Theory and Applications of Categories, Vol. 25, No. 1, 2011, pp. 1–37.

A CATEGORY OF QUANTUM CATEGORIES

DIMITRI CHIKHLADZE

Abstract. Quantum categories were introduced in [5] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures.

1. Introduction

Quantum categories are defined within a monoidal category V. When V is the opposite category of modules over a commutative ring, a quantum category is the same as a bialgebroid. Bialgebroids can be thought of as “several object” generalisations of bialgebras. They were considered for the first time in M. Takeuchi’s paper [17] and appeared later in the work of many authors in different fields, e.g. [10], [13], [20], [3]. A quantum category in a general monoidal category was defined by B. Day and R. Street [5] incorporating both bialgebroids, in the way mentioned above, and ordinary categories, by taking the monoidal category V to be the category of sets. In this paper we approach quantum categories using the bicategorical version of the formal theory of (co)monads [14]. One benefit of this approach is that it makes clear the connection between quantum categories and ordinary categories, which enables us to reproduce ordinary category theory for quantum categories.

The paper is organised in the following way. In Section 2 we review the formal theory of (co)monads in a bicategory. In Sections 3 and 4, we deal with a bicategory ComodV, which is defined from our base monoidal category V, and which plays an important role in our theory. In sections 5 we give a monadic definition of a quantum category and show explicitly how it translates to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. In Section 6 we define the notion of functor between quantum categories, obtaining a category qCatV. When V is the category of sets, then a quantum functor is the same as an ordinary functor, and the qCatV is the category of categories. In Section 7 making use of the functoriality of the qCat we give examples of constructions on quantum categories. In the Appendix we introduce framed string diagrams designed to ease computations involving the quantum structures.

The author gratefully acknowledges support of an International Macquarie University Scholarship.

Received by the editors 2009-10-02 and, in revised form, 2011-01-19.

Transmitted by R.J. Wood. Published on 2011-01-24.

2000 Mathematics Subject Classification: 18D35.

Key words and phrases: quantum category, monoidal category, comonad.

\begin{matrix}c \\ ◯ \end{matrix} Dimitri Chikhladze, 2011. Permission to copy for private use granted.

2. Monoidal comonads

Let B be a bicategory. We write as if B were a 2-category, regarding associativity and unitivity isomorphisms as identities.

Recall that a comonad in B [14], [1] is a pair (B, g), where B is an object of B and g = (g, δ : g ⇒ gg, ϵ : g ⇒ 1_{g}) is a comonoid in the homcategory B(B, B). A map of comonads (k, κ) : (B, g) → (A, g′) consists of a morphism k : B → A and a 2-cell κ : kg ⇒ g′k satisfying:

(kg [figure: kδ] kgg [figure: κg] g′kg [figure: g′κ] g′g′k) = (kg [figure: κ] g′k [figure: g′δ] g′g′k),

(kg [figure: kϵ] k) = (kg [figure: κ] g′k [figure: ϵk] k).

A comonad map transformation τ : (k, κ) ⇒ (k′, κ′) : (A, g) → (B, g′) is a 2-cell τ : k ⇒ k′ satisfying:

(kg [figure: τg] k′g [figure: κ′] g′k′) = (kg [figure: κ] g′k [figure: gτ] g′k′).

Comonads in B, comonad maps and comonad map transformations form a bicategory ComndB under the obvious composition.

B is said [14] to admit the Eilenberg-Moore construction for comonads if the inclusion B → ComndB, taking an object B to (B, 1), has a right biadjoint ComndB → B. The value of this right biadjoint at (B, g) is called an Eilenberg-Moore object of (B, g). It will be denoted by denoted B^{g}. There is a pseudonatural equivalence

B(X, B^{g}) ≃ ComndB((X, 1), (B, g))

The objects of the right side are called g-coalgebras. Taking X = B^{g} and evaluating at the identity, we obtain a universal g-coalgebra (u, γ) : (B^{g}, 1) → (B, g). Every comonad map k : (B, g) → (A, g′) induces a map k̂ : B^{g} → A^{g′} between Eilenberg-Moore objects so that there is an isomorphism:

\xymatrix{B^{g} \ar[r]^{k̂} \ar[d]_{u} & A^{g′} \ar[d]^{u} \\ A \ar[r]_{k} & B} ; ≅  (1)

By an equivalence between suitable categories, comonad structures on k : B → A correspond to diagrams (2) in B.

Let B be a monoidal bicategory [4]. We specify n-ary tensor product pseudofunctors

B^{n} [figure: ⊗_{n}] B

by choosing bracketing for the tensor product to be from the left. So, the expression B_{1} ⊗ … ⊗ B_{n} refers to ⊗_{n}(B_{1} ⊗ … ⊗ B_{n}).

A monoidale E in B consists of an object E together with morphisms p : E ⊗ E → E and j : I → E called the multiplication and the unit respectively, and invertible 2cells expressing associativity and unitivity, subject to coherence conditions. The n-ary multiplication map

E^{n} [figure: p_{n}] E.

is defined by consecutive multiplications from the left.

2.1. Example. Let an object B be the right bidual to an object A in B, with the biduality counit e : A ⊗ B → I and the biduality unit n : I → B ⊗ A. B ⊗ A becomes a monoidale with product p = 1 ⊗ e ⊗ 1 : B ⊗ A ⊗ B ⊗ A → B ⊗ A and unit j = n : I → B ⊗ A.

A monoidal morphism (f, ϕ_{2}, ϕ_{0}) : E → D between monoidales consists of a morphism f : E → D and 2-cells ϕ_{2} : p(f ⊗ f) ⇒ fp, ϕ_{0} : j ⇒ fj satisfying three axioms. The composition of monoidal morphisms (f, ϕ_{2}, ϕ_{0}) : E → D and (f′, ϕ_{2}, ϕ_{0}) : D → F is defined to be (f′f, ϕ_{2}, ϕ_{0}) : E → F, where

ϕ_{2} = (p(f′ ⊗ f′)(f ⊗ f) [figure: ϕ_{2}(f⊗f)] f′p(f ⊗ f) [figure: f′ϕ_{2}] f′fp)
ϕ_{0} = (j [figure: ϕ_{0}] f′j [figure: f′ϕ_{0}] f′fj).

A monoidal morphism is called strong when ϕ_{2} and ϕ_{0} are isomorphisms. Monoidales in B, monoidal morphisms between them and obvious 2-cells form a bicategory MonB.

There is a biequivalence

MonComndB ∼ ComndMonB,  (2)

where the left hand side is defined using the monoidal structure on ComndB inherited from B.

A monoidal comonad is an object of ComndMonB, or equally, an object of MonComndB. Explicitly, a monoidal comonad consists of a monoidale E, a comonad g on E and 2-cells ϕ_{2} : p(g ⊗ g) ⇒ gp, ϕ_{0} : j ⇒ gj such that (g, ϕ_{2}, ϕ_{0}) is a monoidal morphism and (p, ϕ_{2}) : (E ⊗ E, g ⊗ g) → (E, g) and (j, ϕ_{0}) : (I, 1) → (E, g) are comonad maps. A morphism of monoidal comonads (k, κ) : (E, g) → (E′, g′) is a map of underlying comonads such that κ : kg ⇒ g′k is a map of monoidal morphisms.

Mon(−) can be made into a pseudofunctor from the tricategory of monoidal bicategories and monoidal pseudofunctors to the tricategory of bicategories and pseudofunctors. Since the inclusion i : B → ComndB is a strong monoidal pseudofunctor the right biadjoint to it is a monoidal pseudofunctor too. It follows that if i has a right biadjoint, then Mon(i) : MonB → MonComndB has a right biadjoint too. Using the the biequivalence (2) we infer that the canonical inclusion MonB → ComndMonB has a right biadjoint. This proves [12], [11]:

2.2. Proposition. If B admits the Eilenberg-Moore construction for comonads, then so does Mon(B).

Explicitly an Eilenberg-Moore object of a monoidal comonad (E, g) is obtained in the following way. Let E^{g} be the Eilenberg-Moore object for the underlying comonad in B with (u, γ) : (E^{g}, 1) → (E, g) the universal coalgebra. Then p(u ⊗ u) : E^{g} ⊗ E^{g} → E becomes a g-coalgebra with coaction

p(u ⊗ u) [figure: p(γ⊗γ)] p(g ⊗ g)(u ⊗ u) [figure: ϕ_{2}p] gp(u ⊗ u),

and j : I → E becomes a g-coalgebra with the coaction

j [figure: ϕ_{0}] gj.

