Theory and Applications of Categories, Vol. 30, No. 47, 2015, pp. 1624–1646.

THE WAVES OF A TOTAL CATEGORY

R.J. WOOD

Abstract. For any total category 𝒦 , with defining adjunction ⋁ ⊣ Y : 𝒦 [figure] set^{𝒦 ^{op}} , the expression W (A)(K) = set^{set^{𝒦 ^{op}}} (𝒦 (A, ⋁ −), [K, −]), where [K, −] is evaluation at K, provides a well-defined functor W : 𝒦 [figure] 𝒦̂ = set^{𝒦 ^{op}} . Also, there are natural transformations β : W ⋁ [figure] 1𝒦̂ and γ : ⋁ W [figure] 1_{𝒦} satisfying ⋁ β = γ ⋁ and βW = W γ. A total 𝒦 is totally distributive if ⋁ has a left adjoint. We show that 𝒦 is totally distributive iff γ is invertible iff W ⊣ ⋁. The elements of W (A)(K) are called waves from K to A.

Write 𝒦̃(K, A) for W (A)(K). For any total 𝒦 there is an associative composition of waves. Composition becomes an arrow •: 𝒦̃∘_{𝒦} 𝒦̃ [figure] 𝒦̃. Also, there is an augmentation ¯ (−) : 𝒦̃(−, −) [figure] 𝒦 (−, −) corresponding to a natural δ: W [figure] Y constructed via β. We show that if 𝒦 is totally distributive then •: 𝒦̃∘_{𝒦} 𝒦̃ [figure] 𝒦̃is invertible and then 𝒦̃supports an idempotent comonad structure. In fact, 𝒦̃∘_{𝒦} 𝒦̃= 𝒦̃∘𝒦̃𝒦̃so that • is the coequalizer of •𝒦 and 𝒦 •, making 𝒦̃a taxon in the sense of Koslowski [KOS]. For a small taxon 𝒯 , the category of interpolative modules iMod(1, 𝒯 ) is totally distributive [MRW]. Here we show, for any totally distributive 𝒦 , that there is an equivalence 𝒦 [figure] iMod(1, 𝒦̃).

1. Preliminaries

1.1. Most of our notations and conventions are carried over from [MRW]. For a category 𝒦 , we write |𝒦 | for the set of objects of 𝒦 . If K and A are objects of 𝒦 , 𝒦 (K, A) denotes the set of arrows from K to A. We assume the existence of full categories of sets called set and SET, both toposes, with set a (full) subcategory of SET, and |set| an object of SET. The sets in set are called small sets. We assume that set has all sums indexed by objects of set and SET has all sums indexed by objects of SET. We will write i: set [figure]SET for the inclusion. We write CAT for the 2-category of category objects in SET. If a category 𝒦 in CAT has all its hom-sets 𝒦 (K, A) in set, we say that 𝒦 is locally small. For 𝒦 and 𝒜 in CAT, we write PRO(𝒦 , 𝒜) for the category

The author gratefully acknowledges financial support from the Canadian NSERC. Diagrams typeset using M. Barr’s diagxy package

Received by the editors 2015-07-14 and, in revised form, 2015-11-09. Transmitted by Ross Street. Published on 2015-11-30. 2010 Mathematics Subject Classification: 18A30. Key words and phrases: adjunction, totally cocomplete, totally distributive, taxon, i-module, proar-

row equipment.

\begin{matrix}c \\ ◯ \end{matrix} R.J. Wood, 2015. Permission to copy for private use granted.

of all functors 𝒜^{op} × 𝒦 [figure]SET We write P : 𝒦 [figure] 𝒜 for an object of PRO(𝒦 , 𝒜) and say that it is small if P factors through i: set [figure]SET. With the usual compositions, PRO is a bicategory and we have the usual proarrow equipment (−)_{∗}: CAT [figure]PRO.

2. The Yoneda functor and total categories

2.1. The following subsection elaborates on the (−)_{∗} notation, which is used to provide precision in many of the proofs in the paper.

A locally small category 𝒦 has a Yoneda functor Y : 𝒦 [figure]CAT(𝒦 ^{op}, set). We will usually write ˆ(−) : CAT^{coop} [figure]CAT for CAT((−)^{op}, set). Note that if 𝒦 is locally small then 𝒦̂ is locally small if and only if 𝒦 is equivalent to a small category — the “only if” clause being an important result of [F&S] — but, in any event, 𝒦̂ is in CAT. The objects of 𝒦̂ are small profunctors P : 1 [figure] 𝒦 . Henceforth, 𝒦 will denote a locally small category. For A in 𝒦 , A : 1 [figure]𝒦 gives rise to the profunctor A_{∗} : 1 [figure] 𝒦 and A_{∗} = 𝒦 (−, A) = Y (A). Note that A^{∗} = 𝒦 (A, −) : 𝒦 [figure] 1 is right adjoint to A_{∗} as a profunctor. Also, K : 1 [figure]𝒦 gives rise to K̂ : 𝒦̂ [figure]ˆ1 = set, which is easily seen to be evaluation at K, and since 1 is small and 𝒦 is locally small, K̂ has both left and right adjoints. We will write K_{+} : set [figure]𝒦̂for the right adjoint of K̂. We have K_{+}(X)(A) = X^{𝒦 (K,A)} . Since CAT is (cartesian) closed and 𝒦̂= CAT(𝒦 ^{op}, set) = set^{𝒦 ^{op}} we have evaluation 𝒦 ^{op} × 𝒦̂ [figure]set which sends a pair (K, P) to K̂(P) = P (K). However it is convenient for us to write [K, P], functorially, for the common value K̂(P) = P (K) and denote the elements of [K, P] by arrows e: K [figure]P. Such arrows are composable with arrows of the form k: L [figure]K in 𝒦 , where the composite ek: L [figure]P is P (k)(e) = [k, P](e), and with arrows of the form t: P [figure]Q in 𝒦̂, where the composite te: K [figure]Q is t(K)(e) = [K, t](e). It is routine that these composites associate so that tek is well defined, making the arrows e: K [figure]P elements of a module. In fact it may be useful to observe that [−, −] : 𝒦̂ [figure] 𝒦 is an isomorph of the small profunctor Y^{∗} (because Y^{∗}(K, P) = 𝒦̂(K_{∗}, P)) but [−, −] exists as a small profunctor, for any 𝒦 , irrespective of local smallness of 𝒦 .

