from those above, with the simplifying assumptions. The coherence requirements of α and β are the pasting equations

[figure: 1_{E} =; XX^{∘}; I; E; XX^{∘}XX^{∘}; XX^{∘}; EXX^{∘}; XX^{∘}XX^{∘}; XX^{∘}; XX^{∘}E; XX^{∘}; I; E; XX^{∘}; XX^{∘}XX^{∘}; XNX^{∘}; XX^{∘}; XX^{∘}; XX^{∘}; XX^{∘}; XX^{∘}; XX^{∘}; αX^{∘}; Xβ; ≅; X^{∘}X; X^{∘}XX^{∘}X; X^{∘}XN; I; X^{∘}X; N; I; X^{∘}X; N; X^{∘}X; X^{∘}XX^{∘}X; NX^{∘}X; ≅; X^{∘}X; X^{∘}X; X^{∘}X; X^{∘}X; X^{∘}X; X^{∘}X; X^{∘}XX^{∘}X; X^{∘}X; X^{∘}EX; X^{∘}α; βX; 1_{N} =]

where the unlabelled isomorphisms in the squares are given by pseudofunctoriality of ⊗. We will verify the first of these equations, verification of the second being similar, now using X^{∘} = X but continuing to suppress the constraints both for B and for the monoidal structure. Thus we must show that the composite on the left below

[figure: XXX; XX; Xd^{∗}; XXXX; XXX; d^{∗}XX; XXXX; XXX; XXd^{∗}; XXX; XX; d^{∗}X; ≅; XXX; XXXX; XdX; XXX; XX; d^{∗}X; XXX; XX; Xd^{∗}; XX; XXX; Xd; XX; XXX; dX; δ_{1}X; Xδ_{0}^{−1}; XX; XXX; Xt^{∗}X; XX; I; tt; =; XX; XXX; Xt^{∗}X; XX; X; d^{∗}; XXX; XX; d^{∗}X; XXX; XX; Xd^{∗}; XX; X; d^{∗}; ≅; X; XX; d; XX; XXX; dX; XX; XXX; Xd; XXX; XX; Xd^{∗}; XXX; XX; d^{∗}X; XX; I; tt; δ_{1}; δ_{0}^{−1}]

is 1_{E}. Again using pseudofunctoriality of ⊗, we have the equality shown and finally the diagram on the right can be shown to be 1_{E} from the definitions of δ_{0} and δ_{1}.

3.7. If R:X [figure] A is an arrow in B then given pseudo adjunctions X ⊣ X^{∘} and A ⊣ A^{∘} we should expect that adaption of the calculus of mates found in [K&S] will enable us to define R^{∘}:X^{∘} [figure] A^{∘} by the usual formula. In fact, if every object of B has a dual one should expect (−)^{∘} to provide a pseudofunctor (−)^{∘}: B^{oprev} [figure] B between tricategories, where (−)^{rev} denotes dualization with respect to objects of B composed via ⊗, while as usual (−)^{op} denotes dualization with respect to the 1-cells of B. In particular, one should expect (X ⊗ Y)^{∘} ≃ Y^{∘} ⊗ X^{∘}. The point of this paragraph is that the (−)^{∘} of the following proposition arises from the properties already under consideration and is not a new structure as in the similarly denoted operation of [F&S].

3.8. Proposition. For a cartesian bicategory B in which every object is Frobenius, there is an involutory pseudofunctor

(−)^{∘}:B^{op} [figure] B

which is the identity on objects.

Proof. With X^{∘} = X we define

(−)_{A,X}^{∘}:B^{op}(A, X) = B(X, A) [figure] B(A, X)

by the evidently functorial formula

R^{∘} = (X ⊗ E_{A})(X ⊗ R ⊗ A)(N_{X} ⊗ A)

In terms of the one object tricategory (B, ⊗, I) with single object ∗, we can express R^{∘} by the pasting

[figure: ∗; ∗; A; ∗; ∗; ∗; ∗; A; E_{A}; ∗; ∗; I; ∗; ∗; X; ∗; ∗; I; R; ∗; ∗; I; ∗; ∗; X; ∗; ∗; X; N_{X}]

For R:X [figure] A, along with S:A [figure] Y, to give ˜(−)^{∘}:R^{∘}S^{∘} [figure] (SR)^{∘} we consider

