Theory and Applications of Categories, Vol. 28, No. 15, 2013, pp. 403–433.

TENSORS, MONADS AND ACTIONS

Dedicated to the memory of Paweł Waszkiewicz

GAVIN J. SEAL

Abstract. We exhibit sufficient conditions for a monoidal monad 𝕋 on a monoidal category C to induce a monoidal structure on the Eilenberg–Moore category C^{𝕋} that represents bimorphisms. The category of actions in C^{𝕋} is then shown to be monadic over the base category C.

0. Introduction

The original motivation for the current work stemmed from the observation that the category C^{M} of actions of a monoid M in a monoidal category (C, ⊗, E) is monadic over its base. When the base monoidal category is itself an Eilenberg–Moore category C^{𝕋}, the composition of the forgetful functors (C^{𝕋})^{M} → C^{𝕋} and C^{𝕋} → C is monadic again in classical examples: the category of R-modules, seen as a category of R-actions in the category Set^{𝔸} ≅ AbGrp of abelian groups, is monadic over Set, and the category of actions of an integral quantale in the category Set^{ℙ} ≅ Sup of sup-semilattices is monadic over Set [15]. These instances suggest the following underlying principle:

The category (C^{𝕋})^{M} of actions of a monoid M in a monoidal Eilenberg–Moore category C^{𝕋} is monadic over C.

In order to define actions in C^{𝕋}, we first need a tensor (− −) on C^{𝕋} that encodes the “bilinear” nature of the action morphism M X → X. The providential structure is provided by a monoidal monad on (C, ⊗, E) that allows the introduction of morphisms in C that are “𝕋-algebra homomorphisms in each variable”, as originally suggested in [11].

Let us say a word on the technical setting we adopted for this work. In [7, 8, 9, 10], Kock presents the fundamentals of symmetric monoidal monads in a context of closed categories. However, closedness does not appear to play an explicit role in the classical construction of the tensor product on AbGrp or Sup. It also seems reasonable to aim for an action morphism that occurs as an algebraic structure M ⊗ A → A on A, rather than as a morphism of monoids M → [A, A] (where [−, −] would designate an internal hom). Hence, we chose to follow [4] and consider a base category whose monoidal structure is neither assumed to be symmetric, nor closed. Alas, the result we needed in op.cit. is presented with a somewhat obscure proof (in particular, the proposed construction of the unit isomorphism in C^{𝕋} seems a bit brusque). This lead us to our current version of Theorem 2.5.5 that, in turn, provided the necessary ingredients to prove the sought result in Theorem 3.3.3. We note that the hypotheses of Theorem 2.5.5 involve conditions on the tensor (− −) that might not be practical to verify, as the latter is realized via a coequalizer. To remedy this, Theorem 2.6.4 proposes hypotheses that can be tested directly on the base category in the non-closed case (to be compared with Corollary 2.5.7).

Received by the editors 2012-05-01 and, in revised form, 2013-06-28.

Transmitted by Tom Leinster. Published on 2013-07-02.

2010 Mathematics Subject Classification: 18C20, 18D10, 18D35.

Key words and phrases: monoidal category, monad, Eilenberg–Moore category, bimorphism, action.

\begin{matrix}c \\ ◯ \end{matrix} Gavin J. Seal, 2013. Permission to copy for private use granted.

Our work is thus structured as follows. In Section 1, we establish the relevant definitions and notations pertaining to monoidal monads. We also recall that these are fundamentally linked to monoidal structures of Kleisli categories. In Section 2, we recall how bimorphisms and the tensor on C^{𝕋} induced by a monoidal monad 𝕋 are related. Proposition 2.2.2 then exhibits the link between the tensor proposed in [11] with the one studied in [4]. We also review some useful facts about reflexive coequalizers, and recall in 2.5.3 that the tensor in C^{𝕋} of free algebras is the free algebra of their tensor in C (see [4, Proposition 21]). This observation is crucial to establish that the tensor on C^{𝕋} is associative, a fact that is proved in Theorem 2.5.5, and again in Theorem 2.6.4 with alternative hypotheses. We then consider algebraic functors C^{𝕋} → C^{𝕊} induced by monoidal monad morphisms 𝕊 → 𝕋, and show that they are themselves monoidal. Once C^{𝕋} is equipped with the adequate monoidal structure, we turn our attention to actions in Section 3. Our main result is, as mentioned above, that the monadic functors (C^{𝕋})^{M} → C^{𝕋} and C^{𝕋} → C compose to form yet another monadic functor (C^{𝕋})^{M} → C. We conclude by showing that the classical restriction-of-scalars functor between categories of modules is algebraic.

Throughout the text, we illustrate the various notions introduced with the classical examples mentioned above, that is, with structures related to the free abelian group and the powerset monads. Example 2.6.5 also demonstrates that binary coproducts in C^{𝕋} can be interpreted as the tensor induced by binary coproducts in C. Of course, these examples are far from being exhaustive, but we feel that they adequately represent the concepts developed, while hinting at further applications.