Yoneda’s Lemma says that the function ξ_{K,P} : 𝒦̂(K_{∗}, P) [figure][K, P] given by ξ_{K,P} (t) = t(K)(1_{K}) is an isomorphism, natural in K and P. We will write ζ_{K,P} : [K, P] [figure]𝒦̂(K_{∗}, P) for the inverse of ξ_{K,P} and ζ_{K,P} (e) = e_{∗}. Thus e_{∗} is the unique natural transformation that corresponds to element e. We have [K, A_{∗}] = 𝒦 (K, A) and now our e_{∗} notation specializes to the equality:

[figure: 𝒦̂(K_{∗}, A_{∗}); 𝒦 (K, A); Y_{K,A} = (−)_{∗_{K,A}}; [K, A_{∗}]; 𝒦̂(K_{∗}, A_{∗}); ζ_{K,A∗}; [K, A_{∗}]; 𝒦 (K, A); ∥]

2.2. Following [S&W] we say that the locally small 𝒦 is totally cocomplete (usually abbreviated to total ) if Y has a left adjoint, which will then be called X. Furthermore, η:

1𝒦̂ [figure]Y X and ϵ: XY [figure]1_{𝒦} will be used consistently for the unit and counit respectively, so that we speak of the adjunction (η, ϵ : X ⊣ Y). However, just as we write Y (A) = A_{∗} = 𝒦 (−, A) so also we frequently will write X(P) = ⋁ P. Since Y is fully faithful, ϵ is invertible. In terms of the notations introduced so far, we have components ηP : P [figure](⋁ P)_{∗} in 𝒦̂and invertible components ϵA: ⋁ A_{∗} [figure]A in 𝒦 . So for e: K [figure]P, (ηP)(K)(e) = ηP.e can safely be abbreviated to ηe : K [figure](⋁ P)_{∗} which is simply an arrow ηe: K [figure]⋁ P in 𝒦 . The adjunction equations for X ⊣ Y are:

[figure: A_{∗}; (⋁ A_{∗})_{∗}; ηA_{∗}; A_{∗}; A_{∗}; 1_{A∗}; (⋁ A_{∗})_{∗}; A_{∗}; (ϵA)_{∗}; in 𝒦̂, for all A in 𝒦 and; ⋁(⋁ P)_{∗}; ⋁ P; ⋁ ηP; ⋁(⋁ P)_{∗}; ⋁ P; ϵ ⋁ P; ⋁ P; ⋁ P; 1_{⋁ P}; in 𝒦 , for all P in 𝒦̂]

But ϵA is an isomorphism and since (ϵA)_{∗}^{−1} = ηA_{∗} and ηA_{∗} is a_{∗} for a unique a : A [figure]⋁ A_{∗} in 𝒦 , it is consistent with our other abbreviations, to write ηA for this a so that (ηA)_{∗} = ηA_{∗}. Now, the first of the triangle equations can be rewritten as

[figure: A; ⋁ A_{∗}; ηA; A; A; 1_{A}; ⋁ A_{∗}; A; ϵA; in 𝒦 , for all A in 𝒦]

From the second of the triangle equations, we deduce further that the special case η ⋁ P of the abbreviation of the last paragraph satisfies η ⋁ P = ⋁ ηP, for all P in 𝒦̂.

2.3. Remark. For any adjunction (η, ϵ; X ⊣ Y) : 𝒦 [figure]ℒ, it is classical (see, for example, Theorem 1, page 90 of [MAC]) that ϵ is invertible if and only if Y is fully faithful. In this event, ηY = Y ϵ^{−1}. In fact, if Y is also dense as in the case of the Yoneda functor (meaning that 1_{ℒ} is the left Kan extension of Y along Y), then η is uniquely determined by the requirement that ηY = Y ϵ^{−1}.

2.4. For P in 𝒦̂, ⋁ P is the colimit of the discrete fibration, P̄: elP [figure]𝒦 determined by P and is easily seen to be given by ∫^{K}[K, P] · K, where · denotes multiple, so that [K, P] · K is the [K, P]-fold sum of copies of K, and the integral ∫ is the familiar quotient of the corresponding sum ∑. Thus, for each element e : K [figure]P, we have a colimit

injection i_{e} : K [figure]⋁ P, in 𝒦 . Since P is the colimit of elP [figure: P̄] 𝒦 [figure: (−)_{∗}] 𝒦̂, with colimit injections K_{∗} [figure: e_{∗}] P, in 𝒦̂, we have

[figure: K_{∗}; P; e_{∗}; P; (⋁ P)_{∗}; ηP; K_{∗}; (⋁ P)_{∗}; (i_{e})_{∗}]

In fact, for any adjunction with invertible counit, colimit comparisons for the right adjoint are given by instances of the unit. From these observations and the identity above, the following identities follow easily: For each pair K [figure: e] P [figure: t] Q, we have

[figure: K; ⋁ P; i_{e}; ⋁ P; ⋁ Q; ⋁ t; K; ⋁ Q; i_{te}]

For an element f : K [figure]A_{∗}, one and the same as an arrow f : K [figure]A, we have

[figure: K; ⋁ A_{∗}; i_{f}; ⋁ A_{∗}; A; ϵA; K; A; f]

For any element e: K [figure]P, we have

[figure: K; ⋁ K_{∗}; ηK; ⋁ K_{∗}; ⋁ P; ⋁ e_{∗}; K; ⋁ P; i_{e}]

2.5. Remark. Our “sup”-notation while suggestive for those familiar with (constructively) complete distributive latices is less well-adapted when we want to speak of preservation by a functor F : 𝒦 [figure]ℒ of the colimits we call ⋁ P (unless F̂ : ℒ̂ [figure]𝒦̂has a left adjoint). The colimit ⋁ P is more customarily denoted P · 1_{𝒦} , a particular weighted colimit, and is said to be preserved by F when the arrows F i_{x}: F K [figure]F ⋁ K provide an isomorphism P · F [figure: ≃]F (P · 1_{𝒦} ) of weighted colimits. Note that, for total 𝒦 and any functor F : 𝒦 [figure]ℒ for which P · F exists in ℒ, the colimit comparison P · F [figure]F (⋁ P) is given by the natural transformation P [figure]ℒ (F −, F (⋁ P)) determined by the family ⟨F i_{x}: F K [figure]F (⋁ P)⟩_{x∈[K,P]}.

2.6. Remark. In the next section, in Theorem 3.1, we make passing use of the fact that set is cototal (meaning that set^{op} is total). This follows from the fact that, for any functor E: set [figure]set, set^{set}(E, 1_{set}) is small. To prove the latter, proceed as in the proof of Lemma 2.9 of [MRW] which establishes an apparently more complicated result.

3. The waves of a total category

3.1. Theorem. For any total category 𝒦 , there is a functor W : 𝒦 [figure]𝒦̂given by

W (A)(K) = set𝒦̂(𝒦 (A, −). ⋁, [K, −]) = set𝒦̂(A^{∗}. ⋁, K̂)

whose transpose 𝒦 ^{op}×𝒦 [figure]set will be denoted 𝒦̃(−, −) so that W (A)(K) = 𝒦̃(K, A). We adapt [J&J] and call 𝒦̃(K, A) the set of waves from K to A and write ω: K [figure]A for a typical wave. Moreover, there is a natural transformation δ: W [figure]Y, whose A-component δ(A) : W (A) [figure]Y (A) is a natural transformation whose K-component, δ(A)(K), is denoted

δ(K, A) : W (A)(K) = 𝒦̃(K, A) [figure]𝒦 (K, A) = Y (A)(K)

defined by δ(K, A)(K [figure: ω] A) = ωA_{∗}(ηA) ∈ 𝒦 (K, A) and we write δ(K, A)(ω) = \overline{ω}.

Proof. The definition of W, equivalently of 𝒦̃, is manifestly functorial. The force of the claim is that 𝒦̃(K, A) is small, meaning that it is an object of set. In the following calculation we use the fact that set is cototal and that its identity functor is represented by set(1, −). Note that we use ⋀ for a functor that provides cototality.

𝒦̃(K, A) = set𝒦̂(A^{∗}. ⋁, K̂)
≅ set^{set}(A^{∗}. ⋁ .K_{+}, 1_{set})
≅ (set^{set})^{op}(set(1, −), A^{∗}. ⋁ .K_{+})
≅ set(1, ⋀(A^{∗}. ⋁ .K_{+}))
≅ ⋀(A^{∗}. ⋁ .K_{+})

The second to last set is small, since set is a locally small category, hence 𝒦̃(K, A) is small.