[figure: ∗; ∗; Y; ∗; ∗; Y; ∗; ∗; I; E_{Y}; ∗; ∗; I; ∗; ∗; ∗; ∗; I; S; ∗; ∗; I; ∗; ∗; A; ∗; ∗; A; N_{A}; ∗; ∗; ∗; ∗; I; ∗; ∗; A; E_{A}; ∗; ∗; I; ∗; ∗; X; ∗; ∗; I; R; ∗; ∗; I; ∗; ∗; X; ∗; ∗; X; N_{X}]

in which the pasting composite displays R^{∘}S^{∘}. The required ˜(−)^{∘} is obtained as the collapsing of the centre triangles using α^{−1} : (E_{A} ⊗ A)(A ⊗ N_{A}) ≅ s_{A,I} of the pseudo adjunction N_{A}, E_{A}:A ⊣ A. Evidently, ˜(−)^{∘} is invertible. We give the identity constraint for (−)^{∘} as β^{−1} : 1_{X} [figure] (X ⊗ E_{X})(N_{X} ⊗ X) which is again invertible. Finally, having observed that the mate description of R^{∘} = (X ⊗ E_{A})(X ⊗ R ⊗ A)(N_{X} ⊗ A) was given by expanding R:X [figure] A as R:X ⊗ I [figure] I ⊗ A we see by writing R:I ⊗ X [figure] A ⊗ I that we have equally

R^{∘} ≅ (E_{A} ⊗ X)(A ⊗ R ⊗ X)(A ⊗ N_{X})

Thus we may as well give

((−)^{∘})^{op}:B [figure] B^{op}

by the formula

(A [figure: S] X)| [figure] (E_{X} ⊗ A)(X ⊗ S ⊗ A)(X ⊗ N_{A})

so that R^{∘∘} is the pasting

[figure: ∗; ∗; I; ∗; ∗; A; ∗; ∗; A; N_{A}; ∗; ∗; ∗; ∗; I; ∗; ∗; A; E_{A}; ∗; ∗; I; ∗; ∗; X; ∗; ∗; I; R; ∗; ∗; I; ∗; ∗; X; ∗; ∗; N_{X}; ∗; ∗; X; ∗; ∗; I; ∗; ∗; X; E_{X}]

and we have a canonical isomorphism R ≅ R^{∘∘}, again using the α and β constraints of the pseudo adjunctions N_{X}, E_{X}:X ⊣ X of Proposition 3.6.

3.9. Proposition. For an arrow R:X [figure] A in a cartesian bicategory, with X and A Frobenius, if the d̃_{R} and t̃_{R} of the units

[figure: A; A ⊗ A; d_{A}; X; A; R; X; X ⊗ X; d_{X}; X ⊗ X; A ⊗ A; R⊗R; d̃_{R}; A; I; t_{A}; X; A; R; X; I; t_{X}; I; I; ⊤; t̃_{R}]

are invertible then we can construct squares N_{R} and E_{R}

[figure: X; A; R; I; X; t_{X}^{∗}; I; I; 1_{I}; I; A; t_{A}^{∗}; t̃_{R}^{∗}; X ⊗ X; A ⊗ A; R⊗R; X; X ⊗ X; d_{X}; X; A; R; A; A ⊗ A; d_{A}; d̃_{R}^{−1}; I; X ⊗ X; N_{X}; I; A ⊗ A; N_{A}; N_{R} =; X; A; R; X ⊗ X; X; d_{X}^{∗}; X ⊗ X; A ⊗ A; R⊗R; A ⊗ A; A; d_{A}^{∗}; d̃_{R}^{∗}; I; I; 1_{I}; X; I; t_{X}; X; A; R; A; I; t_{A}; t̃_{R}^{−1}; X ⊗ X; I; E_{X}; A ⊗ A; I; E_{A}; E_{R} =; X; A; R; X; X; X; X; A; R; A; A; A; R]

where t̃_{R}^{∗} is the mate of t̃_{R} and d̃_{R}^{∗} is the mate of d̃_{R}, which when tensored with the identity square R, above, satisfy the following equations (in which ⊗ is suppressed):

[figure: XXX; AAA; RRR; X; XXX; N_{X}X; X; A; R; A; AAA; N_{A}A; N_{R}R; X; A; R; XXX; X; XE_{X}; XXX; AAA; RRR; AAA; A; AE_{A}; RE_{R}; X; X; X; ≅; A; A; A; ≅; XXX; AAA; RRR; X; XXX; XN_{X}; X; A; R; A; AAA; AN_{A}; RN_{R}; X; A; R; XXX; X; E_{X}X; XXX; AAA; RRR; AAA; A; E_{A}A; E_{R}R; X; X; X; ≅; A; A; A; ≅; R =; = R]  (2)