1. Basic structures

1.1. Monoidal categories. Let C be a monoidal category, with its tensor denoted by (− ⊗ −) : C × C → C, its unit by E, and its structure natural isomorphisms by

α_{X,Y,Z} : (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z) , λ_{X} : E ⊗ X → X , ρ_{X} : X ⊗ E → X .

When the monoidal category C is symmetric, we denote by σ_{X,Y} : X ⊗ Y → X ⊗ Y the components of its braiding natural isomorphisms (see [13] or [3]). We will denote a monoidal category (C, ⊗, α, E, λ, ρ) or (C, ⊗, α, E, λ, ρ, σ) more briefly by (C, ⊗, E).

1.1.1. Example. The monoidal category recurrent in most of our examples is the category (Set, ×, {⋆}) of sets and maps with its cartesian structure.

1.2. Monoidal monads. Let (C, ⊗, E) be a monoidal category. A monoidal monad 𝕋 = (T, μ, η) is a monad in the 2-category of monoidal categories, monoidal functors, and monoidal transformations; that is, the functor T : C → C comes with a family of maps κ = (κ_{X,Y} : T X ⊗ T Y → T (X ⊗ Y))_{X,Y ∈ob C}, natural in X and Y, that make (T, κ, η_{E}) a monoidal functor, and μ, η monoidal natural transformations (see [13, Section XI.2]).

Similarly, a symmetric monoidal monad 𝕋 = (T, μ, η) is a monad in the 2-category of symmetric monoidal categories, symmetric monoidal functors, and monoidal transformations.

1.2.1. Examples.

(1) The identity monad 𝕀 = (1_{C}, 1, 1) on a monoidal category C is a monoidal monad via the identity natural transformation (1_{X⊗Y} : X ⊗ Y → X ⊗ Y)_{X,Y ∈ob C}. The monoidal monad is symmetric whenever C is.

(2) The free abelian group monad 𝔸 = (A, ∑, (−)) on (Set, ×, {⋆}) is a symmetric monoidal monad via the natural transformation κ whose component κ_{X,Y} : AX × AY → A(X × Y) sends a pair ( ∑_{x∈X} n_{x} · x, ∑_{y∈Y} n_{y} · y) (with all coefficients n_{i} integers) to the element ∑_{x∈X,y∈Y} (n_{x} + n_{y}) · (x, y).

(3) The powerset monad ℙ = (P, ⋃, {−}) on Set (with its cartesian structure) is a symmetric monoidal monad via the natural transformation ι whose component ι_{X,Y} : P X×P Y → P (X×Y) sends a pair of subsets (A, B) to their product A×B ⊆ X×Y.

(4) Any monad 𝕋 on a category C whose monoidal structure is given by finite coproducts (so ⊗ = + and E = ∅ is the initial object in C) is monoidal with respect to the connecting C-morphisms κ_{A,B} : T A + T B → T (A + B).

Monoidal monads correspond to monoidal structures of the Kleisli category, as follows.

1.2.2. Proposition. Given a monad 𝕋 on a monoidal category C, there is a bijective correspondence between the following data:

(1) families κ = (κ_{X,Y} : T X ⊗ T Y → T (X ⊗ Y))_{X,Y ∈ob C} of C-morphisms natural in X and Y making 𝕋 a monoidal monad;

(2) monoidal structures on the Kleisli category C_{𝕋} such that the left adjoint functor F_{𝕋} : C → C_{𝕋} is strict monoidal.

Moreover, (𝕋, κ) is symmetric precisely when the corresponding monoidal structure on C_{𝕋} is symmetric.

Proof. Given a C-morphism κ_{Z,W}, and C-morphisms f : X → T Z, g : Y → T W, one can define the C-morphism f ⊗_{𝕋} g := κ_{Z,W} · (f ⊗ g) : X ⊗ Y → T (Z ⊗ W), thus equipping C_{𝕋} with a tensor ⊗_{𝕋} for which F_{𝕋} is strict monoidal. Conversely, if C_{𝕋} is monoidal with a tensor ⊗_{𝕋}, then strict monoidality of F_{𝕋} forces equality X ⊗_{𝕋} Y = X ⊗ Y for all C-objects X and Y, and one can define κ_{X,Y} := 1_{T X} ⊗_{𝕋} 1_{T Y}. (See for example [4, Proposition 8].)

1.3. Monoidal monad morphisms. Let (𝕊, ι) and (𝕋, κ) be monoidal monads on a monoidal category (C, ⊗, E). A monad morphism ϕ : 𝕊 → 𝕋 is monoidal if it is a monoidal natural transformation.

1.3.1. Example. For any monoidal monad (𝕋, κ) on (C, ⊗, E), the unit η : (𝕀, 1) → (𝕋, κ) is a monoidal monad morphism.

The correspondence of Proposition 1.2.2 extends to morphisms.