The induced morphisms p̂: E^{g} ⊗ E^{g} → E^{g} and ĵ : I → E^{g} define a monoidale structure on E^{g}. This monoidale is the Eilenberg-Moore object of (E, g) in MonB. Moreover, the map u : E^{g} → E is a strong monoidal morphism.

There is an equivalence of categories which establishes a correspondence between monoidal comonad maps (k, κ) : (E, g) → (E′, g′) and diagrams (2), now in MonB.

What we have been discussing so far were standard constructions in a monoidal bicategory. Further we introduce some concepts, which we will later use for our specific purposes.

An opmonoidal morphism (w, ψ_{2}, ψ_{0}) : E → D between monoidales is a monoidal morphism in B^{co}. Thus an opmonoidal morphism consists of a morphism w : E → F and 2-cells ψ_{2} : wp ⇒ p(w ⊗ w), ψ_{0} : hj ⇒ j in B satisfying three axioms.

Monoidal morphisms and opmonoidal morphisms lead us to the setting of a double category [9], [19]. Recall briefly, that a double category has objects and two types of arrows, called horizontal morphisms and vertical morphisms, forming bicategories in the two directions. Also, there is a set of squares, each square having as its sides two horizontal morphisms and two vertical morphisms. Squares can be composed in the two directions.

As suggested, there is a double category with objects the monoidales in B, horizontal arrows the monoidal morphisms and vertical morphisms the opmonoidal morphisms. A square is a 2-cell

[figure: E; f; w; E′; w′; D; f′; D′; σ]

with f and f′ monoidal morphisms and w and w′ opmonoidal morphisms such that:

(w′p(f ⊗f) [figure: ψ_{2}(f⊗f)] p(w′ ⊗w′)(f ⊗f) [figure: p(σ⊗σ)] p(f′ ⊗f′)(w ⊗w) [figure: ϕ_{2}(w⊗w)] f′p(w ⊗w))

= (w′p(f ⊗ f) [figure: w′ϕ_{2}] w′fp [figure: ψ_{2}p] f′wp [figure: σ] f′p(w ⊗ w))

and (w′j [figure: ϕ_{0}] w′fj [figure: σj] f′wj [figure: f′ψ_{0}] f′j) = (w′j [figure: ψ_{0}] j [figure: ϕ_{0}] f′j).

Suppose that (E, g) and (D, g′) are monoidal comonads. An opmorphism of monoidal comonads (h, σ) : (E, g) → (D, g′) is an opmonoidal morphism h : E → D together with a square σ : hg ⇒ g′h, such that (h, σ) : (E, g) → (D, g′) is a map of comonads.

As with the monoidal comonad maps, there is an equivalence of categories which establishes a correspondence between opmorphisms of monoidal comonads h : (E, g) → (D, g′) and diagrams

\xymatrix{E^{g} \ar[r]^{ĥ} \ar[d]_{u} & D^{g′} \ar[d]^{u} \\ E \ar[r]_{h} & D} ; ≅

of opmonoidal morphisms.

A coaction of an opmonoidal morphism h : E → D on a morphism l : E → D of B is a 2-cell λ : lp ⇒ p(h ⊗ l) satisfying two axioms, relating it to the opmonoidal structure on h.

Suppose that (h, σ) : (E, g) → (D, g′) is an opmorphism of monoidal comonads and (l, τ) : (E, g) → (D, g′) is a comonad map. We will say that a left coaction λ of h on l respects the comonad structure if

(lp(g ⊗ g) [figure: λ(g⊗g)] p(h ⊗ l)(g ⊗ g) [figure: p(σ⊗τ)] p(g′ ⊗ g′)(h ⊗ l) [figure: μ(h⊗l)] g′p(h ⊗ l))

= (lp(g ⊗ g) [figure: lμ] lgp [figure: τp] g′lp [figure: g′λ] g′p(h ⊗ l)).

A left coaction of h on l respects comonad structure if and only if it can be lifted to a coaction of l̂: E^{g} → D^{g′} on ĥ : E^{g} → D^{g′}.

There is a similar notion of a right coaction of an opmorphism.

3. The bicategory of comodules

Suppose that V = (V, ⊗, I, c) is a braided monoidal category with finite colimits. Assume that each of the functors X ⊗ − preserves equalizers of coreflexive pairs.

We will work with monoidal bicategory C = ComodV defined in [2]. Objects of C are the comonoids

C = (C, δ : C → C ⊗ C, ϵ : C → I)

in V. The homcategory C(C, D) is the category of Eilenberg-Moore coalgebras for the comonad C⊗−⊗D : V → V. A 1-cell from C to D, depicted C [figure] D, is a comodule from C to D. Recall that this consists of an object M and a coaction map δ : M → C ⊗M ⊗D satisfying two axioms. A 2-cell α : M ⇒ N : C [figure] D is a coaction respecting map M → N. An object of C(C, I) is a left C-comodule, and an object of C(I, C) is a right Ccomodule. A comodule M : C [figure] D becomes a left C-comodule and a right D-comodule via coactions

δ_{l} : M [figure: δ] C ⊗ M ⊗ D [figure: 1⊗1⊗ϵ] C ⊗ M

(3)

δ_{r} : M [figure: δ] C ⊗ M ⊗ D [figure: ϵ⊗1⊗1] M ⊗ D.

The maps δ_{l} and δ_{r} are called left and right coactions on M. If M is a left C-comodule and N is a right D-comodule, then a tensor product M ⊗_{C} N over C is defined by a (coreflexive) equalizer:

[figure: M ⊗_{C} N; M ⊗ N; M ⊗ C ⊗ N.; i; δ_{r}⊗1; 1⊗δ_{l}]

If M is a comodule E [figure] C and N is a comodule C [figure] F, then using the fact that the functor E ⊗ − ⊗ F preserves coreflexive equalizers, M ⊗_{C} N becomes a comodule E [figure] F. Composition in C is defined by N ∘ M = M ⊗_{C} N. It is associative up to canonical isomorphism.

Any comonoid C is a C [figure] C comodule with the coaction

C [figure: δ_{3}] C ⊗ C ⊗ C.

The identity comodule on C is C itself.

As it is a convention to name such bicategories after arrows, ComodV is called the bicategory of comodules. For more on the theory of comodules we refer the reader to [15].

Each comonoid morphism f : C → D determines an adjoint pair in C:

f_{∗} ⊣ f^{∗}C [figure] D

The comodules f^{∗} : C [figure] D and f_{∗} : D [figure] C are both C as objects of V with coactions respectively

C [figure: δ_{3}] C ⊗ C ⊗ C [figure: 1⊗1⊗f] C ⊗ C ⊗ D and

C [figure: δ_{3}] C ⊗ C ⊗ C [figure: f⊗1⊗1] D ⊗ D ⊗ C.

The counit

[figure: C; f^{∗}; f_{∗}; D; D; D; β]

of the adjunction is the map

C ⊗_{C} C ≅ C [figure: f] D.

The unit

[figure: D; f_{∗}; f^{∗}; C; C; C]
α

is induced by the comultiplication δ : C → C ⊗ C as shown on the diagram:

\xymatrix{C ⊗_{D} C \ar[rr]^{eq.} &  & C ⊗ C \ar[r]^{(1⊗f⊗1)(δ⊗1)} \ar[r]_{(1⊗f⊗1)(1⊗δ)} & C ⊗ D ⊗ C \\  & C \ar[ul]^{α} \ar[ur]_{δ}}

In particular, we have comodules ϵ^{∗} : I [figure] C and ϵ_{∗} : C [figure] I. The compositions

C [figure: M] D [figure: ϵ_{∗}] I
I [figure: ϵ_{∗}] C [figure: M] D

reconfirm the fact that M is a left C-comodule and a right D-comodule by (3).

