Theory and Applications of Categories, Vol. 10, No. 19, 2002, pp. 469–485.

OPMONOIDAL MONADS

PADDY MCCRUDDEN

ABSTRACT. Hopf monads are identified with monads in the 2-category OpMon of monoidal categories, opmonoidal functors and transformations. Using Eilenberg-Moore objects, it is shown that for a Hopf monad S, the categories Alg(Coalg(S)) and Coalg(Alg(S)) are canonically isomorphic. The monadic arrows OpMon are then characterized. Finally, the theory of multicategories and a generalization of structure and semantics are used to identify the categories of algebras of Hopf monads.

1. Hopf Monads

The purpose of this note is to put the results of [Moerdijk (1999)] into a 2-categorical framework, highlight their universal nature, and so extend those results.

Propositions 1.1 and 1.2 of this section are from [Moerdijk (1999)]. Let C be a monoidal category, and (S, η, μ) a monad on the underlying category of C. A Hopf monad structure on S consists of natural transformations

[figure: C × C; S×S; ⊗; C; S; C × C; ⊗; C; χ; 1; I; 1; C; S; 1; I; C; ι]

satisfying various axioms; we shall provide these axioms in Example 2.5. Here, the functor I: 1 → C is the functor whose value on the only object of 1 is the unit I of C. Given S-algebras (X, x: SX → X) and (Y, y: SY → Y) the tensor product of the objects X and Y becomes an S-algebra with the follow action.

\xymatrix{S(X ⊗ Y) \ar[r]^{χ} & SX ⊗ SY \ar[r]^{x⊗y} & X ⊗ Y}

Similarly, the arrow ι: SI → I makes I into an S-algebra. The associativity and unit isomorphisms for C are then S-algebra morphisms making the category Alg(S) into a monoidal category in such a way that the forgetful functor U: Alg(S) → C is a strict monoidal functor. The converse of this is also true and we now summarize these remarks as a proposition.

The author thanks McGill University (Canada) and Prof. M. Makkai.

Received by the editors 2002 November15 and, in revised form, 2002 December 2.

Transmitted by Ross Street. Published on 2002 December 4.

2000 Mathematics Subject Classification: 18D10, 18D25, 18D05.

Key words and phrases: Hopf Monad, Eilenberg Moore Algebras, Multicategories, Structure and Semantics.

\begin{matrix}c \\ ◯ \end{matrix} Paddy McCrudden, 2002. Permission to copy for private use granted.

1.1. Proposition. There is a bijection between Hopf monad structures on a monad S and monoidal structures on Alg(S) such that the forgetful functor U: Alg(S) → C is a strict monoidal functor.

Suppose (C, δ, ε) is a coalgebra in C. Observe that the arrows

\xymatrix{SC \ar[r]^{Sδ} & S(C ⊗ C) \ar[r]^{χ} & SC ⊗ SC & SC \ar[r]^{Sε} & SI \ar[r]^{ι} & I}

make SC into a coalgebra in C. In fact, there is a monad Coalg(S) on the category Coalg(C) of coalgebras in C lifting the monad S.

1.2. Proposition. The categories Alg(Coalg(S)) and Coalg(Alg(S)) are canonically isomorphic.

The aim of this note is to analyze and generalize Propositions 1.1 and 1.2. We shall first consider Proposition 1.2.

2. Monads in a 2-category

In this section we recall the notion of a monad in a 2-category and then provide some examples. We then consider monadicity for arrows in a 2-category, and characterize the monadic arrows in the 2-category OpMon. We shall then show that Proposition 1.2 is a consequence of the fact that representable 2-functors preserve Eilenberg-Moore objects.

Let K be a bicategory. A monad [Street (1972)] in K is a lax functor 1 → K from the terminal 2-category 1. It amounts to an object X of K equipped with an arrow T: X → X and two 2-cells,

[figure: X; T; X; T; T; X; μ; X; 1; T; X; η]

called the multiplication and unit respectively, satisfying three axioms that express that μ is associative and unital. We also say that T is a monad on the object X. Given monads S and S′ on X, a monad morphism f: S → S′ is a 2-cell α: S′ ⇒ S respecting the multiplication and unit. Note the change of direction. With the evident compositions there is a category Mnd(X) of monads on X. This category is in fact a subcategory of the underlying category of the 2-category of monads in K [Street (1972)].

2.1. Example. Ordinary Monads. A monad in the 2-category Cat of categories is of course a monad in the usual sense.

2.2. Example. Monoids. For a monoidal category C, there is a bicategory ΣC called the suspension of C. The bicategory ΣC has one object 0 and ΣC(0, 0) = C, and with composition given by the monoidal structure. A monad in ΣC is exactly a monoid in C.