1.3.2. Proposition. Given a monad 𝕋 on a monoidal category C, there is a bijective correspondence between the following data:

(1) monoidal monad morphisms ϕ : 𝕊 → 𝕋;

(2) strict monoidal functors L : C_{𝕊} → C_{𝕋} that commute with the left adjoint functors

from C:

\xymatrix{C_{𝕊} \ar[rr]^{L} &  & C_{𝕋} \\  & C . \ar[ul]^{F_{𝕊}} \ar[ur]_{F_{𝕋}}}

Proof. The one-to-one correspondence between monad morphisms and functors between the Kleisli categories is standard (see for example [14, Theorem 2.2.2]): a monad morphism ϕ : 𝕊 → 𝕋 defines a functor L : C_{𝕊} → C_{𝕋} that is the identity on objects and sends a C_{𝕊}- morphism f : X → SY to the C_{𝕋}-morphism ϕ_{Y} ·f : X → T Y; conversely, a functor L as in (2) defines a monad morphism ϕ : 𝕊 → 𝕋 via its components ϕ_{X} := L(1_{SX}) : SX → T Y. One easily verifies that if ϕ is monoidal, then L is strict monoidal, and that the converse holds, too.

2. The monoidal structure of C^{𝕋}

The prototypical tensor product that we wish to study is provided by the tensor product of R-modules. The role of this tensor is to represent bilinear maps. In our setting, the monoidal structure of the monad allows the introduction of the notion of such a “morphism in each variable” (as suggested in [11] and [8]).

2.1. Bimorphisms. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E), and denote by C^{𝕋} its category of Eilenberg–Moore algebras. For 𝕋-algebras (A, a), (B, b), and (C, c), we say that a C-morphism f : A ⊗ B → C is a C^{𝕋}-bimorphism^{1} (or simply a bimorphism) if the diagram

\xymatrix{T A ⊗ T B \ar[d]_{a⊗b} \ar[r]^{κ} & T (A ⊗ B) \ar[r]^{T f} & T C \ar[d]^{c} \\ A ⊗ B \ar[rr]^{f} &  & C}  (2.1.i)

1These bimorphisms should not be confused with the C-morphisms that are at the same time monic and epic. Since we do not consider the latter, there is little reason to avoid the bimorphism terminology.

commutes. The set of all bimorphisms f : (A, a) ⊗ (B, b) → (C, c) is denoted by C^{𝕋}(A, B; C).

2.1.1. Examples.

(1) For the identity monad 𝕀 = (1_{C}, 1, 1) on a monoidal category C, with the identity natural transformation, C^{𝕀}-bimorphisms are just C-morphisms since C^{𝕀} ≅ C.

(2) For the free abelian group monad 𝔸, the category Set^{𝔸} is isomorphic to AbGrp, the category of abelian groups. Via the natural transformation κ of Example 1.2.1(2), a Set^{𝔸}-bimorphism f : A × B → C is a map that is additive in each variable.

(3) For the powerset monad ℙ on Set with the natural transformation ι of Example 1.2.1(3), the Eilenberg–Moore category Set^{ℙ} is isomorphic to Sup, the category of complete sup-semilattices. With this interpretation, a Set^{ℙ}-bimorphism f : A × B → C is a map that preserves suprema in each variable.

The next result shows in what sense a bimorphism captures the idea of a “morphism in each variable”.

2.1.2. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E). For 𝕋-algebras (A, a), (B, b), and (C, c), a C-morphism f : A⊗B → C is a C^{𝕋}-bimorphism if and only if both diagrams

[figure: A ⊗ T B; 1⊗b; κ·(η⊗1); T (A ⊗ B); T f; T C; c; A ⊗ B; f; C; and; T A ⊗ B; a⊗1; κ·(1⊗η); T (A ⊗ B); T f; T C; c; A ⊗ B; f; C]

commute.

Proof. If f is a C^{𝕋}-bimorphism, then one can append each of the following commutative diagrams

[figure: A ⊗ T B; η⊗1; 1⊗b; T A ⊗ T B; a⊗b; A ⊗ B; and; T A ⊗ B; 1⊗η; a⊗1; T A ⊗ T B; a⊗b; A ⊗ B]

to the left of (2.1.i) to obtain the respective diagrams of the statement. Conversely, if the two diagrams of the statement commute, then

f · (a ⊗ b) = f · (1_{A} ⊗ b) · (a ⊗ 1_{T B})
= c · T f · κ_{A,B} · (η_{A} ⊗ 1_{T B}) · (a ⊗ 1_{T B})
= c · T f · κ_{A,B} · (T a ⊗ T 1_{B}) · (η_{T A} ⊗ 1_{T B})
= c · T (f · (a ⊗ 1_{B})) · κ_{T A,B} · (η_{T A} ⊗ 1_{T B})
= c · T (c · T f · κ_{A,B} · (1_{T A} ⊗ η_{B})) · κ_{T A,B} · (η_{T A} ⊗ 1_{T B})
= c · T f · μ_{A⊗B} · T κ_{A,B} · T (1_{T A} ⊗ η_{B}) · κ_{T A,B} · (η_{T A} ⊗ 1_{T B})
= c · T f · μ_{A⊗B} · T κ_{A,B} · κ_{T A,T B} · (T 1_{T A} ⊗ T η_{B}) · (η_{T A} ⊗ 1_{T B})
= c · T f · κ_{A,B} · (μ_{A} ⊗ μ_{B}) · (η_{T A} ⊗ T η_{B}) = c · T f · κ_{A,B} .