For k: L [figure]K, ω: A^{∗}. ⋁ [figure]K̂, and a: A [figure]B, the following unambiguous composite of natural transformations:

B^{∗}. ⋁ [figure: a^{∗}. ⋁] A^{∗}. ⋁ [figure: ω] K̂ [figure: k̂] L̂

defines the wave

aωk = 𝒦̃(k, a)(ω) = (L [figure: k] K [figure: ω] A [figure: a] B)

(Both (−)^{∗} and ˆ(−) have variance (−)^{coop}.) For naturality of δ it suffices to show that \overline{aωk} = a\overline{ω}k. Here, and elsewhere, the following configuration diagram is helpful. For a P in 𝒦̂and an arrow f : A [figure]⋁ P in 𝒦 , (ωP)(f) is a K-element of P as in

[figure: P; ⋁ P; K; P; (ωP)(f); K; A; ω; A; ⋁ P; f]  (1)

(Observe that the right hand vertical arrow of Diagram 1 can equally be displayed as f : A [figure](⋁ P)_{∗}.) In terms of Diagram 1 we see ((aωk)(P))(f) as:

[figure: P; P; L; P; ((aωk)(P))(f)=((ωP)(fa))k; L; K; k; K; P; (ωP)(fa); P; ⋁ P; K; P; (ωP)(fa); K; A; ω; A; ⋁ P; fa; ⋁ P; ⋁ P; A; ⋁ P; fa; A; B; a; B; ⋁ P; f]

Since \overline{aωk} = (ωB_{∗})(ηB.a)k, to show \overline{aωk} = a\overline{ω}k is to show

((ωB_{∗})(ηB.a))k = a((ωA_{∗})(ηA))k

for which the case k = 1_{K} clearly suffices. By naturality of η (here standing for ϵ^{−1}) we can rewrite the condition as

(ωB_{∗})(⋁ a_{∗}.ηA) = a((ωA_{∗})(ηA))

which follows from

[figure: [K, A_{∗}]; [K, B_{∗}]; [K,a_{∗}]; 𝒦 (A, ⋁ A_{∗}); [K, A_{∗}]; ωA_{∗}; 𝒦 (A, ⋁ A_{∗}); 𝒦 (A, ⋁ B_{∗}); 𝒦 (A,⋁ a_{∗}); 𝒦 (A, ⋁ B_{∗}); [K, B_{∗}]; ωB_{∗}]

an instance of naturality of ω, evaluated at ηA.

3.2. Remark. The definition of a wave ω: K [figure]A as a natural transformation A^{∗}. ⋁ [figure]K̂ provides that it has, for each P ∈ 𝒦̂, a P -component which is a function ωP : 𝒦 (A, ⋁ P) [figure][K, P]. If, instead of a total category 𝒦 , we were speaking of a complete lattice 𝒦 with elements K and A, we would have K totally below A, written K << A, if and only if,

(∀P ∈ 𝒦̂)(A ≤ ⋁ P implies K ∈ P)

where now 𝒦̂is the lattice of down-sets of 𝒦 . The reader familiar with [LAW] will see the connection between waves and the totally below relation. In the study of continuous posets the latter relation, with universal quantification restricted to up-directed down-sets, has long been called the way below relation. In their paper on continuous categories, [J&J], Johnstone and Joyal generalized the way below relation to wavy arrows with a definition analogous to ours. Since the word wave already connotes a directional phenomenon and physical wavy arrows would not fly very well, we speak more simply of waves.

Whenever possible we will abbreviate (ωP)(f) by ω(f). We illustrate this by exhibiting naturality of ω in P in terms of Diagram 1:

[figure: P; ⋁ P; K; P; ω(f); K; A; ω; A; ⋁ P; f; Q; ⋁ Q; P; Q; t; P; ⋁ P; ⋁ P; ⋁ Q; ⋁ t; K; Q; ω((⋁ t)f)]

for a t: P [figure]Q in 𝒦̂.

The reader is encouraged to reprove the equation \overline{aω} = a\overline{ω} (part of the proof of Theorem 3.1) in terms of Diagram 1.

3.3. Remark. Any full set of natural transformations between a parallel pair of functors is the value of an end and here we see that

𝒦̃(K, A) = ∫_{P} set(𝒦 (A, −). ⋁ P , [K, P]) ≅ ∫_{P} [K, P]^{𝒦 (A,⋁ P)}

where the end is over P in 𝒦̂. Of course, we can see 𝒦̃as a small profunctor from 𝒦 to 𝒦 and the display above reveals the following diagram as a right extension diagram in the bicategory of profunctors. (Recall that ⋁_{∗}(K, P) = 𝒦 (K, ⋁ P). We noted in subsection 2.1 that [−, −] ≅ Y^{∗}.)

[figure: 𝒦̂; 𝒦; Y^{∗}≅[−,−]; 𝒦̂; 𝒦; ^{⋁^{∗}}; 𝒦; 𝒦; 𝒦̃]

The totally below relation for a sup-lattice is an order ideal and is given by an analogous right extension.

3.4. Proposition. For any wave ω: K [figure]A in a total category 𝒦 , any P in 𝒦̂, and any arrow f : A [figure]⋁ P in 𝒦 , we have the following equation:

[figure: P; (⋁ P)_{∗}; ηP; K; P; ω(f); K; A; \overline{ω}; A; (⋁ P)_{∗}; f]  (2)

Proof.

[figure: P; ⋁ P; K; P; ω(f); K; A; ω; A; ⋁ P; f; (⋁ P)_{∗}; ⋁(⋁ P)_{∗}; P; (⋁ P)_{∗}; ηP; P; ⋁ P; ⋁ P; ⋁(⋁ P)_{∗}; K; (⋁ P)_{∗}; ω(⋁ ηP.f); ⋁ P; ⋁(⋁ P)_{∗}; η ⋁ P; ⋁ P; ⋁(⋁ P)_{∗}; ⋁ ηP; A; ⋁ A_{∗}; ηA; ⋁ A_{∗}; ⋁(⋁ P)_{∗}; ⋁ f_{∗}; A_{∗}; ⋁ A_{∗}; K; A_{∗}; ω(ηA); K; A; ω; A; ⋁ A_{∗}; ηA; (⋁ P)_{∗}; ⋁(⋁ P)_{∗}; A_{∗}; (⋁ P)_{∗}; f_{∗}; A_{∗}; ⋁ A_{∗}; ⋁ A_{∗}; ⋁(⋁ P)_{∗}; ⋁ f_{∗}; K; (⋁ P)_{∗}; ω(⋁ f_{∗}.ηA)]

Consider the diagram on the left above. Amongst other things, it shows by naturality of ω that ηP.ω(f) = ω(⋁ ηP.f). But it also shows that ⋁ ηP.f = ⋁ f_{∗}.ηA. So we have ηP.ω(f) = ω(⋁ f_{∗}.ηA). Turning to the diagram on the right, we see again by naturality of ω that ω(⋁ f_{∗}.ηA) = f_{∗}.ω(ηA). But ω(ηA) = \overline{ω} and the element f_{∗}.\overline{ω} is equally the arrow f\overline{ω} so that we have ηP.ω(f) = f.\overline{ω} as required.