The monoidal structure on C extends the monoidal structure on V. The tensor product of comonoids C = (C, δ, ϵ) and C′ = (C′, δ′, ϵ′) is C ⊗C′ with comultiplication and counit:

(1 ⊗ c ⊗ 1)(δ ⊗ δ′) : C ⊗ C′ → C ⊗ C′ ⊗ C ⊗ C′
ϵ ⊗ ϵ′ : C ⊗ C′ → I.

The monoidal unit of C is I, which is a comonoid in an obvious way. On 1-cells, the tensor product of comodules M : C [figure] D and N : C′ [figure] D′ is M ⊗ N, which is a comodule C ⊗ C′ [figure] D ⊗ D′ with coaction:

M ⊗ N [figure: δ⊗δ] C ⊗ M ⊗ D ⊗ C′ ⊗ N ⊗ D′ [figure: c_{142536}] C ⊗ C′ ⊗ M ⊗ N ⊗ D ⊗ D′.

Here and below a morphism named c subscripted with a permutation is an isomorphism coming from the braiding.

We often encounter comodules going between tensor products of comonoids, like M : C_{1}⊗C_{2}⊗… C_{n} [figure] D_{1}⊗D_{2}⊗… D_{m}. Such a comodule inherently is a left C_{i}-comodule, for 1 ≤ i ≤ n, and a right D_{i}-comodule, for 1 ≤ i ≤ m. Conversely, given left C_{i}-comodule and right D_{i}-comodules structures on M compatible in a certain way, M becomes a C_{1} ⊗ C_{2} ⊗ … C_{n} [figure] D_{1} ⊗ D_{2} ⊗ … D_{m} comodule. This enables us to describe a comodule just by giving left and right coactions. A map M → N is a comodule map between comodule M, N : C_{1} ⊗ C_{2} ⊗ … C_{n} [figure] D_{1} ⊗ D_{2} ⊗ … D_{m} if and only if it is a left C_{i}- comodule map for all 1 ≤ i ≤ n and a right D_{i}-comodule map for all 0 ≤ i ≤ m.

C is a right autonomous monoidal bicategory. The bidual of a comonoid C = (C, δ, ϵ) is the comonoid with the opposite comultiplication C^{o} = (C, cδ, ϵ). Unit and counit are comodules e : C^{o} ⊗ C [figure] I and n : I [figure] C ⊗ C^{o}, both of which are C as objects of V and the coactions on them are respectively

C [figure: δ_{3}] C ⊗ C ⊗ C [figure: 1⊗c] C ⊗ C ⊗ C and

C [figure: δ_{3}] C ⊗ C ⊗ C [figure] c ⊗ 1C ⊗ C ⊗ C.

It follows that C^{o} ⊗ C is a monoidale in C. The multiplication is p = 1 ⊗ e ⊗ 1 and the unit is j = n. Still more explicitly, the multiplication C^{o} ⊗ C ⊗ C^{o} ⊗ C [figure] C^{o} ⊗ C is p = C ⊗ C ⊗ C with coaction

C^{⊗3} [figure: δ_{3}⊗δ_{3}⊗δ_{3}] C^{⊗9} [figure: c_{146725839}] C^{⊗9}

and the unit I [figure] C^{o} ⊗ C is j = C with coaction

C [figure: δ_{3}] C^{⊗3} [figure: c_{213}] C^{⊗3}

Let M and N be comodules I [figure] C^{o} ⊗ C. Regard these as comodules C [figure] C by the equivalence

C(I, C^{o} ⊗ C) ≃ C(C, C).  (4)

The composite

I [figure: M⊗N] C^{o} ⊗ C ⊗ C^{o} ⊗ C [figure: p] C^{o} ⊗ C

is M ⊗_{C} N with right C^{o} ⊗ C-coaction the unique map δ_{l} : M ⊗_{C} N → (M ⊗_{C} N) ⊗ C ⊗ C making

\xymatrix{M ⊗_{C} N \ar[r]^{i} \ar[d]_{δ_{l}} & M ⊗ N \ar[d]^{δ_{l}} \\ (M ⊗_{C} N) ⊗ C ⊗ C \ar[r]_{i⊗⊗C⊗C} & M ⊗ N ⊗ C ⊗ C}

commute.

The equivalence (4) is a monoidal equivalence, where the monoidal structure on the left side comes from the pseudomonoid structure on C^{o} ⊗ C and the monoidal structure on the right is defined to be the composition in C.

Next we prove some technical lemmas, which we use in Section 5.

3.1. Lemma. Suppose that β is a 2-cell:

[figure: A ⊗ C ⊗ C^{o} ⊗ B; A⊗e⊗B; M; A ⊗ B; N; D; β]

Let α : M → N be the map in V determined by the pasting composite

[figure: A ⊗ B; A⊗B; A⊗ϵ^{∗}⊗B; A ⊗ C ⊗ C^{o} ⊗ B; A⊗e⊗B; M; A ⊗ B; A⊗B; A ⊗ B; N; D; A⊗ϵ⊗B; β]  (5)

It satisfies

\xymatrix{M \ar[r]^{δ_{l}} & C ⊗ C ⊗ M \ar[r]^{1⊗ϵ⊗1} \ar[r]_{ϵ⊗1⊗1} & C ⊗ M \ar[r]^{1⊗α} & C ⊗ N}  (6)

The 2-cell β is uniquely determined by a left A ⊗ B- right D-comodule map α which satisfies (6).

Proof. The comodule N ∘ (A ⊗ e ⊗ B) is C ⊗ N with coaction

C ⊗ N [figure: δ_{3}⊗δ] C ⊗ C ⊗ C ⊗ A ⊗ B ⊗ N ⊗ D [figure: c_{4135267}] C ⊗ A ⊗ B ⊗ C ⊗ C ⊗ N ⊗ D.

The left C^{o} and C coactions on C ⊗ N both are the cofree coactions, i.e. they are determined by the comultiplications.

The basic property of a cofree comodule is that any comodule map β : M ⇒ C ⊗ N to a cofree comodule is uniquely determined by its corestriction to N, by which is meant the map α = (ϵ ⊗ N)β : M ⇒ N in V. Specifically, β can be recovered from α as

M [figure: δ_{l}] C ⊗ M [figure: 1⊗α] C ⊗ N.

It follows that in the setting of the lemma β can be reconstructed from α in two ways. The condition (6) asserts that these two reconstructions are the same.

It is easily seen that β is a left A-, B- right D- comodule map if and only if α is. The lemma is proved.

3.2. Lemma. Let β be a 2-cell:

[figure: I; n; M; C^{o} ⊗ C; N; D; β]

Let α : M → N be a map in V determined by the pasting composite:

[figure: I; 1; j; M; I; ϵ^{∗}; C^{o} ⊗ C; N; D; δ; β]

It satisfies:

\xymatrix{C \ar[r]^{α} & N \ar[r]^{δ_{l}} & C ⊗ C ⊗ N \ar[r]^{1⊗ϵ⊗1} \ar[r]_{ϵ⊗1⊗1} & C ⊗ N}  (7)

The 2-cell β is uniquely determined by a right D-comodule map α which satisfies (7).

Proof. The 2-cell β is a map C → C ⊗_{C^{o}⊗C} N. This is induced by a map β′ : C → C ⊗N satisfying

(1 ⊗ ((ϵ ⊗ 1)δ_{l}))β′ = (cδ ⊗ 1)β′  (8)

(1 ⊗ ((1 ⊗ ϵ)δ_{l}))β′ = (δ ⊗ 1)β′  (9)

We have α = (ϵ ⊗ 1)β′. From α we can recover β′ in two ways: using (8) it can be shown that β′ can be reconstructed from α as the top composite in (7), or using (9) it can be shown that β′ can be reconstructed from α as the bottom composite in (7). So, the map α defined from β satisfies (7). Conversely, β can be defined from a map α which satisfies (7).