Proof. The vertical edges of the diagrams have been clarified in Proposition 3.6. For the rest it suffices for each equation to expand N_{R} and E_{R}, verify the following equalities

[figure: X; XXX; XX; X; d^{∗}; XX; XXX; dX; X; XX; d; X; XXX; XXX; XX; Xd^{∗}; δ_{0}; XX; AA; RR; d̃_{R}^{−1}R; XXX; AAA; RRR; Rd̃_{R}^{∗}; XX; AA; RR; AA; AAA; dA; AAA; AA; Ad^{∗}; =; XX; AA; RR; d̃_{R}^{∗}; X; A; R; d̃_{R}^{−1}; XX; AA; RR; XX; X; d^{∗}; AA; AAA; dA; AAA; AA; Ad^{∗}; AA; A; d^{∗}; δ_{0}; A; AA; d; X; XX; d]

[figure: X; XXX; XX; X; d^{∗}; XX; XXX; Xd; X; XX; d; X; XXX; XXX; XX; d^{∗}X; δ_{1}; XX; AA; RR; Rd̃_{R}^{−1}; XXX; AAA; RRR; d̃_{R}^{∗}R; XX; AA; RR; AA; AAA; Ad; AAA; AA; d^{∗}A; =; XX; AA; RR; d̃_{R}^{∗}; X; A; R; d̃_{R}^{−1}; XX; AA; RR; XX; X; d^{∗}; AA; AAA; Ad; AAA; AA; d^{∗}A; AA; A; d^{∗}; δ_{1}; A; AA; d; X; XX; d]

and use such further equalities as

[figure: XX; AA; RR; X; XX; d; X; A; R; A; AA; d; d̃_{R}^{−1}; X; A; R; XX; X; Xt; XX; AA; RR; AA; A; At; Rt̃_{R}^{−1}; X; X; X; ≅; A; A; A; ≅; XX; AA; RR; X; XX; t^{∗}X; X; A; R; A; AA; t^{∗}A; t̃_{R}^{∗}R; X; A; R; XX; X; d^{∗}; XX; AA; RR; AA; A; d^{∗}; d̃_{R}^{∗}; X; X; X; ≅; A; A; A; ≅; R =; R =]

3.10. Every object X of a bicategory with finite products is, essentially uniquely, a pseudo comonoid via d_{X} and t_{X}. It follows that every object X in a cartesian bicategory B is a (pseudo) comonoid (via d_{X} and t_{X}) since M has finite products and the inclusion functor i:M [figure] B is strongly monoidal. (It is the identity on objects and we observe from Proposition 3.24 of [CKWW] that f × g [figure: ≃] f ⊗ g in B.) Similarly, for R:X [figure] A in B, R has an essentially unique comonoid structure in G, via (d_{X}, d̃_{R}, d_{A}) and (t_{X}, t̃_{R}, t_{A}), since G has finite products. In fact, given d_{X} and d_{A}, d̃_{R} is uniquely determined and given t_{X} and t_{A}, t̃_{R} is uniquely determined. This fact can be reinterpreted to say that R:X [figure] A has an essentially unique lax comonoid homomorphism structure via d_{R} = (d_{X}, d̃_{R}, d_{A}) and t_{R} = (t_{X}, t̃_{R}, t_{A}) which is then a comonoid homomorphism if and only if the 2-cells d̃_{R} and t̃_{R} are invertible. Thus being a comonoid homomorphism is a property of an arrow in a cartesian bicategory.

3.11. Theorem. For an arrow R : X [figure] A in a cartesian bicategory, with X and A Frobenius, the following are equivalent:

(1) R is a map;

(2) R is a comonoid homomorphism;

(3) R ⊣ R^{∘}.

Proof. (1) implies (2) follows from the fact that d and t are pseudonatural on maps and (3) implies (1) is trivial. So, assuming (2), that R is a comonoid homomorphism, construct N_{R} and E_{R} as in Proposition 3.9 and define (suppressing ⊗ as usual)

[figure: η_{R} =; X; XXX; N_{X}X; XXX; XXA; XXR; XXX; XAA; XRR; XXX; X; XE_{X}; A; XXA; N_{X}A; XXA; XAA; XRA; XAA; X; XE_{A}; X; A; R; X; X; X; N_{X}R; ≅; XE_{R}; X; X; X; ≅; ϵ_{R} =; A; XXA; N_{X}A; XXA; XAA; XRA; XAA; X; XE_{A}; A; A; A; XXA; AAA; RRA; XAA; AAA; RAA; X; A; R; A; AAA; N_{A}A; AAA; A; AE_{A}; A; A; A; N_{R}A; ≅; ≅; RE_{A}]

where we note that both three-fold vertical composites are the arrow R^{∘}, N_{X}R = 1_{N_{X}} ⊗1_{R} and RE_{A} = 1_{R} ⊗ 1_{E_{A}} are isomorphisms while XE_{R} = 1_{1_{X}} ⊗ E_{R} and N_{R}A = N_{R} ⊗ 1_{1_{A}}.