3.5. Definition. For any total category 𝒦 , there are natural transformations

β: W X [figure]1𝒦̂ and γ: XW [figure]1_{𝒦}

where βP : 𝒦̃(−, ⋁ P) [figure]P has β(K, P) = βP (K) : 𝒦̃(K, ⋁ P) [figure][K, P] given by

β(K, P)(ω: K [figure]⋁ P) = ωP (1_{⋁ P}) = ω(1) : K [figure]P

and γA: ⋁ 𝒦̃(−, A) [figure]A in 𝒦 is defined to be ϵA. ⋁ δA so that

γ = ϵ.Xδ

3.6. Remark. It should be remarked that, for any wave ω : K [figure]A, any P, and any f : A [figure]⋁ P, we have

ωP (f) = βP (f.ω)  (3)

It follows that, for parallel waves ψ and ω, ψ = ω if and only if, for all P and all f : A [figure]⋁ P, βP (f.ψ) = βP (f.ω).

3.7. Proposition. For any wave ω: K [figure]A, seen as an element ω: K [figure]W A, in any total category 𝒦 , we have the following equation:

[figure: K; ⋁ W A; i_{ω}; K; A; ω̄; ⋁ W A; A; γA]

Proof.

[figure: K; ⋁ W A; i_{ω}; K; ⋁ Y A; i_{δ.ω}=iω̄; ⋁ W A; ⋁ Y A; ⋁ δA; ⋁ Y A; A; ϵA; ⋁ W A; A; γA; K; A; ω̄]

We have used the second and third identities of sub-section 2.4 and δ.ω = ω̄. 3.8. Lemma. For any total category,

[figure: W; W XY; W ϵ^{−1}; W; Y; δ; W XY; Y; βY]

Proof. For a total category 𝒦 , the equation of the statement is equivalent to the following equation, for all K and A in 𝒦 ,

[figure: 𝒦̃(K, A); 𝒦̃(K, ⋁ A_{∗}); 𝒦̃(K,ϵ^{−1}A); 𝒦̃(K, A); 𝒦 (K, A) = [K, A_{∗}]; δ(K,A); 𝒦̃(K, ⋁ A_{∗}); 𝒦 (K, A) = [K, A_{∗}]; β(K,A_{∗})]

which follows immediately from the definitions. 3.9. Lemma. For any total category 𝒦 ,

[figure: W X; 1𝒦̂; β; W X; Y X; δX; 1𝒦̂; Y X; η]

Proof. Apply the equation of Lemma 3.8 to X, note ϵ^{−1}X = ηX, and invoke naturality of β.

3.10. Proposition. For any total category 𝒦 , β: W X [figure]1𝒦̂and γ: XW [figure]1_{𝒦} satisfy

Xβ = γX: XW X [figure] X  and  βW = W γ: W XW [figure] W

Proof. To show the first equation, observe that Xβ is the unique natural transformation satisfying Y Xβ.ηW X = η.β. From Lemma 3.9 we have η.β = δX so it suffices to show that Y γX.ηW X = δX. From Definition 3.5 it follows that we must show Y ϵX.Y XδX.ηW X = δX, which is shown by

[figure: W X; Y X; δX; Y XW X; W X; ηW X; Y XW X; Y X; Y X; Y X; 1_{Y X}; Y XW X; Y XY X; Y XδX; Y XY X; Y X; Y ϵX; Y X; Y XY X; ηY X]

For the second equation, observe that the description of β gives βW : W XW [figure] W, with βW (A) = β(W A) so that we have

βW A(K) = β(K, W A) : 𝒦̃(K, ⋁(W A)) [figure][K, W A] = 𝒦̃(K, A)

and, for a wave ω: K [figure]⋁(W A), we have the wave

β(K, W A)(ω) = ω(W A)(1_{⋁(W A)}) = ω(1) : K [figure]A  (4)

Also, we have

W γA(K) = 𝒦̃(K, γA) : 𝒦̃(K, ⋁(W A)) [figure]𝒦̃(K, A)

and hence, for a wave ω: K [figure]⋁(W A), we have the wave

𝒦̃(K, γA)(ω) = γA.ω: K [figure]A  (5)

We must show that the parallel waves ω(1) and γA.ω, of Equations 4 and 5 respectively, are equal. So take P in 𝒦̂, f : A [figure]⋁ in 𝒦 , and consider the following diagram:

[figure: W A; ⋁(W A); K; W A; ω(1); K; ⋁(W A); ω; ⋁(W A); ⋁(W A); 1; W (⋁ P); ⋁(W (⋁ P)); W A; W (⋁ P); W f; W A; ⋁(W A); ⋁(W A); ⋁(W (⋁ P)); ⋁(W f); P; ⋁ P; W (⋁ P); P; βP; W (⋁ P); ⋁(W (⋁ P)); ⋁(W (⋁ P)); ⋁ P; ⋁(W (⋁ P)); ⋁ P; γ ⋁ P; ⋁(W (⋁ P)); ⋁ P; ⋁(βP); ⋁(W A); A; γA; A; ⋁ P; f]

Observe that we have used the first, established, equation Xβ = γX to assert ⋁(βP) = γ ⋁ P. We now interpret the diagram in the spirit of Diagram 1 and conclude:

βP.W f.ω(1) = ω(⋁(βP). ⋁(W f)) = ω(f.γA) = (γA.ω)(f)

The element ω(1) : K [figure]W A = 𝒦̃(−, A) is of course the wave ω(1) : K [figure]A and, since W f = 𝒦̃(−, f), W f.ω(1) is the wave f.ω(1) : K [figure]⋁ P. It follows that the left side of the display above is β(f.ω(1)) and, by Equation 3 of Remark 3.6, this is simply (ω(1))(f) so that we have shown

(ω(1))(f) = (γA.ω)(f)

which proves that ω(1) = γA.ω. This completes the proof of βW = W γ.

Our somewhat ponderous treatment of elements and the (−)_{∗} notation was developed to provide precision for the somewhat subtle notation in the proof of the next Proposition.

3.11. Proposition. For waves K [figure: ω] L [figure: ψ] A in a total category 𝒦̂,

\overline{ψ}ω = ψ\overline{ω}

Proof. We must show that, for any P in 𝒦̂and any f : A [figure]⋁ P,

((\overline{ψ}ω)P)(f) = ((ψ\overline{ω})P)(f)

Consider

[figure: K; L; ω; P; (⋁ P)_{∗}; ηP; L; P; ψ(f); L; A; \overline{ψ}; A; (⋁ P)_{∗}; f; L; (⋁ P)_{∗}; e; (⋁ L_{∗})_{∗}; (⋁ P)_{∗}; (⋁ ψ(f)_{∗})_{∗}; L_{∗}; (⋁ L_{∗})_{∗}; ηL_{∗}=(ηL)_{∗}; L_{∗}; P; ψ(f)_{∗}; P; (⋁ P)_{∗}; ηP; L_{∗}; (⋁ P)_{∗}; e_{∗}; L_{∗}; ⋁ L_{∗}; K; L_{∗}; \overline{ω}; K; L; ω; L; ⋁ L_{∗}; ηL; P; ⋁ P; L_{∗}; P; ψ(f)_{∗}; L_{∗}; ⋁ L_{∗}; ⋁ L_{∗}; ⋁ P; ⋁ ψ(f)_{∗}; K; L; \overline{ω}; L; P; ψ(f)]