It is easily checked that β is a right D-comodule map if and only if α is. The lemma is proved.

The maps ξ_{2} = 1 ⊗ ϵ ⊗ 1 : C ⊗ C ⊗ C → C ⊗ C and ξ_{0} = δ : C → C ⊗ C define a monoidal morphism structure on ϵ^{∗} = ϵ^{∗} ⊗ ϵ^{∗} : I [figure] C^{o} ⊗ C. For any n ≥ 0 we have a 2-cell:

[figure: I; ϵ^{∗}; I; (C^{o} ⊗ C)^{⊗n}; p_{n}; I; ϵ^{∗}; C^{0} ⊗ C; ξ_{n}]  (10)

3.3. Lemma. For any n, the function defined on the set of 2-cells

C((C^{o} ⊗ C)^{⊗n}, C^{o} ⊗ C)(M, N ∘ p_{n})

with values in

C(I, C^{o} ⊗ C)(M ∘ ϵ^{∗}, N ∘ ϵ^{∗})  (11)

taking

[figure: (C^{o} ⊗ C)^{⊗n}; p_{n}; M; C^{o} ⊗ C; A; C^{o} ⊗ C; β]

to the pasting composite

[figure: I; I; ϵ^{∗}; (C^{o} ⊗ C)^{⊗n}; p_{n}; M; I; ϵ^{∗}; C^{o} ⊗ C; A; C^{o} ⊗ C; ξ_{n}; β]

is injective.

Proof. The n = 0 case follows from Lemma 3.1. For n = 1 the function forgets the left C^{o} ⊗ C comodule structure which clearly is injective. For n ≥ 2, the 2-cell ξ_{n} can be written as a pasting composite of the 2-cells ξ_{2} ⊗ (C^{o} ⊗ C)^{⊗n−1}, …, ξ_{2} ⊗ (C^{o} ⊗ C)^{⊗2}, ξ_{2}. Pasting from the left by each of these is an injective function by Lemma 3.2, hence pasting from the left by ξ_{n} is injective too.

4. Comonads in the bicategory of comodules

We will use the lower case Greek letters ϵ and δ for counits and comultiplications of both comonads in C and the comonoids. Although these are not the same, below it will become clear that such notation is not confusing.

Let E be a comonoid. There is an equivalence of categories between comonads on E in the bicategory C of comodule and comonoid maps with codomain E. If ϵ : G → E is a comonoid map, then the adjunction

[figure: ϵ_{∗} ⊣ ϵ^{∗} : E; G]

induces a comonad on E. Conversely, if G is a comonad on E with comonad comultiplication δ : G → G⊗_{E} G and comonad counit ϵ : G → E, then G itself becomes a comonoid with comultiplication and counit

G [figure: δ] G ⊗_{E} G [figure: i] G ⊗ G

G [figure: ϵ] E [figure: ϵ] I

while ϵ : G → E becomes a comonoid map. In fact, the comonoid G is the Eilenberg-Moore object of (E, G) with the universal G-coalgebra

[figure: G; ϵ_{∗}; ϵ_{∗}; E; g; E; δ]

4.1. Proposition. C admits the Eilenberg-Moore construction for comonads.

It follows from Proposition 2.2 that MonC also admits the comonad Eilenberg-Moore construction. To wit, given a monoidal structure on a comonad G, the comonoid G becomes a monoidale in C, while ϵ_{∗} : G [figure] E becomes a strong monoidal morphism.

The correspondence between comonads and comonoid maps lifts to a correspondence between monoidal comonads on the monoidale E and monoidales G in C together with a comonoid map G → E such that ϵ_{∗} : G [figure] E is a strong monoidal morphism.

5. Quantum Categories

Essentially following [5] we define a quantum category in V. In [5] it was shown that a quantum category in Set is the same as a small category and a quantum category in V ect^{op} is the same as a bialgebroid [17], [10], [20]. Most of the section after the definition is dedicated to proving Statement 5.6, which translates that definition to a set of axioms close to the definitions of bialgebroid in the literature.

5.1. Definition. A quantum graph (C, A) in V consists of a comonoid C and a comonad A on C^{o} ⊗ C.

[figure: C^{o} ⊗ C; A; A; C^{o} ⊗ C; A; C⊗C; C^{o} ⊗ C; ϵ; δ]

5.2. Definition. A quantum category (C, A) in V consists of a comonoid C together with a monoidal comonad A on C^{o} ⊗ C.

A quantum category has an underlying quantum graph and 2-cells

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; A⊗A; p; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; C^{o} ⊗ C; A; C^{o} ⊗ C; I; j; j; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{2}; μ_{0}]

which make A into a monoidal morphism and both of which are comonad maps.

If V is the category of sets with the monoidal structure given by the cartesian product, then a quantum category in V is the same as an ordinary category. We do not consider this fact in details since this was done in [3]. When V is the opposite category of modules over a commutative ring, then a quantum category in V is the same as a bialgebroid. This will become clearer with the Statement 5.6. We continue with a general monoidal category V.

By Section 4, a quantum graph amounts to comonoids C, A and a comonoid map ϵ : A → C^{o} ⊗ C. The latter itself amounts to comonoid maps s : A → C^{o} and t : A → C satisfying:

\xymatrix{ & A ⊗ A \ar[r]^{s⊗t} & C ⊗ C \ar[dd]^{c} \\ A \ar[dr]_{δ} \ar[ur]^{δ} \\  & A ⊗ A \ar[r]_{t⊗s} & C ⊗ C}

By s and t we can express ϵ as

A [figure: δ] A ⊗ A [figure: s⊗t] C ⊗ C.

C is called the object of objects of the quantum graph. A is called the object of arrows. The maps s and t are called the source and the target maps respectively.

We regard A as a comodule C [figure] C using the right C^{o} ⊗ C-coaction on it. In terms of s and t left and right C-coactions on A are

A [figure: δ_{3}] A ⊗ A ⊗ A [figure: 1⊗s⊗ϵ] A ⊗ C [figure: c^{−1}] C ⊗ A and

A [figure: δ_{3}] A ⊗ A ⊗ A [figure: 1⊗ϵ⊗t] A ⊗ C.

The tensor product H = A⊗_{C} A of A with itself over C is called the object of composable arrows for the quantum graph. It is defined by the equalizer

\xymatrix{H \ar[r]^{i} & A ⊗ A \ar[r]^{1⊗(c^{−1}(1⊗s⊗ϵ)δ_{3})} \ar[r]_{(1⊗ϵ⊗t)δ_{3})⊗1} & A ⊗ C ⊗ A.}

The composite comodule

C^{o} ⊗ C ⊗ C^{o} ⊗ C [figure: A⊗A] C^{o} ⊗ C ⊗ C^{o} ⊗ C [figure: p] C^{o} ⊗ C

is H with left C^{o} ⊗ C ⊗ C^{o} ⊗ C-coaction the unique map δ_{l} : H → C ⊗ C ⊗ C ⊗ C ⊗ H making

\xymatrix{H \ar[r]^{δ_{l}} \ar[d]_{i} & C ⊗ C ⊗ C ⊗ C ⊗ H \ar[d]^{1⊗1⊗1⊗1⊗i} \\ A ⊗ A \ar[r]_{δ_{l}} & C ⊗ C ⊗ C ⊗ C ⊗ A ⊗ A}

commute and right C^{o} ⊗ C-coaction the unique map δ_{r} : H → C ⊗ C ⊗ H making

\xymatrix{H \ar[r]^{δ_{r}} \ar[d]_{i} & H ⊗ C ⊗ C \ar[d]^{i⊗1⊗1} \\ A ⊗ A \ar[r]_{δ_{r}} & A ⊗ A ⊗ C ⊗ C}

commute. We regard H as a comodule C [figure] C using the right C^{o} ⊗ C-coaction on it. The map ν_{2} : H → A determined by the pasting composite

[figure: I; I; ϵ^{∗}; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; I; ϵ^{∗}; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{2}; ξ_{2}]  (12)

is called the composition map of the quantum category. The map ν_{0} : C → H determined by the pasting composite

[figure: I; j; I; j; I; ϵ^{∗}; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{0}; ξ_{0}]  (13)

is called the unit map of the quantum category.