When η_{R} and ϵ_{R} are pasted at R^{∘} the result is

[figure: X; XXX; N_{X}X; XXX; XXA; XXR; XXX; XAA; XRR; XXX; X; XE_{X}; A; XXA; N_{X}A; XAA; X; XE_{A}; X; A; R; X; X; X; N_{X}R; ≅; XE_{R}; X; X; X; ≅; A; A; A; XXA; AAA; RRA; XAA; AAA; RAA; X; A; R; A; AAA; N_{A}A; AAA; A; AE_{A}; A; A; A; N_{R}A; RE_{A}; ≅; =; XXX; AAA; RRR; X; XXX; N_{X}X; X; A; R; A; AAA; N_{A}A; N_{R}R; X; A; R; XXX; X; XE_{X}; XXX; AAA; RRR; AAA; A; AE_{A}; RE_{R}; X; X; X; ≅; A; A; A; ≅; =; R]

the first equality from functoriality of ⊗, the second equality being the first equation of (2) of Proposition 3.9. To complete the proof that we have an adjunction η_{R}, ϵ_{R}:R ⊣ R^{∘} we must show that when η_{R} is pasted to ϵ_{R} at R the result is R^{∘}. To aid readability we draw as commutative as many regions as possible. Consider:

[figure: A; XXA; N_{X}A; A; A; A; A; AAA; N_{A}A; N_{R}A; XXA; XAA; XRA; AAA; XXA; RRA; AAA; XAA; RAA; XAA; X; XE_{A}; AAA; A; AE_{A}; X; A; R; A; X; R^{∘}; A; A; A; X; XXX; N_{X}X; A; XXA; N_{X}A; XXA; XXX; XXR; XXA; XAA; XRA; XAA; XXX; XRR; XXX; X; XE_{X}; XAA; X; XE_{A}; X; X; X; XE_{R}; X; X; X; A; X; R^{∘}]

FROBENIUS OBJECTS IN CARTESIAN BICATEGORIES  39

(which is the requisite pasting rotated 90 degrees counterclockwise). Rearrange it as below:

[figure: XXA; AAA; RRA; A; XXA; N_{X}A; A; A; A; A; AAA; N_{A}A; N_{R}A; AAA; XXAAA; N_{X}AAA; A; AAA; N_{A}A; A; XXA; N_{X}A; XXA; XXAAA; XXN_{A}A; XXAAA; XAAAA; XRAAA; XXA; XXAAA; XXN_{A}A; XXA; XAA; XRA; XAA; XAAAA; XAN_{A}A; XXXXA; XXXAA; XXXRA; XXAAA; XXXXA; XXRRA; XXAAA; XAAAA; XRAAA; XAAAA; XXXAA; XRRAA; XXA; XAA; XRA; XXXXA; XXA; XE_{X}XA; XXXXA; XXXAA; XXXRA; XXXAA; XAA; XE_{X}AA; XAA; X; XE_{A}; XXXAA; XAA; XE_{X}AA; XXXAA; XXX; XXXE_{A}; XXX; X; XE_{X}; X; X; X; XXX; X; XE_{X}; XXX; XAA; XRR; XAA; X; XE_{A}; XE_{R}; XXXXA; XAAAA; XRRRA; XXA; XXXXA; N_{X}XXA; XXA; XXA; XXA; XAAAA; XAA; XAAE_{A}; XAA; XAA; XAA]

(3)

The following prism commutes:

[figure: XXA; AAA; RRA; A; XXA; N_{X}A; A; A; A; A; AAA; N_{A}A; N_{R}A; XXXXA; XXAAA; XXRRA; XXA; XXXXA; XXN_{X}A; XXA; XXA; XXA; XXA; XXAAA; XXN_{A}A; XXN_{R}A; A; XXA; N_{X}A; A; XXA; N_{X}A; XXA; XXXXA; N_{X}XXA; AAA; XXAAA; N_{X}AAA; XAA; XAAAA; XAN_{A}A; XXA; XAA; XRA; XXA; XAA; XRA; XXAAA; XAAAA; XRAAA; XXXXA; XAAAA; XRRRA; XRN_{R}A]