2.3. Example. Categories. Recall the bicategory Span(Set) of spans of sets. An object is a set, an arrow is a span A ← B → C of sets. The set B is called the head of the span, and the arrows are called the left and right legs. A 2-cell is a function between the heads commuting with the legs. Composition is given by pullback, and the identity span is given by the span of identity functions. A monad in Span(Set) is exactly a category [Bénabou (1967)].

2.4. Example. Multicategories. Multicategories were introduced in [Lambek (1969)] and have experienced a resurgence of interest of late. A multicategory is a monad in the Kleisli bicategory Span_{T} (Set) of spans, where T is the cartesian free monoid monad on the category Set of sets [Leinster (1997), Hermida (1999)]. We shall return to multicategories in Section 3.

For monoidal categories C and D, an opmonoidal functor F: C → D is a functor F: C → D along with natural transformations

[figure: C × C; F ×F; ⊗; C; F; D × D; ⊗; D; χ; 1; I; 1; C; F; 1; I; D; ι]

satisfying the following three axioms.

[figure: C^{2}; ⊗; C; F; C^{3}; 1⊗; ⊗1; F^{3}; C^{2}; ⊗; F^{2}; D; D^{3}; ⊗1; D^{2}; ⊗; a; χ; χ1; C^{2}; ⊗; F^{2}; C; F; C^{3}; 1⊗; F^{3}; D^{2}; ⊗; D; D^{3}; ⊗1; 1⊗; D^{2}; ⊗; χ; 1χ; a]

[figure: C^{2}; ⊗; C; F; C; 1I; 1; F; C; 1; F; D; D; 1; D; 1; r; C^{2}; ⊗; F^{2}; C; F; C; 1I; F; D^{2}; ⊗; D; D; 1; 1I; D; 1; χ; 1ι; r]

[figure: C^{2}; ⊗; C; F; C; I1; 1; F; C; 1; F; D; D; 1; D; 1; l; C^{2}; ⊗; F^{2}; C; F; C; I1; F; D^{2}; ⊗; D; D; 1; I1; D; 1; χ; ι1; l]

Given parallel opmonoidal functors F and G, a transformation of opmonoidal functors from F to G is a natural transformation α: F ⇒ G satisfying the following two axioms.

[figure: C^{2}; G^{2}; ⊗; C; G; F; C^{2}; ⊗; C; χ; α; C^{2}; G^{2}; F^{2}; ⊗; C; F; C^{2}; ⊗; C; α^{2}; χ]

[figure: 1; 1; I; C; G; F; 1; I; C; ι; α; 1; 1; I; C; F; 1; I; C; ι]

With the evident compositions there is a 2-category OpMon whose objects are monoidal categories, whose arrows are opmonoidal functors and whose 2-cells are transformations of opmonoidal functors. By [Kelly (1974b), Section 10.8], the 2-category OpMon is isomorphic to the 2-category Alg_{c}(T) of algebras, colax morphisms and transformations for some 2-monad T on Cat.

2.5. Example. Hopf monads. A Hopf Monad is exactly a monad in the 2-category OpMon.

Following Example 2.5, we suggest the more descriptive term opmonoidal monad for Hopf monad, which we shall use for the remainder of this paper.

2.6. Example. Bialgebras. Let B be a bialgebra in a symmetric monoidal category C. Let S: C → C denote the endofunctor of C whose value on an object X is X ⊗ B. The multiplication and unit of B induce natural transformations μ: SS ⇒ S and η: 1 ⇒ S making S into a monad on C. The comultiplication, counit and symmetry induce natural transformations

[figure: C × C; S×S; ⊗; C; S; C × C; ⊗; C; χ; 1; I; 1; C; S; 1; I; C; ι]

making S into an opmonoidal monad.

Recall that adjunctions and extensions may be be defined in any 2-category [Kelly & Street (1974), Street & Walters (1978)]. Let K be a 2-category and G: L → K an arrow in K. A left adjoint of G is an arrow F: K → L and two 2-cells η: 1 ⇒ GF and ε: F G ⇒ 1, called the unit and counit respectively, satisfying εF ∘ F η = 1 and Gε ∘ ηG = 1. One calls F the left adjoint of G. If H: L → N is an arrow with the same domain as G then the right extension (also called the right Kan extension) of H along G is an arrow Ran_{G}H: K → N and a 2-cell

[figure: L; G; H; K; Ran_{G}H; N]

such that for any J: K → N, the function

K(K, N)(J, Ran_{F} H) → K(L, N)(JG, H)

given by composing with this 2-cell is a bijection. Left adjoints and right extensions may not always exist, but if they do, they are unique up to canonical isomorphism.