This shows commutativity of (2.1.i).

2.1.3. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E). Then

C^{𝕋}(−, −; −) : (C^{𝕋} × C^{𝕋})^{op} × C^{𝕋} → Set

is a functor defined by

C^{𝕋}(g, h; k)(f) := k · f · (g ⊗ h)

for all C^{𝕋}-morphisms g : A′ → A, h : B′ → B, k : C → C′, and bimorphisms f : A⊗B → C.

Proof. The given functor is well-defined. Indeed, for C-morphisms f, g, h, k as in the claim (and 𝕋-algebras (A, a), (B, b), (C, c), (A′, a′), (B′, b′), (C′, c′)), one has

c′ · T (k · f · (g ⊗ h)) · κ_{A′,B′} = k · c · T f · κ_{A,B} · (T g ⊗ T h)

= k · f · (a ⊗ b) · (T g ⊗ T h) = k · f · (g ⊗ h) · (a′ ⊗ b′) ,

so that C^{𝕋}(g, h; k)f is indeed in C^{𝕋}(A′, B′; C′). Functoriality is immediate by monoidality of C.

2.2. Tensor products. Consider a monoidal monad (𝕋, κ) on a monoidal category (C, ⊗, E). The tensor product q_{A,B} : T (A ⊗ B) → (A B, a ⋊ ⋉ b) of 𝕋-algebras (A, a) and (B, b) is given by the following coequalizer diagram

\xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T κ} \ar[r]_{T (a⊗b)} & T (A ⊗ B) \ar[r]^{q} & A B}

in C^{𝕋}. We often assume that q_{A,B} is implicit and speak of the tensor product (A B, a ▷◁ b), or simply A B. For C^{𝕋}-morphisms g : (A, a) → (A′, a′), h : (B, b) → (B′, b′), one has

q_{A′,B′} · T (g ⊗ h) · μ_{A⊗B} · T κ_{A,B} = q_{A′,B′} · μ_{A′,B′} · T T (g ⊗ h) · T κ_{A,B}
= q_{A′,B′} · μ_{A′,B′} · T κ_{A′,B′} · T (T g ⊗ T h)
= q_{A′,B′} · T (a′ ⊗ b′) · T (T g ⊗ T h)
= q_{A′,B′} · T (g ⊗ h) · T (a ⊗ b) ,

so there is a unique C^{𝕋}-morphism g h : A B → A′ B′ with (g h)·q_{A,B} = q_{A′,B′}·T (g⊗h), and the diagram

\xymatrix{T (A ⊗ B) \ar[r]^{q} \ar[d]_{T (g⊗h)} & A B \ar[d]^{g h} \\ T (A′ ⊗ B′) \ar[r]^{q} & A′ B′}

commutes.

We say that C^{𝕋} has tensors when q_{A,B} exists for all 𝕋-algebras A and B. In particular, if C^{𝕋} has reflexive coequalizers, then it has tensors. For C^{𝕋} to moreover be a monoidal category, we also need associativity and unit isomorphisms in C^{𝕋} that make the corresponding coherence diagrams commute. This is the subject of Theorem 2.5.5.

2.2.1. Convention. For the sake of convenience, from now on we say that a diagram of the form

\xymatrix{X \ar[r]^{g} \ar[r]_{h} & Y \ar[r]^{f} & Z}

commutes if f · g = f · h.

2.2.2. Proposition. Given a monoidal monad (𝕋, κ) on a monoidal category (C, ⊗, E) and 𝕋-algebras (A, a), (B, b), the following statements are equivalent for all C-morphisms f : T (A ⊗ B) → C:

(i) the diagram \xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T κ} \ar[r]_{T (a⊗b)} & T (A ⊗ B) \ar[r]^{f} & C} commutes;

(ii) the diagram \xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T (κ·(η·a⊗1))} \ar[r]_{μ·T (κ·(1⊗η·b))} & T (A ⊗ B) \ar[r]^{f} & C} commutes.

Proof. Suppose that f · μ_{A⊗B} · T κ_{A,B} = f · T (a ⊗ b). Then one immediately obtains

f · μ_{A⊗B} · T (κ_{A,B} · (η_{A} · a ⊗ 1_{T B})) = f · T (a ⊗ b) = f · μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B} · b)) .