Replace the top three squares of (3) above by the two front faces of the prism. Employ a similar commuting prism to replace the bottom three squares of (3) and obtain:

[figure: XXA; XXXXA; N_{X}XXA; A; XXA; N_{X}A; A; XXA; N_{X}A; XXA; XXXXA; XXA; XXA; XXA; XXXXA; XAAAA; XRRRA; XXA; XXXXA; XXN_{X}A; XXA; XAA; XRA; XAA; XAAAA; XAN_{A}A; XRN_{R}A; XXA; XAA; XRA; XXXXA; XXA; XE_{X}XA; XXXXA; XAAAA; XRRRA; XAAAA; XAA; XE_{A}AA; XE_{R}RA; XAA; X; XE_{A}; XAAAA; XAA; XAAAA; XAA; XAAE_{A}; XAA; X; XE_{A}; XAA; XAA; XAA]

[figure: =; A; XXA; N_{X}A; A; A; A; A; XXA; N_{X}A; XXA; XXA; XXA; XXA; XAA; XRA; XXA; XXA; XXA; XXA; XAA; XRA; XAA; XAA; XAA; XAA; X; XE_{A}; XAA; XAA; XAA; XAA; X; XE_{A}; X; X; X; = R^{∘}]

where the penultimate equality is obtained from the second equation of (2) of Proposition 3.9 by tensoring it on the left by X and on the right by A and applying the result to the two middle squares of the penultimate pasting.

3.12. From Theorem 3.11 it follows that for a map f:X [figure] A, with X and A Frobenius in a cartesian bicategory, we have f^{∗} ≅ f^{∘} and we may as well write f^{∗} = f^{∘} for our specified right adjoints in this event and use the explicit formula for f^{∘} when it is convenient to do so.

3.13. Theorem. If A is a Frobenius object in a cartesian bicategory B, then, for all T in B, the hom-category M(T, A) is a groupoid.

We will break the proof of Theorem 3.13 into a sequence of lemmas and employ the notation of 2.3.

3.14. Lemma. With reference to the 2-cell δ_{1} in Definition 3.1,

dd^{∗} [figure: ≃] (p^{∗} ∧ r^{∗})(p ∧ r) and (d^{∗} ⊗ X)(X ⊗ d) [figure: ≃] p^{∗}p ∧ p^{∗}r ∧ r^{∗}r

and these canonical isomorphisms identify δ_{1} with (ππ, πρ, ρρ). Here the components are horizontal composites of the local product projection 2-cells. For example, πρ is

[figure: A ⊗ A; A; p∧r; A ⊗ A; A; r; A; A ⊗ A; p^{∗}∧r^{∗}; A; A ⊗ A; p^{∗}; ρ; π]

We will write

δ = (ππ, πρ, ρρ):(p^{∗} ∧ r^{∗})(p ∧ r) [figure] p^{∗}p ∧ p^{∗}r ∧ r^{∗}r:A ⊗ A [figure] A ⊗ A  (4)

Proof. We have

p ∧ r ≅ d^{∗}(p ⊗ r)d ≅ d^{∗}(p, r) ≅ d^{∗}1_{A⊗A} = d^{∗}

and

p^{∗} ∧ r^{∗} ≅ d^{∗}(p^{∗} ⊗ r^{∗})d ≅ d^{∗}(p ⊗ r)^{∗}d ≅ (p, r)^{∗}d ≅ 1_{A⊗A}^{∗}d = d

so that dd^{∗} ≅ (p^{∗} ∧ r^{∗})(p ∧ r). To exhibit the other isomorphism of the statement we will write d_{3} : A ⊗ A [figure] (A ⊗ A) ⊗ (A ⊗ A) ⊗ (A ⊗ A) for the three-fold diagonal map (1_{A⊗A}, 1_{A⊗A}, 1_{A⊗A}) and then

p^{∗}p ∧ p^{∗}r ∧ r^{∗}r ≅ d_{3}^{∗}(p^{∗}p ⊗ p^{∗}r ⊗ r^{∗}r)d_{3} ≅ d_{3}^{∗}(p^{∗} ⊗ p^{∗} ⊗ r^{∗})(p ⊗ r ⊗ r)d_{3} ≅ (d^{∗} ⊗ A)(A ⊗ d)

Of course δ = (ππ, πρ, ρρ) in (4) of the Lemma is invertible if and only if A is Frobenius. We will write

ν = ρπ:(p^{∗} ∧ r^{∗})(p ∧ r) [figure] r^{∗}p:A ⊗ A [figure] A ⊗ A

for the “other” horizontal composite of projections and for A Frobenius we define μ as the unique 2-cell (ν.δ^{−1}) making commutative