2.7. Lemma. [Street (1972), Theorem 4] If G: L → K is an arrow in a 2-category K with a left adjoint F, and if ε: F G ⇒ 1 is the counit of the adjunction, then any arrow H: L → N has a right extension along G given by the arrow HF: K → L and the 2-cell Hε: HF G ⇒ H. This extension is preserved by any 2-functor. □

Let G: L → K be an arrow such that the right extension R = Ran_{G}G of G along itself exists. There are 2-cells μ: RR ⇒ R and η: 1 ⇒ R making R a monad in K [Street (1972), Section 2]. We call R the monad generated by G. When G has a left adjoint F we also say that R is generated by the adjunction F ⊣ G.

We now turn to the notion of algebras for a monad. Let T be a monad in K and let Y be an object of K. A Y -based T -algebra consists of an arrow M: Y → X equipped with a 2-cell

[figure: X; T; Y; M; M; X; μ]

satisfying two axioms that express that μ: T ∘ M ⇒ M is associative and unital. If M and N are Y -based T -algebras, then a Y -based T -algebra morphism f : M → N is a 2-cell f: M ⇒ N in K satisfying one axiom, expressing that f respects the action of T. With the evident compositions there is a category Alg(Y, T) whose objects are Y -based T -coalgebras, and whose arrows are Y -based T -algebra morphisms, and Alg(Y, T) is the value at the object Y of a 2-functor Alg(−, T) : K^{op} → Cat. An Eilenberg-Moore object of T is a representation

K(−, Alg(T)) ≅ Alg(−, T)

of this 2-functor. This means that Alg(T) is a T -algebra (U: Alg(T) → X, μ) with a 1- and a 2-dimensional universal property. The 1-dimensional property states that if M is a Y -based T -algebra then there exists a unique arrow c_{M}: Y → Alg(T), called the comparison, such that the following equation holds.

[figure: X; T; Y; c_{M}; Alg(T); U; U; X; X; T; Y; M; M; X]

The 2-dimensional property states that if f: M → N is a morphism of Y -based T - algebras, then there exists a unique 2-cell c_{f}: c_{M} ⇒ c_{N} such that f = U ∘ c_{f}. In general, Eilenberg-Moore objects need not exist; one says that K admits the construction of Eilenberg-Moore objects if for all monads T in K there exists a representation of Coalg(−, T). The existence of Eilenberg-Moore objects is a completeness property of K. Eilenberg-Moore objects are unique up to isomorphism, not just equivalence.

2.8. Example. Eilenberg-Moore Objects in Cat. It is well known that Eilenberg-Moore objects exist in Cat. Explicitly, the category Alg(T) has algebras (X, x : T X → X) as objects, and algebra morphisms as arrows. The arrow U: Alg(T) → X is the forgetful functor, and the action

[figure: X; T; Alg(T); U; U; X; μ]

has component x at the object (X, x). Of course this category of algebras was known before its universal property.

Suppose S is a monad in a 2-category K and F: K → L is a 2-functor. Then the arrow FS is canonically a monad in L, and furthermore, F takes S-algebras to FS-algebras. In particular, if the Eilenberg-Moore object Alg(S) of S exists, then FAlg(S) is canonically an FS algebra. We say that F preserves the Eilenberg-Moore object of S if FAlg(S) is an Eilenberg-Moore object of FS, and we say that F preserves Eilenberg-Moore objects if it preserves all Eilenberg-Moore objects that exist. Any finite limit preserving functor preserves Eilenberg-Moore objects.

Recall that an opmonoidal functor is called strong if the natural transformations χ and ι are isomorphisms.

2.9. Proposition. The 2-category OpMon admits Eilenberg-Moore objects and the forgetful 2-functor U: OpMon → Cat preserves them. Moreover, for any monad S on C in OpMon, the universal arrow U: Alg(S) → C is strong monoidal.

Proof. The 2-category OpMon is isomorphic to the 2-category Alg_{c}(D) of strict Dalgebras, colax morphism, and appropriate 2-cells for some 2-monad D on Cat. Lack [Lack (1998)] shows that such 2-categories admit Eilenberg-Moore objects, that the forgetful 2-functor preserves them, and that the universal arrow U: Alg(S) → C is strong monoidal.

2.10. Example. Reflection of Eilenberg-Moore objects. We show that the 2-functor U: OpMon → Cat does not reflect Eilenberg-Moore objects. Let (Set_{∗}, +) and (Set_{∗}, ×) be the category of pointed sets equipped with the monoidal structures deriving from coproduct and product respectively. The terminal object is also initial, providing canonical arrows X + Y → X × Y and 0 → 1. These arrows equip the identity functor with the structure of an opmonoidal functor M : (Set_{∗}, +) → (Set_{∗}, ×). The identity natural transformation

[figure: (Set_{∗}, ×); 1; (Set_{∗}, +); M; M; (Set_{∗}, ×); 1]

makes M into an algebra for the identity monad on (Set_{∗}, ×). Clearly this is an Eilenberg-Moore object in Cat. It is not, however, an Eilenberg-Moore object in OpMon since M is not strong monoidal.