The first diagram provides the configuration for calculating (\overline{ψ}ω)(f) and the extra information needed to apply Proposition 3.4. It names the element e = f\overline{ψ} and the next diagram provides another calculation for e_{∗} enabling us to also conclude e = ⋁ ψ(f)_{∗}.ηL. In the third diagram we use naturality of ω to recalculate ω(e) and also note ψ(f)_{∗}.\overline{ω} = ψ(f).\overline{ω}. But ψ(f).\overline{ω} = (ψ\overline{ω})(f). The step by step details are:

(\overline{ψ}ω)(f) = ω(f\overline{ψ}) by definition of \overline{ψ}ω

= ω(ηP.ψ(f)) by Proposition 3.4
= ω(⋁ ψ(f)_{∗}.ηL) as noted in the second diagram above
= ψ(f)_{∗}.\overline{ω} by naturality of ω
= ψ(f).\overline{ω} as noted in the third diagram above
= (ψ\overline{ω})(f) by definition of ψ\overline{ω}

3.12. Theorem. The waves of a total category 𝒦 admit an associative composition ψ • ω given by the common value \overline{ψ}ω = ψ\overline{ω} and this composition is preserved by \overline{(−)}. In somewhat informal terminology, 𝒦̃is a semicategory with the same objects as 𝒦 and \overline{(−)} : 𝒦̃ [figure]𝒦 is an identity-on-objects semifunctor.

Proof. For composable waves we have

(χ • ψ) • ω = (χ • ψ)\overline{ω} = (\overline{χ}ψ)\overline{ω} = \overline{χ}ψ\overline{ω} = \overline{χ}(ψ\overline{ω}) = \overline{χ}(ψ • ω) = χ • (ψ • ω)

Moreover, by definition, \overline{\overline{ψ}ω} = \overline{ψ • ω} = \overline{ψ\overline{ω}} and, by Theorem 3.1, \overline{\overline{ψ}ω} = \overline{ψ}.\overline{ω} = \overline{ψ\overline{ω}} so that \overline{ψ • ω} = \overline{ψ}.\overline{ω}.

3.13. Remark. It follows immediately that an n-fold composite of waves ω_{1} • ⋯ • ω_{i} • ⋯ • ω_{n} is given by each of the n expressions, \overline{ω}_{1}. · · · .ω_{i}. · · · .\overline{ω}_{n}.

3.14. Lemma. For data K [figure: ω] M [figure: f] L [figure: ψ] A in a total category,

ψf • ω = ψ • fω

Proof.

ψf • ω = \overline{ψf}ω = (\overline{ψ}f)ω = \overline{ψ}(fω) = ψ\overline{fω} = ψ • fω

If follows immediately that a composable pair of waves K [figure: ω] M [figure: ψ] A has the same

composite as a composable pair of waves K [figure: τ] L [figure: σ] A, if there exists M [figure: f] L such that

[figure: K; L; τ; M; K; ω; M; L; f; A; L; σ; M; A; ψ; M; L; f]

Temporarily, denote by R the relation on composable pairs of waves from K to A given by ⟨ω, ψ⟩R⟨τ, σ⟩ if there exists an arrow f as above. Write ψ ⊗ ω for the equivalence class of the pair ⟨ω, ψ⟩ with respect to the equivalence relation generated by R. It is clear that composition of waves can be seen as a function defined on such equivalence classes so that •(ψ ⊗ ω) = ψ • ω. The equivalence relation generated by R is the one whose quotient is given by:

∫ ^{L} 𝒦̃(K, L) × 𝒦̃(L, A)

This integral is of course the value of the composite profunctor 𝒦̃∘_{𝒦} 𝒦̃: 𝒦 [figure] 𝒦 at the pair ⟨K, A⟩, although we have no reason to suppose, for a general total 𝒦 , that these values are in set. Regardless, we have •: 𝒦̃∘_{𝒦} 𝒦̃ [figure]𝒦̃.

4. Waves and total distributivity

4.1. Lemma. For 𝒦 a total category, the following are equivalent:

i) The functor X: 𝒦̂ [figure]𝒦 has a left adjoint;

ii) The natural transformation γ: XW [figure]1_{𝒦} of Lemma (3.10) is invertible;

iii) The β: W X [figure]1𝒦̂ of Lemma (3.10) is the counit of an adjunction α, β: W ⊣ X.

Proof. ii) implies iii) Let α = γ^{−1} : 1_{𝒦} [figure]XW. Then from βW = W γ we have βW.W α = 1_{W} and from Xβ = γX we have Xβ.αX = 1_{X} so that α, β : W ⊣ X is an adjunction with counit β.

iii) implies i) is trivial.

i) implies ii) From L ⊣ X = ⋁ we have

W (A)(K) = 𝒦̃(K, A)
= set𝒦̂(𝒦 (A, ⋁ −), K̂)
≅ set𝒦̂(𝒦̂(L(A), −), K̂) by the hypothesized adjunction
≅ K̂(L(A))  by Yoneda’s Lemma
= L(A)(K)

In more detail, suppose that we have an adjunction ⟨ι, κ; L ⊣ X : 𝒦̂ [figure]𝒦 ⟩, with unit ι. Since we then have L ⊣ X ⊣ Y and Y is fully faithful, it follows that L is fully faithfull and ι: 1_{𝒦} [figure]XL is an isomorphism. Define μ: W [figure]L by the following display: where β: W X [figure]1𝒦̂is as described in Definition 3.5.

[figure: W; W XL; W ι; W; L; μ; L; W XL; βL]

We evaluate μ(A)(K):

𝒦̃(K, A) [figure: 𝒦̃(K,ιA)] 𝒦̃(K, ⋁ LA) [figure: β(K,LA)] [K, LA]

and see that, for ω: K [figure]A, μ(ω) = ω(ιA). Next, define ν : L [figure]W, for λ: K [figure]L(A) and f : A [figure]⋁ P by ν(λ)(f) = κP.Lf.λ. Clearly

μ(ν(λ)) = ν(λ)(ιA) = κLA.LιA.λ = λ

while

ν(μ(ω))(f) = κP.Lf.ω(ιA) = ω(⋁ κP. ⋁ Lf.ιA) = ω(⋁ κP.ι ⋁ P.f) = ω(f)

shows that ν(μ(ω)) = ω so that μ and ν are inverse isomorphisms.

Consider the following diagram:

[figure: XW; 1_{𝒦}; γ; XW; XL; XL; 1_{𝒦}; ι; XW XL; XW; XW XL; XW ι; XβL; γXL; XW; XL; Xμ]

We have equality in the top region by the definition of μ. The parallel pair of arrows are equal by Proposition 3.10. We have equality in the lower region by naturality. It follows that we have Xμ = ι.γ and since μ and ι are invertible, so too is γ.

In [MRW] a total category 𝒦 is said to be totally distributive if X : 𝒦̂ [figure]𝒦 has a left adjoint. We prefer to restate Lemma 4.1 as

4.2. Theorem. For 𝒦 a total category, the following are equivalent:

i) The total category 𝒦 is totally distributive;

ii) The natural transformation γ: XW [figure]1_{𝒦} is invertible;

iii) The natural transformation β : W X [figure]1𝒦̂ provides the counit for an adjunction α, β: W ⊣ X.

Henceforth, we reserve α for γ^{−1} when 𝒦 is totally distributive. Observe now that if 𝒦 is totally distributive and ω: K [figure]A is a wave in 𝒦 then at first glance we seem to have also ω(α) : K [figure]A, where expanding the notation ω(α) = ω(W A)(αA) as below

[figure: W A; ⋁ W A ,; K; W A; ω(W A)(αA); K; A; ω; A; ⋁ W A ,; αA]

since K-elements of W A are waves from K to A. However,

4.3. Lemma. For any wave ω: K [figure]A in a totally distributive category, ω(α) = ω

Proof. We have

ω(α) = α.ω(1_{⋁ W A}) = γ(α.ω) = (γα).ω = 1.ω = ω

where the second equality is an instance of the equation βW = W γ of Proposition 3.10.