[figure: (p^{∗} ∧ r^{∗})(p ∧ r); r^{∗}p; ν; (p^{∗} ∧ r^{∗})(p ∧ r); p^{∗}p ∧ p^{∗}r ∧ r^{∗}r; δ; p^{∗}p ∧ p^{∗}r ∧ r^{∗}r; r^{∗}p; μ]  (5)

We remark that a local product of maps is not generally a map. (In the case of the bicategory of relations a local product of maps is a partial map.) Observe though that if A is such that the maps d: A [figure] A ⊗ A and t: A [figure] I have right adjoints in M then A is a cartesian object in M in the terminology of [CKW] and [CKVW]. In this case p ∧ r:A ⊗ A [figure] A is the map that provides “internal” binary products for A.

For maps f, g:T [figure] A we write, as in 2.3, A(f, g) for the composite f^{∗}g and observe that the following three kinds of 2-cells are in natural bijective correspondence

[figure: T; A; f; α; T; A; g; T; T; 1_{T}; α̂; T; T; A(f,g); T; A; g^{∗}; α^{∗}; T; A; f^{∗}]

We have 3.15. Lemma. The hom-category M(T, A) can be equivalently described as the category whose objects are the maps f : T [figure] A and whose hom-sets M(T, A)(f, g) are the sets M(T, T)(1_{T} , A(f, g)) with composition given by pasting composites of the form

[figure: T; T; 1_{T}; α̂; T; T; A(f,g); T; T; 1_{T}; β̂; T; T; A(g,h); T; T; A(f,h); f^{∗}ϵ_{g}h]

Proof. It is a simple exercise with mates to show that the pasting composite displayed is β̂α. We note that ˆ1_{f} = η_{f}.

3.16. Lemma. For objects f, h, g, k of M(T, A), the whisker composite

[figure: T; A ⊗ A; (g,k); A ⊗ A; A ⊗ A; (p^{∗}∧r^{∗})(p∧r); A ⊗ A; A ⊗ A; p^{∗}p∧p^{∗}r∧r^{∗}r; A ⊗ A; T; (f,h)^{∗}; δ=(ππ,πρ,ρρ)]

being in the notation of 2.3

(p^{∗} ∧ r^{∗})(p ∧ r)((f, h)(g, k)) [figure: δ((f,h)(g,k))] (p^{∗}p ∧ p^{∗}r ∧ r^{∗}r)((f, h)(g, k))

is

[figure: T; A; g∧k; A; T; f^{∗}∧h^{∗}; T; T; A(f,g)∧A(f,k)∧A(h,k); (ππ,πρ,ρρ)]

In fact, (p ∧ r)(g, k) ≅ g ∧ k and (f, h)^{∗}(p^{∗} ∧ r^{∗}) ≅ f^{∗} ∧ h^{∗}.

Proof. We have

(p ∧ r)(g, k) ≅ d^{∗}(p ⊗ r)d(g, k) ≅ d^{∗}(p ⊗ r)((g, k) ⊗ (g, k))d ≅ d^{∗}(g ⊗ k)d ≅ g ∧ k

while

(f, h)^{∗}(p^{∗} ∧ r^{∗}) ≅ (f, h)^{∗}d^{∗}(p^{∗} ⊗ r^{∗})d ≅ ((p ⊗ r)d(f, h))^{∗}d ≅ ((f ⊗ h)d)^{∗}d ≅ f^{∗} ∧ h^{∗}

On the other hand, precomposing with maps and postcomposing with right adjoints preserves local products so that we have

(f, h)^{∗}(p^{∗}p ∧ p^{∗}r ∧ r^{∗}r)(g, k) ≅ (f, h)^{∗}(p^{∗}p)(g, k) ∧ (f, h)^{∗}(p^{∗}r)(g, k) ∧ (f, h)^{∗}(r^{∗}r)(g, k)

≅ f^{∗}g ∧ f^{∗}k ∧ h^{∗}k

Assembling these results in hom-notation gives the statement.

The whisker composite in Lemma 3.16 should be thought of as the instantiation of δ at ((f, h)(g, k)) and we have been deliberately selective in mixing our notations in the concluding diagram of the statement; (ππ, πρ, ρρ) being more informative than δ((f, h)(g, k)). If we instantiate the rest of diagram (5) at ((f, h)(g, k)), which is to say whisker with (f, h)^{∗}(−)(g, k), then the result is clearly the lower triangle below.