We now consider the characterization of monadic arrows in OpMon. First we recall the definition of a monadic arrow in a 2-category. Let G: L → K be an arrow in a 2category K such that the right extension of G along itself exists, and let R be the monad generated by G. The universal 2-cell R ∘ G ⇒ G is an action of this monad on G. We say that G is monadic if R exists, the Eilenberg-Moore object of R exists, and the comparison c_{G} : L → Alg(R) is an equivalence.

2.11. Lemma. Let G: L → K be a monadic arrow. Then G has a left adjoint and the monad generated by the adjunction is canonically isomorphic to the monad generated by G. □.

2.12. Lemma. An opmonoidal functor G: D → C has a left adjoint if and only if it is strong and the functor G has a left adjoint in Cat. Furthermore, G is an equivalence in OpMon if an only if it is strong and G is an equivalence in Cat.

Proof. See [Kelly (1974a), Theorem 1.5].

2.13. Proposition. An opmonoidal functor G: D → C is monadic in OpMon if and only if

(i) The functor G is monadic in Cat;

(ii) The opmonoidal functor G is strong.

Proof. Suppose G is monadic. Then by Lemma 2.11, G has a left adjoint in OpMon, and by Lemma 2.12, G is strong. By Proposition 2.9, the functor G is monadic in Cat.

Conversely, suppose the conditions hold. Then by (ii) and Lemma 2.12, G has a left adjoint in OpMon. Thus by Lemma 2.7, the monad in Cat generated by the functor G is equal to the underlying functor of the monad R generated by the opmonoidal functor G. It follows that the underlying functor of the comparison c: D → Alg(R) in OpMon is the comparison in Cat. By (i) this functor is an equivalence. Since the universal Alg(R) → C in Cat reflects isomorphisms and G is strong, the comparison is strong. Thus Lemma 2.12, the comparison is an equivalence in OpMon.

Of course the monadic arrows in Cat are characterized in elementary terms by Beck’s Theorem [Mac Lane (1971), Chapter VI]. We now return to the motivating proposition.

2.14. Proposition. Suppose S is an opmonoidal monad. Then the categories Coalg(Alg(S)) and Alg(Coalg(S)) are canonically isomorphic.

Proof. This follows immediately from the fact that representable 2-functors preserve Eilenberg-Moore and the observation that for any monoidal category D, the category Coalg(D) is canonically isomorphic to the category OpMon(1, D) of opmonoidal functors from the terminal monoidal category.

We now consider the corresponding proposition in the braided case. Recall that if F: C → D is an opmonoidal functor between braided monoidal categories, then F is said to be braided if for all pairs of objects X and Y of C the following diagram commutes.

\xymatrix{F (X ⊗ Y) \ar[r]^{χ} \ar[d]_{F c} & F X ⊗ F Y \ar[d]^{c} \\ F (Y ⊗ X) \ar[r]_{χ} & F Y ⊗ F X}

A braided transformation of braided opmonoidal functors is a transformation of opmonoidal functors. With the evident compositions there is a 2-category BrOpMon whose objects are braided monoidal categories, whose arrows are braided opmonoidal functors and whose 2-cells are braided transformations. Again, by [Kelly (1974b), Section 10.8], the 2-category BrOpMon is isomorphic to the 2-category Alg_{c}(T) of algebras, colax morphisms and transformations for some 2-monad T on Cat.

2.15. Example. Braided Hopf monads. A monad in the 2-category BrOpMon is exactly a braided Hopf monad.

Following Example 2.15, we suggest the more descriptive term braided opmonoidal monad for braided Hopf monad.

2.16. Proposition. The 2-category BrOpMon admits Eilenberg-Moore objects and the forgetful 2-functor U: BrOpMon → Cat preserves them. □

2.17. Proposition. A braided opmonoidal functor G: D → C is monadic in BrOpMon if and only if G is monadic as an opmonoidal functor. □

Suppose C is a coalgebra in a braided monoidal category D. We say that C is cocommutative if the following diagram commutes.

\xymatrix{ & C \ar[dr]^{δ} \ar[dl]_{δ} \\ C ⊗ C \ar[rr]^{c} &  & C ⊗ C}

Define CocomCoalg(D) to be the full sub-category of Coalg(D) consisting of the cocommutative coalgebras.

2.18. Proposition. Suppose S is a braided opmonoidal monad. Then the categories CocomCoalg(Alg(S)) and Alg(CocomCoalg(S)) are canonically isomorphic.

Proof. This follows immediately from the fact that representable 2-functors preserve Eilenberg-Moore objects and the observation that for any braided monoidal category D the category CocomCoalg(D) is canonically isomorphic to the category BrOpMon(1, D) of opmonoidal functors from the terminal braided monoidal category.