We should also point out that the isomorphisms αA: A [figure]⋁ W A, for all A in a totally distributive category, can be interpreted as saying that every object A is the colimit of all waves ω : K [figure]A. This follows from the fact that each W A, like any object in 𝒦̂, is the (not-necessarily small) colimit of its elements and that ⋁ W A is the colimit of the same diagram in 𝒦 . Each element ω : K [figure]A gives rise to a colimit injection i_{ω}: K [figure]⋁ 𝒦̃(−, A).

4.4. Theorem. In a totally distributive category, every wave ω: K [figure]A can be factored as a composite of waves:

[figure: K; M; ω_{1}; M; A; ω_{0}; K; A; ω]

and the factorization is unique up to tensor, in the sense that if also ω = τ_{0} • τ_{1} then τ_{0} ⊗ τ_{1} = ω_{0} ⊗ ω_{1}.

Proof. Let A be in 𝒦 , totally distributive. So γA is invertible and we have

W ⋁ W A [figure: ≃^{W γA}] W A

which in a more convenient notation is

𝒦̃(−, ⋁ W A) [figure: ≃^{𝒦̃(−,γA)}] 𝒦̃(−, A)

and still more conveniently is

𝒦̃(−, ∫ ^{L} L · 𝒦̃(L, A)) [figure: ≃^{𝒦̃(−,γA)}] 𝒦̃(−, A) .

Because W is a left adjoint, W = 𝒦̃(−, □) preserves colimits and we have

∫ ^{L} 𝒦̃(−, L) · 𝒦̃(L, A) [figure: ≃] 𝒦̃(−, ⋁ W A) [figure: ≃^{𝒦̃(−,γA)}] 𝒦̃(−, A)

which, evaluating at K in 𝒦 , gives bijections

∫ ^{L} 𝒦̃(K, L) × 𝒦̃(L, A) [figure: ≃] 𝒦̃(K, ⋁ W A) [figure: ≃^{𝒦̃(K,γA)}] 𝒦̃(K, A) .

This much establishes that 𝒦̃∘_{𝒦} 𝒦̃≅ 𝒦̃but we want to show that •: 𝒦̃∘_{𝒦} 𝒦̃ [figure]𝒦̃ is an isomorphism. To this end, consider a composable pair of waves:

[figure: K; L; ψ; L; A; χ]

The invertible integral comparison, as in 2.5 sends χ ⊗ ψ to the wave

K [figure: ψ] L [figure: i_{χ}]⋁ W A

To act on this wave by γA is to form γA(i_{χ}ψ) = (γA.i_{χ})ψ = χ̄ψ = χ • ψ, where the second equality is Proposition 3.7.

4.5. Remark. It might be wondered if the components of δ : W [figure]Y exhibit the wave sets 𝒦̃(K, A) of a total category 𝒦 as subsets of the arrow sets 𝒦 (K, A). This is not true is general, even for set, the very best behaved total category according to [R&W]. For 𝒦 = set, one can take W (A)(K) = A and δ(a: 1 [figure]A) = K [figure: !] 1 [figure: a] A. Evidently, δ(A)(∅) is not monic.

5. An equivalence

We continue with 𝒦 a totally distributive category. By Theorem 4.4, •: 𝒦̃∘_{𝒦} 𝒦̃ [figure]𝒦̃ is invertible and, by Theorem 3.12, • is associative. It follows that •^{−1} : 𝒦̃ [figure]𝒦̃∘_{𝒦} 𝒦̃ is coassociative. Moreover, it follows immediately from the definition of • in terms of \overline{(−)} : 𝒦̃ [figure]𝒦 that \overline{(−)} provides a counit for •^{−1} and thus (𝒦̃, •^{−1}, \overline{(−)}) is a (small) idempotent comonad on the totally distributive 𝒦 in the bicategory PRO of profunctors.

It will be convenient to make some further remarks in terms of the full and locally small subbicategory of PRO, which we call MAT, determined by the discrete objects. It follows that the objects of MAT are the objects of SET. For (possibly large) sets X and A, a profunctor M : X [figure] A amounts to a matrix of (possibly large) sets M(a, x), for all pairs of elements a in A and x in X. We write Mat(X, A) for the full subcategory of MAT(X, A) determined by the set-valued matrices. Given M and a composable matrix N : A [figure] Y, the composite profunctor N ∘ M : X [figure] Y amounts to the matrix product NM given by:

NM(y, x) = ∑_{a} N(y, a) × M(a, x)

Our reason for delving into MAT is that we first defined wave composition • at the level of composable pairs of waves and it is helpful here to temporarily return to that point of view. Indeed, if we simply regard 𝒦̃as a matrix 𝒦̃: |𝒦 | [figure]|𝒦 | then the matrix product 𝒦̃𝒦̃is evidently the matrix of composable pairs of waves and we can see • as •: 𝒦̃𝒦̃ [figure]𝒦

It is standard that a category ℳ can be seen as a monad ℳ : |ℳ | [figure] |ℳ |in MAT and that a profunctor Q̃ : ℳ [figure] ℒ can be seen as a matrix Q̃ : |ℳ | [figure] |ℒ |, together with associating, associative and unitary, actions ρ: Q̃ℳ [figure]Q̃ and λ: ℒ Q̃ [figure]Q̃. 2-cells between parallel profunctors can then be seen as matrix 2-cells that are equivariant with respect to the actions.

The bicategory MAT, like the bicategory PRO admits local coequalizers that are stable under composition (from either side). Given Q̃ and a composable profunctor P̃ : ℒ [figure] 𝒦 , the composite profunctor P̃ ∘_{ℒ} Q̃: ℳ [figure] 𝒦 can be computed in MAT as the local coequalizer (coequalizer in MAT(|ℳ |, |𝒦 |):

[figure: P̃ℒ Q̃; P̃Q̃; ρQ̃; Q̃λ; P̃Q̃; P̃ ∘_{ℒ} Q̃]

where ρ is the right action of ℒ on P̃ and λ is the left action of ℒ on Q̃.

In particular, the composite of 𝒦̃with itself, as a profunctor, 𝒦̃∘_{𝒦} 𝒦̃is a local coequalizer in MAT and our •: 𝒦̃∘_{𝒦} 𝒦̃arises as below:

[figure: 𝒦̃𝒦 𝒦̃; 𝒦̃ ; 𝒦̃; ρ𝒦̃; 𝒦̃λ; 𝒦̃ ; 𝒦̃; 𝒦̃∘_{𝒦} 𝒦̃; 𝒦̃ ; 𝒦̃; 𝒦̃; •; 𝒦̃∘_{𝒦} 𝒦̃; 𝒦̃; •]

invertibility of •: 𝒦̃∘_{𝒦} 𝒦̃ [figure]𝒦̃is equivalent to saying that

[figure: 𝒦̃𝒦 𝒦̃; 𝒦̃ ; 𝒦̃; ρ𝒦̃; 𝒦̃λ; 𝒦̃ ; 𝒦̃; 𝒦̃; •]

is a coequalizer.