Conversely, suppose that f ·μ_{A⊗B} ·T (κ_{A,B} ·(η_{A} ·a⊗1_{T B})) = f ·μ_{A⊗B} ·T (κ_{A,B} ·(1_{T A} ⊗η_{B} ·b)) holds. One first notes

f · T (a ⊗ b) = f · μ_{A⊗B} · T η_{A⊗B} · T (a ⊗ b)
= f · μ_{A⊗B} · T (κ_{A,B} · (η_{A} ⊗ η_{B}) · (a ⊗ b))
= f · μ_{A⊗B} · T (κ_{A,B} · (η_{A} · a ⊗ 1_{T B}) · (1_{T A} ⊗ η_{B} · b))
= f · μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B} · b) · (1_{T A} ⊗ η_{B} · b))
= f · μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B} · b)) ,

so

f · T (a ⊗ 1_{B}) = f · μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B})) and
f · T (a ⊗ b) = f · μ_{A⊗B} · T (κ_{A,B} · (η_{A} · a ⊗ 1_{T B})) .

These equalities, together with the fact that

κ_{A,B} = κ_{A,B} · (μ_{A} ⊗ μ_{B}) · (η_{T A} ⊗ T η_{B})
= μ_{A⊗B} · T κ_{A,B} · κ_{T A,T B} · (1_{T T A} ⊗ T η_{B}) · (η_{T A} ⊗ 1_{T B})
= μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B})) · κ_{T A,B} · (η_{T A} ⊗ 1_{T B})

yield

f · μ_{A⊗B} · T κ_{A,B} = f · μ_{A⊗B} · T (μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B})) · κ_{T A,B} · (η_{T A} ⊗ 1_{T B}))
= f · μ_{A⊗B} · T (κ_{A,B} · (1_{T A} ⊗ η_{B})) · μ_{T A⊗B} · T (κ_{T A,B} · (η_{T A} ⊗ 1_{T B}))
= f · T (a ⊗ 1_{B}) · μ_{T A⊗B} · T (κ_{T A,B} · (η_{T A} ⊗ 1_{T B}))
= f · μ_{A⊗B} · T (κ_{A,B} · (η_{A} · a ⊗ 1_{T B})) = f · T (a ⊗ b) ,

as required.

2.2.3. Corollary. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors. Then for 𝕋-algebras (A, a) and (B, b), the diagram

\xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T (κ·(η·a⊗1))} \ar[r]_{μ·T (κ·(1⊗η·b))} & T (A ⊗ B) \ar[r]^{q} & A B}

is a coequalizer in C^{𝕋} (with q_{A,B} as defined in 2.2).

Proof. This is an immediate consequence of the previous Proposition.

2.2.4. Remarks.

(1) Corollary 2.2.3 confirms that the coequalizer suggested to define a tensor in [11, Remark, Section 1] is the same as the one appearing in [4, Proposition 16].

(2) As remarked by Linton [12], the study of bimorphisms trivializes for certain monads. For example, suppose that 𝕋 is a monoidal monad on (Set, ×, {⋆}) whose 𝕋-algebras X have two nullary operations 0, 1 : {⋆} → X and a binary one (−)∗(−) : X ×X → X such that

0 ∗ x = 0 and 1 ∗ x = x

for all x ∈ X. If f : A × B → C is a bimorphism, then

f(a, 1) = 1 and f(0, b) = 0

for all a ∈ A, b ∈ B, so that 1 = f(0, 1) = 0. Hence, if c ∈ C, then c = 1 ∗ c = 0 ∗ c = 0. That is, the only bimorphisms f : A × B → C are those for which C is a singleton. (See also Remark 2.3.5.) This observation applies in particular to bimorphisms in the category of semirings.

2.3. Representing bimorphisms. A major motivation to the introduction of tensor products in categories of R-modules is the representation of bimorphisms. Here, we show that the tensor product of 2.2 plays that role with respect to the bimorphisms of 2.1.

2.3.1. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors. Then

C^{𝕋}(− −, −) : (C^{𝕋} × C^{𝕋})^{op} × C^{𝕋} → Set

is a functor.

Proof. Immediate by functoriality of (− ⊗ −) and T (see 2.2 and Proposition 2.1.3).

The relation between bimorphisms and the tensor product is the subject of the following results.

2.3.2. Lemma. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E). For 𝕋algebras (A, a), (B, b), (C, c), the following statements are equivalent for any C-morphism f : A ⊗ B → C:

(i) f is a bimorphism;

(ii) The diagram \xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T κ} \ar[r]_{T (a⊗b)} & T (A ⊗ B) \ar[r]^{c·T f} & C} commutes.

Proof. If f is a bimorphism, then

c · T f · μ_{A⊗B} · T κ_{A,B} = c · T c · T T f · T κ_{A,B} = c · T f · T (a ⊗ b) .

Conversely, if the diagram in (ii) commutes, then

c · T f · κ_{A,B} = c · T f · μ_{A⊗B} · η_{T (A⊗B)} · κ_{A,B} = c · T f · T (a ⊗ b) · η_{T A⊗T B} = f · (a ⊗ b) ,

so f is a bimorphism.

2.3.3. Lemma. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors. For 𝕋-algebras (A, a), (B, b), and (C, c), the following statements hold.