3. Multicategories

In this section we analyze Proposition 1.1. The main tool used in this section will be multicategories and their morphism. The symbol C will always denote a strict monoidal category.

Recall from Example 2.4 that a multicategory is a monad in the bicategory Span_{T} (Set) where T is the cartesian free monoid monad on the category Set of sets. We now make explicit this definition. A multicategory M has a set of objects, and for each sequence ⟨X_{1}, …, X_{n}⟩ of objects and each object X there is a set M(⟨X_{1}, …, X_{n}⟩, X) of arrows. Here n may be 0, and in this case we write ∅ for the sequence ⟨X_{1}, …, X_{n}⟩. We write elements of M(⟨X_{1}, …, X_{n}⟩, X) as f: ⟨X_{1}, …, X_{n}⟩ → X. There is a specified arrow 1: ⟨X⟩ → X, and a composition operation taking an arrow f: ⟨X_{1}, …, X_{n}⟩ → X and a sequence

f_{1} : ⟨X_{11}, …, X_{1m_{1}}⟩ → X_{1}, …, f_{n}: ⟨X_{n1}, …, X_{nm_{n}}⟩ → X_{n}

of arrows, to an arrow

f⟨f_{1}, …, f_{n}⟩: ⟨X_{11}, …, X_{nm_{n}}⟩ → X

subject to axioms expressing that composition is associative and unital.

3.1. Example. Monoidal categories. It is well known that any monoidal category D gives rise to a multicategory M_{D}. The objects of M_{D} are those of D, for a sequence ⟨X_{1}, …, X_{n}⟩ of objects and an object X the set M_{D}(⟨X_{1}, …, X_{n}⟩, X) is the set D(X_{1} ⊗ ⋯ ⊗ X_{n}, X), and the set M_{D}(∅, X) is the set D(I, X). Here, the object X_{1} ⊗ ⋯ ⊗ X_{n} denotes the tensor product of these objects with all the brackets to the left. Of course, other bracketings give rise to canonically isomorphic objects. The compositions and units are evident.

3.2. Example. Categories over C. Let D be any monoidal category, and A a monoid in D. Recall that the slice category D/A is again monoidal. The objects are pairs (X, f: X → A), and an arrow g: (X, f) → (X′, f′) is an arrow g: X → X′ in D such that f′g = f. Given objects (X, f) and (X′, f′) of D/M, define (X, f) ⊗ (X′, f′) to be (X ⊗ X′, μ ∘ (f ⊗ f′)). This is the value at the pair ((X, f), (X′, f′)) of a functor ⊗: D/A × D/A → D/A which makes D/A a monoidal category; the unit object is the pair (I, η: I → M) and the associativity and unit isomorphism are those of D.

Now suppose C is a strict monoidal category. Thus C is a monoid in Cat, and so by the above paragraph there is a monoidal category Cat/C. Let Cat/C also denote the multicategory M_{Cat/C} as described in Example 3.1.

Given a multicategory M, there is a category M_{0} which has the same set of objects as M and for each pair X and Y of objects of M_{0} the homset M_{0}(X, Y) is M(⟨X⟩, Y). Compositions and units are inherited from M. We call M_{0} the underlying category of M. Note that the underlying category of a multicategory of the form M_{D} for a monoidal category D, is the underlying category of D.

3.3. Example. Monads on C. Let C be a strict monoidal category and write ⊗: C^{n} → C for the functor whose value on an n-tuple X_{1}, …, X_{n} of objects of C is the object X_{1} ⊗ ⋯ ⊗ X_{n}. We interpret ⊗: C^{0} → C as the functor I: 1 → C, and we interpret ⊗: C^{1} → C as the identity functor. We shall now define an multicategory Mnd(C) whose underlying category is the usual category Mnd(C). The objects of Mnd(C) are monads on C. For a sequence ⟨T_{1}, …, T_{n}⟩ of monads and a monad S, the set Mnd(⟨T_{1}, …, T_{n}⟩, S) is the set of natural transformations

[figure: C^{n}; ⊗; ∏ T_{i}; C; S; C^{n}; ⊗; C; α]

satisfying the following two equations.

[figure: C^{n}; ∏ T_{i}; ⊗; C; S; 1; C^{n}; ⊗; C; α; η; C^{n}; ∏ T_{i}; ∏ 1; ⊗; C; 1; C^{n}; ⊗; C; ∏ η]

[figure: C^{n}; ∏ T_{i}; ⊗; C; S; S; C; S; C^{n}; ⊗; C; μ; α; C^{n}; ∏ T_{i}; ⊗; C; S; C^{n}; ⊗; C; S; C^{n}; ⊗; C; ∏ μ; α; α]

The unit arrow 1: ⟨S⟩ → S is the identity natural transformation. If α_{1} : ⟨T_{11}, …, T_{1m_{1}}⟩ → S_{1}, …, α_{n}: ⟨T_{n1}, …, T_{nm_{n}}⟩ → S_{n}, and α: ⟨S_{1}, …, S_{n}⟩ → S are arrows then α⟨α_{1}, …, α_{n}⟩ is the following pasted composite.