Below on the left, we recall the relation on composable pairs ⟨ω, ψ⟩ that generates the equivalence relation whose quotient is 𝒦̃∘_{𝒦} 𝒦̃. We usually denote the equivalence class of ⟨ω, ψ⟩ by ψ ⊗ω but, provisionally, we will now call it ψ ⊗_{𝒦} ω. While 𝒦̃is not a monad on |𝒦 | we can can regard • : 𝒦̃𝒦̃ [figure]𝒦̃as both a left and a right associative action of 𝒦̃on itself. Accordingly, we can, as on the right below, contemplate the relation on composable pairs of waves determined by the existence of a wave ϕ rather than an arrow f making commutative triangles. For the relation on the right we will write, provisionally, ψ ⊗𝒦̃ω for the equivalence class of ⟨ω, ψ⟩ in the generated equivalence relation.

[figure: K; L; τ; M; K; ω; M; L; f; A; L; σ; M; A; ψ; M; L; f; K; L; τ; M; K; ω; M; L; ϕ; A; L; σ; M; A; ψ; M; L; ϕ]  (6)

It should be clear that just as composition of profunctors can be computed as a local coequalizer in MAT, so too the following MAT local coequalizer

[figure: 𝒦̃ ; 𝒦̃ ; 𝒦̃; 𝒦̃ ; 𝒦̃; •𝒦̃; 𝒦̃•; 𝒦̃ ; 𝒦̃; 𝒦̃∘𝒦̃𝒦̃]

computes the matrix whose value 𝒦̃∘𝒦̃𝒦̃(K, A) is the set of composable pairs of waves modulo the equivalence relation with equivalence classes given by the ψ ⊗𝒦̃ω.

5.1. Theorem. If 𝒦 is a total category for which •: 𝒦̃∘_{𝒦} 𝒦̃ [figure]𝒦̃is invertible, in particular if 𝒦 is a totally distributive category, then 𝒦̃∘𝒦̃𝒦̃= 𝒦̃∘_{𝒦} 𝒦̃so that

[figure: 𝒦̃ ; 𝒦̃ ; 𝒦̃; 𝒦̃ ; 𝒦̃; •𝒦̃; 𝒦̃•; 𝒦̃ ; 𝒦̃; 𝒦̃; •]

is a local coequalizer in MAT and 𝒦̃is a taxon in the sense of Koslowski [KOS].

Proof. It suffices to show that, for any composable pair of waves ⟨ω, ψ⟩, ψ⊗𝒦̃ω = ψ⊗_{𝒦} ω. If we consider the diagram on the right of (6) and replace the wave ϕ by the arrow \overline{ϕ}, the definition of wave composition shows immediately that the result is a diagram as on the left of (6). Formally, we have ψ ⊗𝒦̃ω ⊆ ψ ⊗_{𝒦} ω. On the other hand, consider the diagram on the left of (6) and construct the following diagram by factoring σ as σ_{0} • σ_{1}:

[figure: K; N; σ_{1}fω; M; K; ω; M; N; σ_{1}f; N; A; σ_{0}; M; N; σ_{1}f; M; A; ψ; K; N; σ_{1}fω; K; L; τ; N; L; σ_{1}; N; A; σ_{0}; N; L; σ_{1}; A; L; σ]

The construction shows that ψ ⊗_{𝒦} ω ⊆ ψ ⊗𝒦̃ω.

In [KOS], Koslowski constructed not only a 2-category of taxons and semi-functors (with the main effort involving the 2-cells) but also a bicategory of taxons and interpolative modules. The latter when constructed within MAT are like profunctors between categories except that the unitary conditions for modules — which are not stateable for modules between taxons — are replaced by requiring that the actions be local coequalizers in MAT. These ideas were repeated in [MRW] from a perspective closer to ours here. Nevertheless, we attempt to make this account reasonably self-contained with respect to the definitions.

5.2. Definition. A taxon 𝒯 consists of a set |𝒯 | (possibly large) and a matrix 𝒯 : |𝒯 | [figure] |𝒯 | together with a (matrix) 2-cell •: 𝒯 𝒯 [figure]𝒯 for which

[figure: 𝒯 𝒯 𝒯; 𝒯 𝒯; •𝒯; 𝒯 •; 𝒯 𝒯; 𝒯; •]

is a coequalizer. For taxons 𝒮 and 𝒯 , an i-module (short for interpolative module) from 𝒮 to 𝒯 is a matrix P : |𝒮 | [figure] |𝒯 | together with actions ρ: P 𝒮 [figure]P and λ: 𝒯 P [figure]P for which

[figure: P 𝒮 𝒮; P 𝒮; ρ𝒮; P •; P 𝒮; P; ρ; and; 𝒯 𝒯 P; 𝒯 P; •P; 𝒯 λ; 𝒯 P; P; λ]

are coequalizers. A 2-cell t: P [figure]Q: 𝒮 [figure] 𝒯 is a matrix 2-cell t: P [figure]Q: |𝒮 | [figure] |𝒯 | which is equivariant with respect to the actions. Given i-modules P : 𝒮 [figure] 𝒯 and Q : 𝒯 [figure] 𝒰 , the composite i-module Q ∘_{𝒯} P : 𝒮 [figure] 𝒰 is given by the coequalizer

[figure: Q𝒯 P; QP; ρ𝒯; 𝒯 λ; QP; Q ∘_{𝒯} P]

With the other, expected, composites and the obvious constraints, taxons, i-modules, and their 2-cells constitute a bicategory that we call iMOD. We write iMod(𝒮 , 𝒯 ) for the full subcategory of iMOD(𝒮 , 𝒯 ) determined by the set-valued i-modules.

Our intention now is to show that, for 𝒦 a totally distributive category, there is an equivalence 𝒦 [figure]iMod(1, 𝒦̃). To give a small i-module P : 1 [figure] 𝒦̃is to give a small matrix P : 1 [figure] |𝒦̃| and a left action λ: 𝒦̃P [figure]P for which

[figure: 𝒦̃𝒦̃P; 𝒦̃P; •P; 𝒦̃λ; 𝒦̃P; P; λ]

is a coequalizer. On the other hand, to give a module P : 1 [figure] 𝒦 is to give a matrix P : 1 [figure] |𝒦 | and a left action 𝒦 P [figure]P which is associative and unitary. (Of course such a module is but a functor P : 𝒦 ^{op} [figure]set, equally a small profunctor P : 1 [figure] 𝒦 .) From an i-module (P, λ) : 1 [figure] 𝒦̃we now construct a module (P, λ_{1}) : 1 [figure] 𝒦 . Consider the following diagram, in which we have written @ for the left action of 𝒦 on 𝒦̃.

[figure: 𝒦 𝒦̃𝒦̃P; 𝒦 𝒦̃P; 𝒦 •P; 𝒦 𝒦̃λ; 𝒦 𝒦̃P; 𝒦 P; 𝒦 λ; 𝒦̃𝒦̃P; 𝒦̃P; •P; 𝒦̃λ; 𝒦̃P; P; λ; 𝒦 𝒦̃𝒦̃P; 𝒦 𝒦̃P; @𝒦̃P; 𝒦 𝒦̃P; 𝒦̃P; @P; 𝒦 P; P; λ_{1}]

The top row is 𝒦 applied to the coequalizer diagram for (P, λ). It is also a coequalizer since coequalizers in MAT are preserved by composition. We have equality in the bottom square on the left by naturality of @. The definition of • in terms of the equivariant \overline{(−)} provides equality in the top square on the left. These considerations uniquely determine λ_{1} as displayed. Straightforward diagram chases show that λ_{1} is unitary and associative. Thus the construction (P, λ) ↦ (P, λ_{1}) defines an object function from 𝒦̃-i-modules to 𝒦 -modules. Moreover, if (Q, λ) is another i-module from 1 to 𝒦̃and t : P [figure]Q is a matrix 2-cell which is equivariant for the 𝒦̃actions then another, easy, diagram shows that t is also equivariant for the 𝒦 actions.