5.3. Lemma. The 2-cells μ_{2} and μ_{0} determine a monoidal morphism structure on the comodule A : C^{o} ⊗ C [figure] C^{o} ⊗ C if and only if (A, ν_{2}, ν_{0}) is a monoid in C(C, C).

Proof. It follows from Lemma 3.3 that μ_{2} and μ_{0} determine a monoidal morphism structure on A if and only if the pasting composites (12) and (13) determine a monoidal morphism structure on ϵ^{∗}A. Using the equivalence C(I, C^{o} ⊗ C) ≃ C(C, C), the 2-cells (12) and (13) determine a monoidal morphism structure on ϵ^{∗}A if and only if (A, ν_{2}, ν_{0}) is a monoid in C(C, C).

Consider the pasting composite

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; A⊗A; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{2}; δ]  (14)

It is a map H → H ⊗_{(C^{o}⊗C)} A. Let γ_{r} be the composite of this with the canonical injection H ⊗_{(C^{o}⊗C)} A → H ⊗ A.

5.4. Lemma. The 2-cell μ_{2} is a comonad morphism if and only if the following diagrams commute

\xymatrix{H \ar[r]^{ν_{2}} \ar[d]_{γ_{r}} & A \ar[d]^{δ} \\ H ⊗ A \ar[r]_{ν_{2}⊗1} & A ⊗ A}  (15)

\xymatrix{H \ar[r]^{ν_{2}} \ar[d]_{i} & A \ar[d]^{ϵ} \\ A ⊗ A \ar[r]_{ϵ⊗ϵ} & I}  (16)

Proof. The map μ_{2} is a comonad map if:

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; A⊗A; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; C^{o} ⊗ C; A; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{2}; μ_{2}; δ]

=

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; C^{o} ⊗ C; A⊗A; A; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{2}; δ]

and

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; A⊗A; p; C^{⊗4}; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; C^{o} ⊗ C; C⊗C; C^{o} ⊗ C; ϵ; ≅; C^{o} ⊗ C ⊗ C^{o} ⊗ C; A⊗A; p; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; C^{o} ⊗ C; C⊗C; A; C^{o} ⊗ C; μ_{2}; ϵ; =]

By Lemma 3.3 these equalites between pasting diagrams are equivalent to the following equalities obtained by suitably pasting to them the 2-cell (10) for n = 2.

[figure: I; I; A⊗A; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; I; A; C^{o} ⊗ C; A; C^{o} ⊗ C; μ_{2}; ν_{2}; δ]

=

[figure: I; I; A⊗A; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; I; A⊗A; A; C^{o} ⊗ C; A; C^{o} ⊗ C; ν_{2}; δ]

and

[figure: I; A⊗A; I; C^{⊗4}; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; I; C⊗C; C^{o} ⊗ C; ϵ; ≅; I; A⊗A; I; C^{o} ⊗ C ⊗ C^{o} ⊗ C; p; I; C⊗C; A; C^{o} ⊗ C; ν_{2}; ϵ; =]

It is easy to translate these equalities into commutative diagrams. The first of them translates to (15). The second translates to the commutativity of

\xymatrix{A \ar[r]^{δ} & A ⊗ A \ar[r]^{s⊗t} & C ⊗ C \\ H \ar[u]^{ν_{2}} \ar[rr]_{i} &  & A ⊗ A \ar[u]_{s⊗t}}  (17)

This reduces to the commutativity of (17). Indeed, in the diagram

\xymatrix{A \ar[r]^{δ_{l}} & A ⊗ C ⊗ C \ar[r]^{ϵ⊗1⊗1} & C ⊗ C \\ H \ar[u]^{ν_{2}} \ar[r]_{δ_{l}} & H ⊗ C ⊗ C \ar[u]_{ν_{2}⊗1⊗1} \ar[r]_{i⊗1⊗1} & A ⊗ A ⊗ C ⊗ C \ar[u]_{ϵ⊗ϵ⊗1⊗1}}

the left square commutes since ν_{2} is a right C^{o} ⊗ C-comodule map and the right square commutes given (16). A little calculation shows that the outer part is exactly (17).

The 2-cell μ_{0} is a map μ_{0} : C → C ⊗_{(C^{o}⊗C)} A. Let γ_{r} be the composite of this with the canonical injection C ⊗_{(C^{o}⊗C)} A → C ⊗ A.

5.5. Lemma. The 2-cell μ_{0} is a comonad morphism if and only if the following diagrams commute

\xymatrix{C \ar[r]^{ν_{0}} \ar[d]_{γ_{r}} & A \ar[d]^{δ} \\ C ⊗ A \ar[r]_{ν_{0}⊗1} & A ⊗ A}  (18)

\xymatrix{ & C \ar[dl]_{ν_{0}} \ar[dr]^{ϵ} \\ A \ar[rr]_{ϵ} &  & I}  (19)

Proof. The map μ_{0} is a comonad morphism if:

[figure: I; j; j; j; C^{o} ⊗ C; A; C^{o} ⊗ C; C^{o} ⊗ C; A; I; j; j; C^{o} ⊗ C; A; A; C^{o} ⊗ C; C^{o} ⊗ C; A; =; μ_{0}; μ_{0}; μ_{0}; δ]

and

[figure: I; j; j; C^{o} ⊗ C; A; C⊗C; C^{o} ⊗ C; I; j; j; C^{o} ⊗ C; C⊗C; C^{o} ⊗ C; =; μ_{0}; ϵ; 1_{C}]

Using Lemma 3.3 these are equivalent to the equalities:

[figure: I; I; j; j; I; A; C^{o} ⊗ C; C^{o} ⊗ C; A; I; I; j; I; A; A; C^{o} ⊗ C; C^{o} ⊗ C; A; =; ν_{0}; μ_{0}; ν_{0}; δ]

and

[figure: I; I; j; I; A; C⊗C; C^{o} ⊗ C; I; I; j; I; C⊗C; C^{o} ⊗ C; =; ν_{0}; ϵ; 1_{C}]

The first of these translates to the commutativity of (18). The second translates to

\xymatrix{A \ar[r]^{δ} \ar[d]_{ν_{0}} & A ⊗ A \\ C \ar[r]_{δ} & C ⊗ C \ar[u]^{s⊗t}}  (20)

Which reduces to the commutativity of (20). Indeed, in the diagram

\xymatrix{A \ar[r]^{δ_{3}} & A ⊗ A ⊗ A \ar[r]^{s⊗t} & A ⊗ C ⊗ C \ar[r]^{ϵ⊗1⊗1} & C ⊗ C \\ C \ar[u]^{ν_{0}} \ar[r]_{δ_{3}} & C ⊗ C ⊗ C \ar[r]_{c⊗1} & C ⊗ C ⊗ C \ar[r]_{ϵ⊗1⊗1} \ar[u]^{ν_{0}⊗1⊗1} & C ⊗ C \ar[u]_{1}}

the left square commutes since ν_{0} is a right C^{o} ⊗ C-comodule map, and the right square commutes given (19), while the outer part can be seen to be (20).