[figure: C^{n_{1}+···+m_{n}}; ⊗; ∏ ⊗; ∏ T_{ij}; C; S; C^{n}; ⊗; ∏ S_{i}; C^{n}; ⊗; C^{n_{1}+···+m_{n}}; ∏ ⊗; ⊗; C; α; ∏ α_{i}]

One may show that this is an arrow in Mnd(C) and that this composition is associative and unital.

For multicategories M and N, a multicategory morphism F: M → N consists of a function F from the set of objects of M to the set of objects of N along with a family of functions

F: M(⟨X_{1}, …, X_{n}⟩, X) → N(⟨F X_{1}, …, F X_{n}⟩, F X)

which respect the composition and units. Given a multicategory morphism F: M → N there is an evident underlying functor F_{0} : M_{0} → N_{0} between the corresponding underlying categories.

3.4. Example. Semantics. In this example we shall define a multicategory morphism Alg: Mnd(C) → Cat/C whose underlying functor is the usual semantics functor [Street (1972), Section 2]. For an object S of Mnd(C), the object Alg(S) is defined to be the pair (Alg(S), U: Alg(S) → C). Suppose α: ⟨T_{1}, …, T_{n}⟩ → S is an arrow in Mnd(C). It is not difficult to show that the 2-cell

[figure: C^{n}; ⊗; ∏ T_{i}; C; S; ∏ Alg(T_{i}); ∏ U; ∏ U; C^{n}; ⊗; C; α; ∏ μ]

makes

\xymatrix{∏ Alg(T_{i}) \ar[r]^{∏ U} & C^{n} \ar[r]^{⊗} & C}

into an ∏ Alg(T_{i})-based S-algebra. It follows by the universal property of Alg(S) that there is a unique functor Alg(α): ∏ Alg(T_{i}) → Alg(S) such that the above natural transformation is equal to the following diagram.

[figure: C; S; ∏ Alg(T_{i}); Alg(α); Alg(S); U; U; C]

In particular,

\xymatrix{∏ Alg(T_{i}) \ar[r]^{Alg(α)} \ar[d]_{∏ U} & Alg(S) \ar[d]^{U} \\ C^{n} \ar[r]_{⊗} & C}

commutes, so that Alg(α): ⟨Alg(T_{1}), …, Alg(T_{n})⟩ → Alg(S) is a morphism in Cat/C. This assignment respects compositions and units, and so indeed defines a multicategory morphism Alg: Mnd(C) → Cat/C. Clearly the underlying functor of this morphism is the usual semantics functor.

Suppose M is any multicategory. A monoid in M is an object A of M equipped with arrows μ: ⟨A, A⟩ → A and η: ∅ → A, called the multiplication and unit respectively, such that the following three equations hold.

μ⟨1, μ⟩ = μ⟨μ, 1⟩
μ⟨1, η⟩ = 1
μ⟨η, 1⟩ = 1

If A and B are monoids in M, then a monoid morphism from A to B is an arrow f: ⟨A⟩ → B such that f⟨μ⟩ = μ⟨f, f⟩. With the evident compositions there is a category Mon(M) and a forgetful functor U: Mon(M) → M_{0} into the underlying category of M.

3.5. Example. Monoids in a monoidal category. For a monoidal category D, the category of monoids Mon(M_{D}) in the multicategory M_{D} is the usual category of monoids Mon(D) in D and the forgetful functor U: Mon(M_{D}) → (M_{D})_{0} is the usual forgetful functor U: Mon(D) → D.

3.6. Example. Strict monoidal categories over C. By Example 3.5, a monoid in the multicategory Cat/C is exactly a monoid in the monoidal category Cat/C. This is exactly a strict monoidal category A with a strict monoidal functor U: A → C. A monoid morphism is a strict monoidal morphism. When A is of the form Alg(T) for some monad on C this amounts to a lifting of the monoidal structure of C to Alg(T).

3.7. Example. Opmonoidal monads. A monoid in the multicategory Mnd(C) is exactly an opmonoidal monad on C and a monoid morphism is exactly a morphism of opmonoidal monads. In order to see this, observe that for a monad S on C, in order to give a multiplication μ: ⟨S, S⟩ → S and a unit η: ∅ → S is to give natural transformations as in the following diagrams.