(1) If f : A⊗B → C is a bimorphism, then there is a unique C^{𝕋}-morphism \overline{f} : A B → C

such that

\xymatrix{T (A ⊗ B) \ar[d]_{q} \ar[r]^{T f} & T C \ar[d]^{c} \\ A B \ar[r]^{\overline{f}} & C}

commutes.

(2) If g : A B → C is a C^{𝕋}-morphism, then g·q_{A,B} ·η_{A⊗B} : A⊗B → C is a bimorphism that induces g, that is, \overline{g · q_{A,B} · η_{A⊗B}} = g.

Proof. Since c · T f is a C^{𝕋}-morphism, the first claim follows directly from Lemma 2.3.2 and the universal property of the coequalizer q_{A,B}.

Given a C^{𝕋}-morphism g : A B → C, one uses that q_{A,B} is a coequalizer and that g · q_{A,B} : (T (A ⊗ B), μ_{A⊗B}) → (C, c) is a C^{𝕋}-morphism to write

g · q_{A,B} · η_{A⊗B} · (a ⊗ b) = g · q_{A,B} · T (a ⊗ b) · η_{T A⊗T B}
= g · q_{A,B} · μ_{A⊗B} · T κ_{A,B} · η_{T A⊗T B}
= g · q_{A,B} · μ_{A⊗B} · η_{T (A⊗B)} · κ_{A,B}
= g · q_{A,B} · μ_{A⊗B} · T η_{A⊗B} · κ_{A,B}

= c · T (g · q_{A,B}) · T η_{A⊗B} · κ_{A,B} = c · T (g · q_{A,B} · η_{A⊗B}) · κ_{A,B} ,

showing that g · q_{A,B} · η_{A⊗B} is a bimorphism. Since

c · T (g · q_{A,B} · η_{A⊗B}) = g · q_{A,B} · μ_{A⊗B} · T η_{A⊗B} = g · q_{A,B} ,

the C^{𝕋}-morphism induced by g · q_{A,B} · η_{A⊗B} is indeed g.

2.3.4. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors. For 𝕋-algebras (A, a), (B, b), and (C, c), there is a bijection

C^{𝕋}(A, B; C) ≅ C^{𝕋}(A B, C)

natural in each variable.

Proof. The required bijection is described in Lemma 2.3.3. Indeed, if g : A B → C is a C^{𝕋}-morphism, then f = g · q_{A,B} · η_{A⊗B} is a bimorphism, to which corresponds the unique C^{𝕋}-morphism g : A B → C. Conversely, if f : A ⊗ B → C is a bimorphism, then there is a C^{𝕋}-morphism \overline{f} : A B → C such that c · T f = \overline{f} · q_{A,B}; according to Lemma 2.3.3, one obtains in return a bimorphism

g = \overline{f} · q_{A,B} · η_{A⊗B} = c · T f · η_{A⊗B} = c · η_{C} · f = f .

For a C^{𝕋}-bimorphism f : A ⊗ B → C, and C^{𝕋}-morphisms g : A′ → A, h : B′ → B, k : C → C′, one has

k · \overline{f} · (g h) · q_{A′,B′} = k · \overline{f} · q_{A,B} · T (g ⊗ h) = k · c · T f · T (g ⊗ h) = c′ · T k · T f · T (g ⊗ h) ,

that is,

\overline{k · f · (g ⊗ h)} = k · \overline{f} · (g h)

(by unicity of the induced C^{𝕋}-morphism). Hence, the bijection is natural.

2.3.5. Remark. In the trivial cases where the only bimorphisms are C-morphisms f : A ⊗ B → I into the terminal object of C (as in Remark 2.2.4(2)), Proposition 2.3.4 shows that A B ≅ I: the identity C^{𝕋}-morphism 1_{A B} corresponds to a bimorphism f : A ⊗ B → A B, so that A B ≅ I.

2.4. Reflexive coequalizers. We recall here some basic results pertaining to reflexive coequalizers, and thus applying to the coequalizer q defined in 2.2.

2.4.1. Proposition. A functor F : X × Y → Z preserves reflexive coequalizers if and only if F (X, −) : Y → Z and F (−, Y) : X → Z preserves reflexive coequalizers for all X ∈ ob X, Y ∈ ob Y.

Proof. The necessity of the statement is immediate since F (X, g) = F (1_{X}, g) and F (f, Y) = F (f, 1_{Y}) for all X ∈ ob X, Y ∈ ob Y. For the sufficiency, see the dual of [5, Corollary 1.2.12].

Since we wish to study the tensor products A B defined via the reflexive C^{𝕋}- coequalizer q_{A,B} : T (A ⊗ B) → A B, we recall below two results pertaining to the existence of such colimits.

2.4.2. Proposition. Let 𝕋 be a monad on C. If C has reflexive coequalizers and T preserves them, then the forgetful functor C^{𝕋} → C creates them.

Proof. See [11, Proposition 3].