Now we are in position to unpack Definition 5.2. Start again with a quantum graph (C, A). There is a unique map γ_{l} : H → A ⊗ A ⊗ H making

\xymatrix{H \ar[r]^{i} \ar[d]_{γ_{l}} & A ⊗ A \ar[r]^{δ⊗δ} & A ⊗ A ⊗ A ⊗ A \ar[d]^{1⊗c⊗1} \\ A ⊗ A ⊗ H \ar[rr]_{1⊗1⊗i} &  & A ⊗ A ⊗ A ⊗ A} ;  (21)

commute. By Lemma 3.1 the 2-cell μ_{2} is determined by the map ν_{2}. The condition of 3.1 says that ν_{2} should respects the right coaction by C^{o} ⊗ C and the left coactions by the first and the fourth terms in C^{o} ⊗ C ⊗ C^{o} ⊗ C and satisfy (6), which now becomes

\xymatrix{H \ar[r]^{γ_{l}} & A ⊗ A ⊗ H \ar[r]^{t⊗ϵ⊗1} \ar[r]_{ϵ⊗s⊗1} & C ⊗ H \ar[r]^{1⊗ν_{2}} & C ⊗ A.}  (22)

Using (22), it can be shown that there exists a unique map γ_{r} making

\xymatrix{H \ar[r]^{δ_{l}} \ar[d]_{γ_{r}} & A ⊗ A ⊗ H \ar[d]^{1⊗1⊗ν_{2}} \\ H ⊗ A \ar[r]_{i⊗1} & A ⊗ A ⊗ A}  (23)

commute. This is the same as the map defined before by (14). Observe that commutativity of the diagram (15) in Lemma 5.4 implies that ν_{2} respects the left coaction by the first and the fourth terms in C^{o} ⊗ C ⊗ C^{o} ⊗ C.

By Lemma (3.2) the 2-cell μ_{0} is determined by the map ν_{0}. The condition of (3.2) says that ν_{0} should respect the right C^{o} ⊗ C coaction and satisfy (7), which now becomes

\xymatrix{C \ar[r]^{ν_{0}} & A \ar[r]^{(s⊗1)δ} \ar[r]_{(t⊗1)δ} & C ⊗ A.}  (24)

The common value of the two composites in (24) is the same as the map γ_{r} defined above. Observe that if (A, ν_{2}, ν_{0}) is a comonoid in C(C, C), then (24) follows from (22).

Assembling the established facts we obtain:

5.6. Statement. [B. Day, R. Street] Giving a quantum category structure on a quantum graph (C, A) is equivalent to giving maps ν_{2} : H → A and ν_{0} : C → A satisfying the following axioms:

Axiom 1: (A, ν_{2}, ν_{0}) is a monoid in C(C, C).

Axiom 2: The diagram (22), in which the map γ_{l} is defined by (21).

Axiom 3: The diagram (15), in which the map γ_{r} is defined by (23) using Axiom 2. Axiom 4: The diagram (16). Axiom 5: The diagram (18), in which the map γ_{r} is defined by (24). Axiom 6: The diagram (19).

For more clarity see the Appendix where these axioms are presented using string diagrams.

By Section 4, a quantum category structure on a quantum graph (C, A) is the same as a monoidale structure on A, such that ϵ : A → C^{o} ⊗ C is strong monoidal. In term of our data this monoidal structure on A can be expressed as follows. The multiplication A ⊗ A [figure] A is H with left and right coactions the maps γ_{l} : H → A ⊗ A ⊗ H and γ_{r} : H → H ⊗ A. The unit I [figure] A is C with the right coaction the map γ_{r} : C → C ⊗ A. The monoidale A is the Eilenberg-Moore object of the comonad A : C^{o} ⊗ C [figure] C^{o} ⊗ C in MonComod(V). Applying the representable pseudofunctor MonC(I, −) : MonC → MonCat to the universal 2-cell

[figure: A; ϵ_{∗}; ϵ_{∗}; C^{o} ⊗ C; A; C^{o} ⊗ C]

we obtain an Eilenberg-Moore construction in the category of monoidal categories and monoidal functors:

[figure: Comod(I, A); C(I, C^{o} ⊗ C); A∘−; C(I, C^{o} ⊗ C)]

Using the equivalence (4) we can transport the monoidal comonad A ∘ − on the category C(I, C^{o} ⊗ C) to a monoidal comonad on the category C(C, C). Thus, a quantum category defines a monoidal comonad on C(C, C), the Eilenberg-Moore object of which is the category of the right A-comodules.

6. The category of quantum categories

Suppose that f : C → C′ is a comonoid map. Let α : f^{∗} ∘ f_{∗} → C and β : C′ → f_{∗} ∘ f^{∗} be the unit and the counit of the adjunction f^{∗} ⊣ f_{∗} as in Section 3. They are maps respectively in C(C, C) and C(C′, C′). By biduality, from α we get a map α : e ⟶ e ∘ (f_{∗} ⊗ f^{∗o}) in C(C ⊗ C^{o}, I) and from β we get a map β : (f_{∗} ⊗ f^{∗o}) ∘ n ⇒ n in C(I, C^{′o} ⊗ C′).

The comodule f^{∗o}⊗f_{∗} : C^{o}⊗C [figure] C^{′o}⊗C′ is an opmonoidal morphism with structure 2-cells

[figure: C^{o} ⊗ C; f^{∗o}⊗f_{∗}; C^{′o} ⊗ C′; C^{o} ⊗ C ⊗ C^{o} ⊗ C; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗f_{∗}; p; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; p; C^{o} ⊗ C; f^{∗o}⊗f_{∗}; C^{′o} ⊗ C′; I; j; j; ω_{2}; ω_{0}]

defined as ω_{2} = f^{∗o} ⊗ α ⊗ f_{∗} and ω_{0} = β.

Suppose that g : C → C′ is another comonoid map. The opmonoidal map f^{∗o} ⊗ f_{∗} acts from the left on f^{∗o} ⊗ g_{∗} by

[figure: C^{o} ⊗ C; f^{∗o}⊗g_{∗}; C^{′o} ⊗ C′; C^{o} ⊗ C ⊗ C^{o} ⊗ C; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗g_{∗}; p; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; p; λ_{l}]

defined as λ_{l} = f^{∗o} ⊗ α ⊗ g_{∗}. Similarly g^{∗o} ⊗ g_{∗} acts from the right on f^{∗o} ⊗ g_{∗} with coaction 2-cell λ_{r} defined similarly.

6.1. Definition. A map between quantum graphs (σ, f) : (C, A) → (C′, A′) consists of a morphism of comonoids f : C → C′ and a 2-cell

[figure: C^{o} ⊗ C; f^{∗o}⊗f_{∗}; A; C^{o} ⊗ C; f^{∗o}⊗f_{∗}; C^{′o} ⊗ C′; A′; C^{′o} ⊗ C; σ]  (25)

such that (f^{∗o} ⊗ f_{∗}, σ) is a comonad map.

6.2. Definition. A functor (f, σ) : (C, A) → (C′, A′) between quantum categories is a map between the underlying quantum graphs such that the 2-cell σ is a square.

In other words a quantum functor is an opmorphism of monoidal comonads of the form (f^{∗o} ⊗ f_{∗}, φ) : (C, A) → (C′, A′). Here are the equalities that σ must satisfy:

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; A⊗A; A′⊗A′; A′; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗f_{∗}; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗f_{∗}; f^{∗o}⊗f_{∗}; p; p; p; σ⊗σ; μ_{2}; ω_{2}]

=

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{o} ⊗ C; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; A⊗A; A′; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗f_{∗}; f^{∗o}⊗f_{∗}; p; p; A; p; f^{∗o}⊗f_{∗}; μ_{2}; σ; ω_{2}]

and

[figure: I; C^{o} ⊗ C; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; j; j; j; A; f^{∗o}⊗f_{∗}; f^{∗o}⊗f_{∗}; A′; I; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; j; j; j; f^{∗o}⊗f_{∗}; A′; =; η; σ; ω_{0}; η; ω_{0}]

A map of quantum graphs (C, A) → (C′, A) amounts to comonoid maps f : C → C′ and φ : A → A′ for which the diagrams

\xymatrix{A \ar[r]^{s} \ar[d]_{φ} & C \ar[d]^{f} & A \ar[r]^{t} \ar[d]_{φ} & C \ar[d]^{f} \\ A′ \ar[r]_{s} & C′ & A′ \ar[r]_{t} & C′}  (26)

commute. The pair (f, φ) is a functor of quantum categories if additionally it satisfies:

\xymatrix{ & A ⊗_{C} A \ar[r]^{ν_{2}} \ar[dl]_{i} & A \ar[dd]^{φ} \\ A ⊗_{C′} A \ar[dr]_{φ⊗_{C′}φ} \\  & A′ ⊗_{C′} A′ \ar[r]_{ν_{2}} & A′}

and

\xymatrix{C \ar[r]^{ν_{0}} \ar[d]_{f} & A \ar[d]^{φ} \\ C′ \ar[r]_{ν_{0}} & A′}

The tensor product A⊗_{C′} A of A with itself over C′ is taken by regarding A as a comodule C′ [figure] C′ with left and right coactions:

A [figure: δ] A ⊗ A [figure: 1⊗f] A ⊗ A′ [figure: 1⊗s] A ⊗ C′ [figure: c] C′ ⊗ A

A [figure: δ] A ⊗ A [figure: 1⊗f] A ⊗ A′ [figure: 1⊗s] A ⊗ C′.