[figure: C × C; S×S; ⊗; C; S; D × D; ⊗; D; χ; 1; I; 1; C; S; 1; I; D; ι]

The axioms that express that these natural transformations make S an opmonoidal functor are exactly the associativity and unit axioms for the monoid. The axioms that express that the natural transformations μ: SS ⇒ S and η: 1 ⇒ S are opmonoidal transformations are exactly the axioms for μ: ⟨S, S⟩ → S and η: ∅ → S to be arrows in the multicategory Mnd(C). A similar argument shows that an opmonoidal monad morphism is exactly a monoid morphism.

Suppose F: M → N is a multicategory morphism and A is a monoid in M. Then clearly the object F A of N is canonically a monoid in N, and this is the value at the monoid A of a functor Mon(F): Mon(M) → Mon(N), and the following diagram commutes.

\xymatrix{Mon(M) \ar[r]^{Mon(F)} \ar[d]_{U} & Mon(N) \ar[d]^{U} \\ M_{0} \ar[r]_{F_{0}} & N_{0}}

3.8. Example. Liftings of monoidal structures to Alg(S). By Examples 3.6 and 3.7 and the above comments, we have the following commutative diagram in Cat.

\xymatrix{OpMonMnd(C) \ar[r]^{Alg} \ar[d]_{U} & Mon(Cat/C) \ar[d]^{U} \\ Mnd(C)^{op} \ar[r]_{Alg} & Cat/C}

This implies that given an opmonoidal structure on a monad S, the category Alg(S) inherits a monoidal structure from C. Of course, we already stated this fact in Section 1 and in Proposition 2.9. We are interested in the converse of this statement.

We say that a multicategory morphism F: M → N is fully faithful when the functions

F: M(⟨X_{1}, …, X_{n}⟩, X) → N(⟨F X_{1}, …, F X_{n}⟩, F X)

are isomorphisms. We shall show that fully faithful morphisms are representably fully faithful.

There is a 2-category Mlt of multicategories [Hermida (1999)]. The objects and arrows of this 2-category have been described above, and we now recall the 2-cells. Suppose F, G :

M → N are parallel multicategory morphisms. A transformation α from F to G consists of a family of arrows α_{X}: ⟨F X⟩ → GX such that for all arrows h: ⟨X_{1}, …, X_{n}⟩ → X the following equation holds.

α_{X}⟨F h⟩ = Gh⟨α_{X_{1}}, …, α_{X_{n}}⟩

Recall that an arrow F: K → L in a 2-category K is said to be representably fully faithful when for all objects P of K, the functor

K(P, F): K(P, M) → K(P, N)

is fully faithful.

3.9. Proposition. If a multicategory morphism is fully faithful, then it is representably fully faithful.

Proof. Suppose H, G: P → M are multicategory morphisms and α: F H → F G is a multicategory transformation. Since the function F: M(⟨HX⟩, GX) → N(⟨F HX⟩, F GX) is an isomorphism for all X, there exists a unique arrow β_{X}: ⟨HX⟩ → GX such that F β_{X} = α_{X}. This is the component at X of the unique multicategory transformation β such that F β = α.

We shall now show that monoidally fully faithful functors induce fully faithful morphisms of multicategories. Suppose F: D → D′ is a monoidal functor. For each sequence ⟨X_{1}, …, X_{n}⟩ of objects of D inductively define an arrow ψ: F X_{1} ⊗ ⋯ ⊗ F X_{n} → F (X_{1} ⊗ ⋯ ⊗ X_{n}) as follows. For n = 0, 1, 2, define the arrow ψ to be ι: I → F I and 1: F X_{1} → F X_{1} and χ: F X_{1} ⊗ F X_{2} → F (X_{1} ⊗ X_{2}) respectively. For n > 2, define ψ to be the following composite.

\xymatrix{F X_{1} ⊗ ⋯ ⊗ F X_{n−1} ⊗ F X_{n} \ar[dr]^{ψ⊗1} &  & F (X_{1} ⊗ ⋯ ⊗ X_{n}) \\  & F (X_{1} ⊗ ⋯ ⊗ X_{n−1}) ⊗ F X_{n} \ar@{-->}[ur]^{χ}}

Composition with this arrow and F induces a function

M_{F} : M_{D}(⟨X_{1}, …, X_{n}⟩, X) → M_{D′}(⟨F X_{1}, …, F X_{n}⟩, F X)

making M_{F} a morphism of multicategories. Similarly, If α: F ⇒ G : D → D′ is a monoidal natural transformation, then for all objects X of D, the arrow α: F X → GX in D′ is an arrow α: ⟨F X⟩ → GX in M_{D′} which is the component at X of a multicategory transformation M_{α}: M_{F} ⇒ M_{G}. This assignment preserves compositions and so defines a 2-functor M_{( )} : MonCat → Mlt.