5.3. Proposition. The construction (P, λ) ↦ (P, λ_{1}) defines a functor J satisfying the following identity with the evident forgetful functors.

[figure: iMod(1, 𝒦̃); Mat(1, |𝒦 |); iMod(1, 𝒦̃); Pro(1, 𝒦 ); J; Pro(1, 𝒦 ); Mat(1, |𝒦 |)]

The functor W : 𝒦 [figure]𝒦̂which sends A to 𝒦̃(−, A) can equally be seen as W : 𝒦 [figure]Pro(1, 𝒦 ). The left 𝒦 action on 𝒦̃(−, A) is just the left action of arrows on waves with codomain A. An element of the matrix of actable pairs with codomain A, namely 𝒦 𝒦̃(−, A) is a pair L [figure: f] K [figure: ω] A. The effect of the action has been denoted ωf and can further be seen as the result of applying the function 𝒦̃(f, A) : 𝒦̃(K, A) [figure]𝒦̃(L, A) to the element ω ∈ 𝒦̃(K, A). It is clear that for a wave L [figure: ψ]K we can define 𝒦̃(ψ, A) : 𝒦̃(K, A) [figure]𝒦̃(L, A) by composition of waves and, at the risk of labouring the obvious, A ↦ 𝒦̃(−, A) defines some functor 𝒦 [figure]iMod(1, 𝒦̃) that we will call W̃ : 𝒦 [figure]iMod(1, 𝒦̃). (That 𝒦̃(−, A) satisfies the requisite coequalizer condition is

easily seen formally by recognizing it as the profunctor composite 1 [figure: A] 𝒦 [figure: 𝒦̃] 𝒦 .) What does require a check, which we leave for the reader, is showing that J applied to the actions 𝒦̃(ψ, A), for ψ a wave returns the actions given by the 𝒦̃(f, A), for f an arrow. More formally:

5.4. Proposition. The functor W̃ satisfies the equality

[figure: 𝒦; Pro(1, 𝒦 ); W; iMod(1, 𝒦̃); 𝒦; W̃; iMod(1, 𝒦̃); Pro(1, 𝒦 ); J]

We further define X̃: iMod(1, 𝒦̃) [figure]𝒦 to be the composite XJ. In the sequel, when we refer to the matrix entries P (K, ∗) of an i-module (P, λ) from 1 to 𝒦̃, we will write P (K, ∗) = [K, (P, λ)] = [K, P] to conform with our usage in 𝒦̂, equally in Pro(1, 𝒦 ).

5.5. Theorem. For 𝒦 totally distributive, the functors W̃: 𝒦 [figure]iMod(1, 𝒦̃) and X̃ provide an adjoint equivalence of categories. Proof. We have the isomorphism α̃ = α: 1_{𝒦} [figure]XW = XJW̃ = X̃W̃, by total distributivity of 𝒦 . We wish to construct a β̃: W̃X̃ [figure]1_{iMod(1,𝒦̃)}. To this end we require, for each (P, λ) in iMod(1, 𝒦̃), a 𝒦̃-equivariant β̃(K, (P, λ)) : 𝒦̃(K, ⋁(P, λ_{1})) [figure][K, (P, λ)]. The requisite functions are provided by the β(K, (P, λ_{1})) : 𝒦̃(K, ⋁(P, λ_{1})) [figure][K, (P, λ_{1})], where we recall that for ω: K [figure]⋁(P, λ_{1}), we have β(ω) = ω(1). However, we need to show that these are equivariant with respect to the 𝒦̃actions. So given L [figure: ψ] K we require that β(ω • ψ) = β(ω)ψ. By the right side we refer to L [figure: ψ] K [figure: β(ω)] P in [L, (P, λ)]. However, writing β(ω) = pχ by interpolativity of (P, λ) as in:

[figure: K; M; χ; K; P; β(ω); P; M; p]

we have

β(ω • ψ)=β(ω\overline{ψ})=(ω\overline{ψ})(1) =^{(a)}ω(1)\overline{ψ} =β(ω)\overline{ψ} =(pχ)\overline{ψ} =^{(b)}p(χ\overline{ψ})=p(χ • ψ)=(pχ)ψ =β(ω)ψ

where (a) follows from the action of 𝒦 on waves and (b) follows from the construction of a 𝒦 module from a 𝒦̃i-module. To complete the proof of the theorem it now suffices to show that the βP : 𝒦̃(−, ⋁ P) [figure][−, P] are invertible when P arises from a left K̃ i-module via J. Because 𝒦̃(−, □) is a left adjoint,it suffices to show that the composite:

∫ ^{L} 𝒦̃(−, L) · [L, P] [figure: ≃] 𝒦̃(−, ∫ ^{L} L · [L, P]) [figure: βP] [−, P]

is invertible, where the first isomorphism is the sup-comparison. Invertibility of this composite is equivalent to each evaluation at K:

∫ ^{L} 𝒦̃(K, L) × [L, P] [figure: ≃] 𝒦̃(K, ⋁ P) [figure: βP] [K, P]

being a bijection. Note that the domain of the composite can be written as the profunctor composite 𝒦̃∘_{𝒦} P, evaluated at K. However, using the interpolativity of P, the argument in Theorem 5.1 shows that 𝒦̃∘_{𝒦} P is equally 𝒦̃∘𝒦̃P, so that the composite above can be written:

(𝒦̃∘𝒦̃P)(K) [figure: ≃] 𝒦̃(K, ⋁ P) [figure: βP] [K, P]  (7)

We will show that this composite is the evaluation at K of the invertible matrix arrow 𝒦̃∘𝒦̃P [figure]P corresponding to the MAT coequalizer λ: 𝒦̃P [figure]P. Consider a composable wave and element K [figure: ω] L [figure: p] P. Integral comparison sends the tensor p ⊗ ω to the wave K [figure: ω] L [figure: i_{p}]⋁ P which β sends to i_{p}ω(1_{⋁ P}) = ω(i_{p}) : K [figure] P. Using the expansion of i_{p} on the right below, as discussed in 2.4:

[figure: L_{∗}; ⋁ L_{∗}; K; L_{∗}; \overline{ω}; K; L; ω; L; ⋁ L_{∗}; ηL; P; ⋁ P; L_{∗}; P; p_{∗}; L_{∗}; ⋁ L_{∗}; ⋁ L_{∗}; ⋁ P; ⋁ p_{∗}; K; L; \overline{ω}; L; P; p; L; ⋁ P; i_{p}]

we see that ω(i_{p}) = p\overline{ω}. However, we can write p = qψ (for unique q ⊗ ψ) and now

p\overline{ω} = (qψ)\overline{ω} = q(ψ\overline{ω}) = q(ψ • ω) = (qψ)ω = pω

This shows that the composite in (7) is the required bijection and completes the proof that β̃ is invertible.

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Department of Mathematics and Statistics Dalhousie University Halifax, NS, B3H 3J5, Canada

Email: rjwood@mathstat.dal.ca
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