Observe that in a quantum functor the map f is determined by the map φ via f = φν_{0}.

The notion of the quantum functor includes the notion of functor between small categories and the notion of weak morphism of bialgebroids [16].

By Section 2 an opmorphism between monoidal comonads is determined by an opmonoidal morphism between the Eilenberg-Moore objects. Thus given a quantum functor (f, φ) : (C, A) → (C′, A′) the comodule φ_{∗} : A [figure] A′ has an opmonoidal morphism structure which lifts the opmonoidal morphism structure on f^{∗o} ⊗f_{∗}^{o} : C^{o} ⊗C [figure] C^{o} ⊗C. By application of the functor MonComod(I, −) : MonComod → MonCat we get an opmonoidal functor between categories of right A-comodules:

Comod(I, A) [figure: MonComod(I,φ_{∗})] Comod(I, A′).

Define composition of quantum functors φ : (C, A) → (C′, A′) and φ′ : (C′, A′) → (C′′, A′′) by

A [figure: φ] A′ [figure: φ′] A′′.

The units for this composition are provided by quantum functors of the form (1, 1).

6.3. Theorem. Quantum categories and functors between them form a category qCatV.

6.4. Definition. A natural transformation τ : (f, φ) ⇒ (g, φ′)

[figure: (A, C); (f,φ); (g,φ′); (A′, C′); τ]

between quantum functors is a 2-cell

[figure: C^{o} ⊗ C; A; f^{∗⊗o}⊗g_{∗}; C^{o} ⊗ C; f^{∗⊗o}⊗g_{∗}; C^{′o} ⊗ C′; A′; C^{′o} ⊗ C′; τ]

making f^{∗o} ⊗ g_{∗} into a comonad map so that both the left coaction of f^{∗o} ⊗ f_{∗} and the right coaction of g^{∗o} ⊗ g_{∗} on f^{∗o} ⊗ g_{∗} respect the comonad structure.

Here are the equalities which the 2-cell τ should satisfy:

24  DIMITRI CHIKHLADZE

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; A⊗A; A′⊗A′; A′; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗f_{∗}; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗f_{∗}; f^{∗o}⊗f_{∗}; p; p; p; σ⊗τ; μ_{2}; λ_{l}]

=

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{o} ⊗ C; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; A⊗A; A′; f^{∗o}⊗f_{∗}⊗f^{∗o}⊗g_{∗}; f^{∗o}⊗g_{∗}; p; p; A; p; f^{∗o}⊗g_{∗}; μ_{2}; τ; λ_{l}]

and

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; A⊗A; A′⊗A′; A′; f^{∗o}⊗g_{∗}⊗g^{∗o}⊗g_{∗}; f^{∗o}⊗g_{∗}⊗g^{∗o}⊗g_{∗}; f^{∗o}⊗g_{∗}; p; p; p; τ⊗σ′; μ_{2}; λ_{r}]

=

[figure: C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{o} ⊗ C ⊗ C^{o} ⊗ C; C^{′o} ⊗ C′ ⊗ C^{′o} ⊗ C′; C^{o} ⊗ C; C^{o} ⊗ C; C^{′o} ⊗ C′; C^{′o} ⊗ C′; A⊗A; A′; f^{∗o}⊗g_{∗}⊗g^{∗o}⊗g_{∗}; f^{∗o}⊗g_{∗}; p; p; A; p; f^{∗o}⊗g_{∗}; μ_{2}; τ; λ_{r}]

A natural transformation τ : (f, φ) → (g, φ′) : (C, A) → (C′, A′) amounts to a comonoid map τ : A → A′ such that:

\xymatrix{A \ar[r]^{s} \ar[d]_{τ} & C \ar[d]^{f} & A \ar[r]^{t} \ar[d]_{τ} & C \ar[d]^{g} \\ A′ \ar[r]_{s} & C′ & A′ \ar[r]_{t} & C′}  (27)

\xymatrix{ & A′ ⊗_{C} A′ \ar[rr]^{i} &  & A′ ⊗_{C′} A′ \ar[dr]^{ν_{2}} \\ A ⊗_{C} A \ar@{-->}[ur]^{φ⊗_{C}τ} \ar[rr]^{ν_{2}} \ar[dr]_{τ⊗_{C}φ} &  & A \ar[rr]^{τ} &  & A′ \\  & A′ ⊗_{C} A′ \ar[rr]_{i} &  & A′ ⊗′_{C} A′ \ar[ur]_{ν_{2}}}

7. Constructions on quantum categories

A coreflexive-equalizer-preserving braided strong-monoidal functor V → W defines a functor between the categories of quantum categories qCatV → qCatW. Thus qCat can be viewed as a 2-functor from the 2-category of braided monoidal categories (satisfying the condition at the beginning of Section 3) and braided strong monoidal functors to the 2category of categories. This functor preserves finite products since we have isomorphisms

qCat(V × W) ≅ qCatV × qCatW
qCat(1) ≅ 1

7.1. Example. When V is a symmetric monoidal category, then the functors − ⊗ − : V × V → V, I : 1 → V are symmetric monoidal. From them we obtain functors

− ⊗ − : qCatV × qCatV → qCatV
I : 1 → qCatV

defining a monoidal structure on qCatV.

7.2. Example. There is a functor Set → V, taking a set S to the S-fold coproduct S · I of the monoidal unit, provided these copowers exist. When a certain distributivity law is satisfied, this functor is strong monoidal. Any coreflexive equalizer in Set (which does not involve an empty set) is split, and thus preserved by any functor. We have qCatSet = Cat [5]. So we get a functor:

Cat → qCatV

To wit any category determines a quantum category in any (sufficiently good) monoidal category.

7.3. Example. Suppose that V has small coproducts, and assume that each of the functors X + − preserves coreflexive equalizers. For any finite set S let S · V stand for the S-fold coproduct of an object V of V. There is a coreflexive-equalizer-preserving braided strong-monoidal functor − · − : Set_{f} × V → V. The preservation of coreflexive equalizers is due to Lemma 0.17 in [6]. We have qCatSet_{f} = Cat_{f}. Thus we obtain a functor

Cat_{f} × qCatV → qCatV  (28)

Let i : Δ → Cat_{f} be the canonical embedding of the simplicial category into the category of finite categories. Precomposing (28) with i × 1 we obtain a functor

Δ × qCatV → qCatV

Let A and B be quantum categories in V. Consider the simplicial set

qCatV(− · A, B).  (29)

The 1-simplexes of this simplicial set are the quantum functors from A to B. The 2simplexes are the quantum natural transformation. For quantum categories A, B and C, there is a map of simplicial sets

qCatV(− · A, B) × qCatV(− · B, C) → qCatV(− · A, C)  (30)

defined on the n-simplexes by

qCatV(n · A, B) × qCatV(n · B, C) [figure: (n·−)×1] qCatV(n · n · A, n · B) × qCatV(n · B, C)

[figure: qCatV(δ_{(n,A)},1)×1] qCatV(n · A, n · B) × qCatV(n · B, C) [figure: comp] qCatV(n · A, C).