The next existence result is not used explicitly in this work, but it can prove helpful for the verification of hypotheses in specific examples.

2.4.3. Proposition. For a regular monad 𝕋 on C (see [1, Definition 20.21]), C^{𝕋} has all colimits that exist in C.

Proof. See [1, Proposition 20.33].

2.5. The monoidal Eilenberg–Moore category. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E). Theorem 2.5.5 presents sufficient conditions for the C^{𝕋}-morphisms

q_{A,B} : T (A ⊗ B) → A B

to induce a monoidal structure on C^{𝕋}. The proof of this result relies in part on the explicit description of the coequalizer q_{T A,T B} : T (T A ⊗ T B) → T A T B defined in 2.5.3.

2.5.1. Lemma. Let 𝕋 be a monad on a category C. For C-morphisms r : Z → T Y, s : Z → Y, and p : T Y → X,

(p · μ_{Y} · T r = p · T s) ⟹ (p · r = p · η_{Y} · s) .

Proof. One simply has

p · r = p · μ_{Y} · η_{T Y} · r = p · μ_{Y} · T r · η_{Z} = p · T s · η_{Z} = p · η_{Y} · s .

2.5.2. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E). The following statements hold:

(1) the diagram

\xymatrix{T (T T A ⊗ T T B) \ar[r]^{μ·T κ} \ar[r]_{T (μ⊗μ)} & T (T A ⊗ T B) \ar[r]^{μ·T κ} & T (A ⊗ B)}

forms a coequalizer in C^{𝕋};

(2) if f : A → A′ and g : B → B′ are C-morphisms, then T (f ⊗ g) : T (A ⊗ B) → T (A′ ⊗ B′) is the unique C^{𝕋}-morphism that makes the diagram

\xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T κ} \ar[d]_{T (T f⊗T g)} & T (A ⊗ B) \ar[d]^{T (f⊗g)} \\ T (T A′ ⊗ T B′) \ar[r]^{μ·T κ} & T (A′ ⊗ B′)}

commute.

Proof. By naturality of μ and the fact that μ · μT = μ · T μ, one has

μ_{A⊗B} · T κ_{A,B} · μ_{T A⊗T B} · T κ_{T A,T B} = μ_{A⊗B} · T μ_{A⊗B} · T T κ_{A,B} · T κ_{T A,T B}
= μ_{A⊗B} · T κ_{A,B} · T (μ_{A} ⊗ μ_{B}) ,

so the diagram in the statement commutes. Suppose that f : (T (T A ⊗ T B), μ_{T A⊗T B}) → (C, c) is a C^{𝕋}-morphism that makes

\xymatrix{T (T T A ⊗ T T B) \ar[r]^{μ·T κ} \ar[r]_{T (μ⊗μ)} & T (T A ⊗ T B) \ar[r]^{f} & C}

commute. Setting \overline{f} := f · T (η_{A} ⊗ η_{B}), one has

\overline{f} · μ_{A⊗B} · T κ_{A,B} = f · μ_{A⊗B} · T κ_{A,B} · T (T η_{A} ⊗ T η_{B})
= c · T f · T κ_{A,B} · T (T η_{A} ⊗ T η_{B})
= c · T f · T η_{T A⊗T B} = f

by using the equality f ·κ_{T A,T B} = f ·η_{T A⊗T B} ·(μ_{A}⊗μ_{B}) following from Lemma 2.5.1. Since μ_{A⊗B} · T κ_{A,B} is an epimorphism, the comparison C^{𝕋}-morphism \overline{f} is uniquely determined.

The diagram in the concluding statement commutes by naturality of μ and κ, and the unicity of the induced map follows from the fact that μ_{A⊗B} · T κ_{A,B} is epic.

2.5.3. Identifying coequalizers. In view of Proposition 2.5.2, we can set

q_{T A,T B} := μ_{A⊗B} · T κ_{A,B} and T A T B := T (A ⊗ B)

for all A, B ∈ ob C. Hence, if f : T A → T A′ and g : T B → T B′ are C-morphisms, then

T f T g = T (f ⊗ g) : T A T B → T A′ T B′ .

If moreover the coequalizer q_{A,B} : T (A ⊗ B) → A B exists in C^{𝕋}, it follows from commutativity of

\xymatrix{T (T A ⊗ T B) \ar[r]^{T (a⊗b)} \ar[d]_{μ·T κ} & T (A ⊗ B) \ar[d]^{q} \\ T (A ⊗ B) \ar[r]^{q} & A B}

that

q_{A,B} = a b .

These identifications will be used without necessarily further mention in the proof of Theorem 2.5.5.

2.5.4. Remark. The following result states that, under certain hypotheses, (C^{𝕋}, , T E) becomes a monoidal category with associativity and unit structure isomorphisms induced by those of (C, ⊗, E). The meaning of this term is made clear directly in the proof in an attempt to avoid a rather cumbersome direct definition.