[figure: (f^{∗} ∧ h^{∗})(g ∧ k); A(h, g); ρπ; (f^{∗} ∧ h^{∗})(g ∧ k); A(f, g) ∧ A(f, k) ∧ A(h, k); (ππ,πρ,ρρ); A(f, g) ∧ A(f, k) ∧ A(h, k); A(h, g); μ((f,h),(g,k); (f^{∗} ∧ h^{∗})(g ∧ k); A(f, g) ∧ A(f, k) ∧ A(h, k); 1_{T}; (f^{∗} ∧ h^{∗})(g ∧ k); Ξ; 1_{T}; A(f, g) ∧ A(f, k) ∧ A(h, k); (α,β,γ)]  (6)

In the top triangle above it is clear that a 1_{T} -element of A(f, g) ∧ A(f, k) ∧ A(h, k) is exactly an “S” shaped configuration in M(T, A) of the form

[figure: f; g; α; f; k; β; h; k; γ]

For A Frobenius we will be interested in lifting 1_{T} -elements of A(f, g) ∧ A(f, k) ∧ A(h, k) though the isomorphism

(ππ, πρ, ρρ):(f^{∗} ∧ h^{∗})(g ∧ k) [figure] A(f, g) ∧ A(f, k) ∧ A(h, k)

As we discussed in 2.3, we do not have precise knowledge of general 1_{T} -elements Ξ of

(f^{∗} ∧ h^{∗})(g ∧ k) = ((p^{∗} ∧ r^{∗})(p ∧ r))((f, h)(g, k))

but those obtained by pasting a 1_{T} -element of (p^{∗} ∧ r^{∗})((f, h), x) to a 1_{T} -element of (p ∧ r)(x, (g, k)), for some x:T [figure] A present no difficulty. (Here, p^{∗} ∧ r^{∗} is the S and p ∧ r is the R of 2.3.) Since

(p^{∗} ∧ r^{∗})((f, h), x) = (f, h)^{∗}(p^{∗} ∧ r^{∗})x ≅ (f^{∗} ∧ h^{∗})x ≅ f^{∗}x ∧ h^{∗}x = A(f, x) ∧ A(h, x)

and

(p ∧ r)(x, (g, k)) = x^{∗}(p ∧ r)(g, k) ≅ x^{∗}(g ∧ k) ≅ x^{∗}g ∧ x^{∗}k = A(x, g) ∧ A(x, k)

(where we have used Lemma 3.16 in each derivation) we see that these special 1_{T} -elements of (f^{∗} ∧ h^{∗})(g ∧ k) are given by (equivalence classes of)“X” shaped configurations in M(T, A) of the form

[figure: f; x; ξ; h; x; ζ; x; g; η; x; k; ω]

It is convenient to write such a 1_{T} -element of (f^{∗} ∧ h^{∗})(g ∧ k) as the following pasting composite

[figure: T; A; g∧k; T; T; 1_{T}; T; A; x; T; A; x; T; T; 1_{T}; T; A; x^{∗}; A; T; f^{∗}∧h^{∗}; T; A; x^{∗}; T; T; 1_{T}; (η,ω); η_{x}; (ξ^{∗},ζ^{∗})]  (7)

Invertibility of δ = (ππ, πρ, ρρ):(f^{∗} ∧h^{∗})(g ∧k) [figure] X(f, g)∧X(f, k)∧X(h, k) tells us that, for every “S” configuration (α, β, γ), there is a unique 1_{T} -element Ξ of (f^{∗} ∧ h^{∗})(g ∧ k) such that δΞ = (α, β, γ). When, as in several classical situations, every 1_{T} -element Ξ comes from an “X” configuration we have motivation for the colloquial name “S”=“X” for the Frobenius condition. (In fact one says “S”=“X”=“Z” when the second “equation” is not derivable from the first but we have Lemma 3.2.)

3.17. Lemma. For a 1_{T} -element Ξ (see (6)) arising from an “X”configuration as in (7), δΞ = (ηξ, ωξ, ωζ) and νΞ = ηζ.

Proof. For δΞ we treat the components separately. For the first, we paste

[figure: T; A; g∧k; T; A; g; A; T; f^{∗}∧h^{∗}; A; T; f^{∗}; π; π]

to (7) and obtain the 1_{T} -element of A(f, g). To see this as a 2-cell f [figure] g paste onto it

[figure: T; A; g; T; T; 1_{T}; T; A; x; T; A; x; T; T; 1_{T}; T; A; x^{∗}; A; T; f^{∗}; T; A; x^{∗}; T; T; 1_{T}; η; η_{x}; ξ^{∗}]

[figure: A; T; f^{∗}; T; A; f; A; A; 1_{A}; ϵ_{f}]

(at f^{∗}) which is the “unhatting” bijection and observe that the result is ηξ:f [figure] g. For the second, first paste (π, ρ) and then paste ϵ_{f}. For the third, first paste (ρ, ρ) and then paste ϵ_{h}. For νξ, paste (ρ, π) to (7) and then paste ϵ_{h} (at h^{∗}).