A monoidal functor F: D → D′ is said to be monoidally fully faithful when it is fully faithful, and the functions

D′(χ, F X): D′(F (X ⊗ Y), F Z) → D′(F X ⊗ F Y, F Z)

D′(ι, F X): D′(F I, F Z) → D′(I, F Z)

are isomorphisms for all objects X, Y and Z of D [McCrudden (1999), Section 3.2]. Monoidally fully faithful functors are representably fully faithful [McCrudden (1999), Proposition 3.2.1].

3.10. Proposition. A monoidal functor F: D → D′ is monoidally fully faithful, if and only if the multicategory morphism M_{F} : M_{D} → M_{D′} is fully faithful. □

3.11. Proposition. Suppose a multicategory morphism F: M → N is fully faithful. The the following diagram is a pullback in the category of categories and functors.

\xymatrix{Mon(M) \ar[r]^{Mon(F)} \ar[d]_{U} & Mon(N) \ar[d]^{U} \\ M_{0} \ar[r]_{F_{0}} & N_{0}}

Proof. Suppose P: A → M_{0} and Q: A → Mon(N) are functors such that F_{0}P = UQ. Then for all objects X of A, the object F_{0}P X is equipped with a multiplication μ: ⟨F_{0}P X, F_{0}P X⟩ → F_{0}P X and a unit ∅ → F_{0}P X, making F_{0}P X a monoid in N. Since the functions

F_{0} : M(⟨P X, P X⟩, P X) → N(⟨F_{0}P X, F_{0}P X⟩, F_{0}P X)
F_{0} : M(∅, P X) → N(∅, F_{0}P X)

are isomorphisms, there are arrows μ: ⟨P X, P X⟩ → P X and η: ∅ → P X which are unique with the property that F_{0}μ = μ and F_{0}η = η respectively. This defines the object function of a functor R: A → Mon(M) which is unique with the desired property.

We defer the proof of the following theorem.

3.12. Theorem. The multicategory morphism Alg: Mnd(C) → Cat/C is fully faithful.

3.13. Corollary. There is a bijection between opmonoidal structures on a monad S on C and liftings of the monoidal structure on C to Alg(S).

Proof. This follows immediately from Theorems 3.12 and 3.11, and Example 3.8.

We now extend this bijection to an equivalence between the category of opmonoidal monads and certain Eilenberg-Moore objects. Underlying this is the equivalence of structure and semantics [Street (1972), Section 2]. Let EM(Cat/C) denote the full submulticategory of Cat/C consisting of the Eilenberg-Moore objects. Observe that the tensor product in Cat/C of Eilenberg-Moore objects is not necessarily and Eilenberg-Moore object.

3.14. Theorem. The morphism Alg: Mnd(C) → Cat/C of multicategories factors through the sub-multicategory EM(Cat/C) and the morphism

Alg: Mnd(C) → EM(Cat/C)

of multicategories is an equivalence.

Proof. For an object (A, U) of EM(Cat/C), define a monad Str(A, U) to be the monad generated by U and its left adjoint. Now suppose

G: ⟨(A_{1}, U_{1}), …, (A_{n}, U_{n})⟩ → (A, U)

is an arrow in EM(Cat/C). Let F denote the left adjoint of U and F_{i} denote the left adjoint of U_{i}, and let R and R_{i} denote the monads so induced. Let Str(G) denote the following 2-cell.

[figure: C^{n}; ⊗; C; R; C^{n}; 1; ∏ F_{i}; ∏ R_{i}; ∏ A_{i}; ∏ U_{i}; ∏ U_{i}; G; A; U; U; C^{n}; ⊗; C; ∏ η; μ]

This is an arrow ⟨R_{1}, …, R_{n}⟩ → R in Mnd(C). This assignment preserves composition and so defines a morphism of multicategories Str: EM(Cat/C) → Mnd(C). For any object S of Mnd(C), there is a canonical invertible arrow Str(Alg(S)) → S in Mnd(C) which is the component of an invertible transformation of multicategories ε: StrAlg ⇒ 1. Also, for any object (A, U) of EM(Cat/C), the comparison functor A ⇒ Alg(Str(A, U)) is an arrow in EM(Cat/C) which is the component of an invertible transformation of multicategories η: 1 → AlgStr. Finally, the triangular identities hold, so that Alg has a left adjoint with invertible unit and counit.

It is now straightforward to prove Theorem 3.12 using Theorem 3.14.

Recall that given a morphism of multicategories F: M → N, there is an induced functor Mon(F): Mon(M) → Mon(N). Similarly, any multicategory transformation α: F ⇒ G induces a natural transformation Mon(α): Mon(F) ⇒ Mon(G). This assignment preserves compositions and identities, and so defines a 2-functor Mon: Mlt → Cat. Since 2-functors preserve equivalences, we have the following corollary as an immediate consequence of Theorem 3.14.

3.15. Corollary. Formation of the categories of algebras induces a equivalence between the category OpMonMnd(C) of opmonoidal monads on C and the category Mon(EM(Cat/C)). □

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