Here δ is the natural transformation

[figure: Cat × qCatV; diag×1; −·−; Cat × Cat × qCatV; −·−·−; qCatV; δ]

obtained by applying qCat− to the obvious natural transformation

[figure: Set × V; diag×1; −·−; Set × Set × V; −·−·−; V; δ]

Using (30) as compositions we can construct a simplicial set enriched category with objects the quantum categories in V and a hom simplicial set from A to B given by (29).

7.4. Theorem. For any (sufficiently good) braided monoidal category V the above defines a simplicial set enriched category qCatV.

7.5. Example. Consider the category of families FamV. An object of FamV is a pair (S, {A_{s}}), where S is a set and {A_{s}} is an S indexed family of objects of V. A morphism (f, {φ_{s}}) : (S, {A_{s}}) → (S′, (A′_{s})) consists of a map f : S → S′ and for each s in S a morphism φ_{s} : A_{s} → A_{f(s)} in V. The monoidal structure on V induces a monoidal structure on FamV in the obvious way. Consider the functor

FamV → V.

taking (S, {A_{s}}) to the coproduct ∐ A_{s}. This functor is monoidal and under mild conditions on V, it preserves coreflexive equalizers. By applying the qCat we obtain a functor

qCatFamV → qCatV.

The functor

FamV → Set

taking (S, {A_{s}}) to the set S determines a functor

qCatFamV → Cat.  (31)

In this way each object in qCatFamV has an underlying category.

Next we show how Hopf group coalgebras introduced in [18] are quantum categories (groupoids [5]).

Let G denote a group. A Hopf G-coalgebra consists of a family of algebras {A_{g}} indexed by elements g of G together with a family of linear maps {A_{gg′} → A_{g} ⊗ A_{g′}} and an antipode satisfying certain axioms. Such a Hopf group coalgebra without an antipode is an object of qCatFamV ect^{op} with the object of objects (1, {I}) and the object of arrows (G, {A_{g}}). The underlying category is the group G. Using the functor (31) from a Hopf group coalgebra we obtain a quantum category in V ect^{op}, in other words a bialgebroid. The antipode would make this bialgebroid into a Hopf algebroid.

8. Appendix: Computations for quantum categories in string diagrams

We introduce framed string diagrams to represent morphisms in a braided monoidal bicategory as an enrichment of the string diagrams of [7]. These diagrams are designed to ease presentation of quantum structures.

A string diagram of [7] has edges labeled by objects of V and nodes labeled by morphisms of V and represents a morphism in V. For example a morphism f : X ⊗ Y → Z in V is represented by a string diagram:

[figure: f; X; Y; Z]

The identity morphism on an object X is represented by

X

The braiding isomorphisms c and c^{−1} are represented respectively by

[figure: X; Y; Y; X]

Composition is by concatenation and tensoring is by juxtaposition.

A framed string diagram besides strings and nodes may have framed regions labeled by comonoids in V. A framed region labeled by a comonoid C has two strings passing through it, of which, the left string is labeled by a right C-comodule and the right string is labeled by a left C-comodule. Such a framed region corresponds to taking tensor product over C. Now we give the description.

Suppose that C is a comonoid in V. Suppose that M is a right C-comodule and N is a left C-comodule with coactions:

[figure: M; C; N; C]

The framed string diagram

M N C

represents the identity morphism on M ⊗_{C} N. The framed string diagram

M N C

represents a canonical injection i : M ⊗_{C} N → M ⊗ N. Observe that we have:

[figure: M; N; C; =; M; N; C]

The framed string diagram

[figure: f; M; N; L]

represents a morphisms M ⊗_{C} N → L. The following is a rule for building a new string diagram from a given one. Suppose that a morphism … → … M ⊗ N … is represented by a framed string diagram

M N . . . . . .  …

and suppose that it factors through a morphism … → … M ⊗_{C} N …; that is, the equality

[figure: M; N; …; …; …; C; =; M; N; …; …; …; C]  (32)

holds, then the latter morphism is represented by

M N . . . . . .  …

Thus every time we want to introduce a new frame in a string diagram using this rule an extra computation establishing (32) should be performed. We also consider overlapping of framed regions. If M is a right C-comodule, N is a comodule from C to C′ and L is a left comodule, then

M N  L C  C′

represents the identity map on M ⊗_{C} N ⊗_{C′} L. If f : N ⊗_{C′} L → K is a left C-comodule map and g : M ⊗_{C} N → P is a right C′-comodule map, then the framed string diagrams

[figure: M; N; L; K; C; C′; f; L; N; M; P; C′; C; g]

represent 1 ⊗_{C} f : M ⊗_{C} N ⊗_{C′} L → M ⊗_{C′} K and g ⊗_{C′} 1 : M ⊗_{C} N ⊗_{C′} L → P ⊗_{C′} L respectively.

A quantum graph consists of comonoids C and A, for comultiplications and counits of which we write

[figure]

and comonoid maps s : A → C, t : A → C related by

[figure: s; t; =; s; t]

A quantum category consists of a quantum graph together with the composition map ν_{2} : A ⊗_{C} A → A and the identity map ν_{0} : C → A:

[figure: ν_{2}]  ν_{0}

which satisfy the six axioms in Statement 5.6. Below we quickly go through all of these axioms using framed string diagrams. A is regarded as a left and right C-comodule by coactions:

[figure: s; t]

The tensor product H = A ⊗_{C} A of A with itself over C is a left and a right C-comodule by coaction:

[figure: s; t]

Axiom 1 says that (A, ν_{2}, ν_{0}) should be a monoid in ComodV(C, C). This means that the following conditions should be satisfied.

[figure: ν_{0}; s; =; ν_{0}; ν_{0}; t; =; ν_{0}]

[figure: \begin{aligned}& ν_{2} \\ s \end{aligned}; =; \begin{aligned}& ν_{2} \\ s \end{aligned}; ν_{2}; t; =; ν_{2}; t]

ν_{2} ν_{2}  =  \begin{aligned}& ν_{2} \\ ν_{2} \end{aligned}

\begin{aligned}& t \\ & ν_{0} \\ ν_{2} \end{aligned} =  s ν_{0} ν_{2}  =

We have:

[figure: t]  =  t  =  s  =  s

Therefore we can form a framed string diagram This is the map γ_{l} : H → A ⊗ A ⊗ H. Axiom 2 is:

\begin{aligned}& ν_{2} \\ s \end{aligned}  =  [figure: ν_{2}; t]

Using Axiom 2 we have:

[figure: t; =; t; =; t]

[figure: s; =; s; =; s]

Thus, we can form a framed string diagram

ν_{2} This is the map γ_{r} : H → H ⊗ A. Axiom 3:

[figure: ν_{2}; ν_{2}; =; ν_{2}]

Axiom 4:

[figure: ν_{2}]  =

Note that these string diagrams look exactly like the string diagrams for two of the bialgebroid axioms. The map γ_{r} : C → C ⊗ A is either of:

[figure: ν_{0}; t; =; ν_{0}; s]

Axiom 5:

[figure: ν_{0}; =; ν_{0}; s; ν_{0}]

Axiom 6:

ν^{0}  =

Axioms for a quantum functor (f, φ) : (A, C) → (A′, C′) in framed string diagrams are:

\begin{matrix}φ \\ s \end{matrix}  =  \begin{matrix}s \\ f \end{matrix}  \begin{matrix}φ \\ t \end{matrix}  =  \begin{matrix}t \\ f \end{matrix}

[figure: ν_{2}; φ]  =  ν_{2} φ  φ

\begin{matrix}ν_{0} \\ φ \end{matrix}  =  \begin{matrix}f \\ ν_{0} \end{matrix}

Axioms for a quantum natural transformation τ : (f, φ) → (g, φ′) in framed string diagrams are:

\begin{matrix}τ \\ s \end{matrix}  =  \begin{matrix}s \\ f \end{matrix}  \begin{matrix}τ \\ t \end{matrix}  =  \begin{matrix}t \\ g \end{matrix}

ν_{2} φ  τ  =  [figure: ν_{2}; τ]  =  τ ν_{2} φ

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Department of Mathematics, Macquarie University, NSW, Australia

Email: d.chikhladze@gmail.com

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