The 2-cell μ of (5) when instantiated as in (6) provides a completion of “S” configurations, as by the dotted arrow below. (It ultimately has the air of a Malcev operation.)

[figure: f; g; f; k; h; k; h; g]

In particular, given a 2-cell α: f [figure] g we have the “S” configuration (1_{f}, α, 1_{g}) and we write α^{†} = μ(1, α, 1).

3.18. Lemma. α^{†} = α^{−1}

Proof. Using the hom-notation of 2.3 it is easy to see that the following diagram commutes and that the clockwise composite is αα^{†}:1_{T} [figure] A(g, g).

[figure: 1_{T}; A(f, g) ∧ A(f, g) ∧ A(g, g); (α,α,1); A(f, f) ∧ A(f, g) ∧ A(g, g); 1_{T}; (1,α,1); A(f, f) ∧ A(f, g) ∧ A(g, g); A(f, g) ∧ A(f, g) ∧ A(g, g); A(f,α)∧A(f,g)∧A(g,g); A(f, g) ∧ A(f, g) ∧ A(g, g); A(g, g); μ; A(f, f) ∧ A(f, g) ∧ A(g, g); A(f, g) ∧ A(f, g) ∧ A(g, g); A(f,α)∧A(f,g)∧A(g,g); A(f, f) ∧ A(f, g) ∧ A(g, g); A(g, f); μ; A(g, f); A(g, g); A(g,α)]  (8)

Let Ξ be the “X” configuration

[figure: f; g; α; g; g; 1_{g}; g; g; 1_{g}; g; g; 1_{g}]

46  R.F.C WALTERS AND R.J. WOOD

From Lemma 3.17 it is clear that δΞ = (α, α, 1). Thus the anti-clockwise composite of diagram (8) is

μ(α, α, 1) = μδΞ = νΞ = 1_{g}

where the last equality is also from Lemma 3.17, so that αα^{†} = 1_{g}.

Similarly, the composite α^{†}α is the clockwise composite in the commutative diagram.

[figure: 1_{T}; A(f, f) ∧ A(f, g) ∧ A(f, g); (1,α,α); A(f, f) ∧ A(f, g) ∧ A(g, g); 1_{T}; (1,α,1); A(f, f) ∧ A(f, g) ∧ A(g, g); A(f, f) ∧ A(f, g) ∧ A(f, g); A(f,f)∧A(f,g)∧A(α,g); A(f, f) ∧ A(f, g) ∧ A(f, g); A(f, f); μ; A(f, f) ∧ A(f, g) ∧ A(g, g); A(f, f) ∧ A(f, g) ∧ A(f, g); A(f,f)∧A(f,g)∧A(α,g); A(f, f) ∧ A(f, g) ∧ A(g, g); A(g, f); μ; A(g, f); A(f, f); A(α,f)]

Notice that (1, α, α) is obtained by applying δ to the “X” configuration

[figure: f; f; 1_{f}; f; f; 1_{f}; f; f; 1_{f}; f; g; α]

The rest of the proof proceeds as above.

This completes the proof of Theorem 3.13.

References

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[LACK] S. Lack. Composing PROPs, Theory and Applications of Categories 13(2004), 147–163.

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[MSW] M. Menni, N. Sabadini, and R.F.C. Walters. A universal property of the monoidal 2-category of cospans of finite linear orders and surjections, Theory and Applications of Categories 18 (2007), 631–653.

[RSW] R. Rosebrugh, N. Sabadini, and R.F.C. Walters. Generic commutative separable algebras and cospans of graphs, Theory and Applications of Categories 15 (2005), 264–177.

[S&W] R. Street and R.F.C. Walters. Yoneda structures on 2-categories, J. Algebra 50 (1978), 350–379.

[Wd] R.J. Wood. Proarrows 1, Cahiers Topologie Géom. Différentielle Catégoriques 23 (1982), 279–290. [W&W] R.F.C. Walters and R.J. Wood. Bicategories of Spans as Cartesian Bicategories, in preparation.

Dipartimento di Scienze delle Cultura

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