2.5.5. Theorem. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that all the coequalizers q_{A,B} exist in C^{𝕋}. If

\xymatrix{T (T A ⊗ T B) T T C \ar[r]^{(μ·T κ) μ} \ar[r]_{T (a⊗b) T c} & T (A ⊗ B) T C \ar[r]^{q c} & (A B) C}

\xymatrix{T T A T (T B ⊗ T C) \ar[r]^{μ (μ·T κ)} \ar[r]_{T a T (b⊗c)} & T A T (B ⊗ C) \ar[r]^{a q} & A (B C)}

are coequalizer diagrams and q_{A,B} q_{C,D} are epimorphisms (for all 𝕋-algebras (A, a), (B, b), (C, c) and (D, d)), then (C^{𝕋}, , T E) is a monoidal category with structure morphisms induced by those of (C, ⊗, E).

Moreover, if (C, ⊗, E) and (𝕋, κ) are symmetric monoidal, then so is (C^{𝕋}, , T E).

Proof. In this proof, we consider 𝕋-algebras (A, a), (B, b), (C, c), (D, d), as well as (A′, a′), (B′, b′), (C′, c′).

Associativity. By 2.5.3 and the definitions of α and κ, both the inner and outer squares in

\xymatrix{T ((T A ⊗ T B) ⊗ T C) \ar[d]_{T α} \ar[r]^{(μ·T κ) μ} \ar[r]_{T (a⊗b) T c} & T ((A ⊗ B) ⊗ C) \ar[d]^{T α} \\ T (T A ⊗ (T B ⊗ T C)) \ar[r]_{(μ·T κ) μ} \ar[r]^{T a T (b⊗c)} & T (A ⊗ (B ⊗ C))}

commute in C^{𝕋}. By hypothesis, q_{A,B} c and a q_{B,C} are coequalizers, so there is a C^{𝕋}-isomorphism \overline{α}_{A,B,C} : (A B) C → A (B C) induced by T α:

\xymatrix{T ((A ⊗ B) ⊗ C) \ar[r]^{q c} \ar[d]_{T α} & (A B) C \ar[d]^{\overline{α}} \\ T (A ⊗ (B ⊗ C)) \ar[r]^{a q} & A (B C) .}

Note that for C^{𝕋}-morphisms f : (A, a) → (A′, a′), g : (B, b) → (B′, b,), h : (C, c) → (C′, c′), the diagram

\xymatrix{T ((A ⊗ B) ⊗ C) \ar[r]^{q c} \ar[d]_{T ((f⊗g)⊗h)} & (A B) C \ar[d]^{(f g) h} \\ T ((A′ ⊗ B′) ⊗ C′) \ar[r]^{q c′} & (A′ B′) C′ ;}

commutes, and there is a similar diagram for f (g h). Naturality of T α, commutativity of the diagrams for (f g) h, f (g h) and \overline{α}_{A,B,C} together with the fact that q_{A,B} c is epic yield naturality of \overline{α}:

\xymatrix{(A B) C \ar[d]_{(f g) h} \ar[r]^{\overline{α}} & A (B C) \ar[d]^{f (g h)} \\ (A′ B′) C′ \ar[r]^{\overline{α}} & A′ (B′ C′) .}

Commutativity of the coherence diagram

\xymatrix{((A B) C) D \ar[r]^{\overline{α}} \ar[d]_{\overline{α} 1} & (A B) (C D) \ar[r]^{\overline{α}} & A (B (C D)) \\ (A (B C)) D \ar[rr]^{\overline{α}} &  & A ((B C) D) \ar[u]_{1 \overline{α}}}

follows from commutativity of the coherence diagram of T (((A ⊗ B) ⊗ C) ⊗ D), the definition and naturality of \overline{α}, the fact that q_{A,B} = a b, and the hypothesis that (q_{A,B} c) d ≅ q_{A,B} q_{C,D} is epic.

Unitariness. For a 𝕋-algebra (A, a), the composite C-morphism

\xymatrix{T E ⊗ A \ar[r]^{1⊗η} & T E ⊗ T A \ar[r]^{κ} & T (E ⊗ A) \ar[r]^{T λ} & T A \ar[r]^{a} & A}

is a bimorphism: on one hand, we have

a · T (a · T λ_{A} · κ_{E,A} · (1_{T E} ⊗ η_{A})) · κ_{T E,A}
= a · T λ_{A} · μ_{E⊗A} · T κ_{E,A} · κ_{T E,T A} · (1_{T T E} ⊗ T η_{A})
= a · T λ_{A} · κ_{E,A} · (μ_{E} ⊗ μ_{A}) · (1_{T T E} ⊗ T η_{A})
= a · T λ_{A} · κ_{E,A} · (μ_{E} ⊗ 1_{T A}) ,

and on the other hand,

a · T λ_{A} · κ_{E,A} · (1_{T E} ⊗ η_{A}) · (μ_{E} ⊗ a)
= a · T λ_{A} · κ_{E,A} · (1_{T E} ⊗ T a · η_{T A}) · (μ_{E} ⊗ 1_{T A})
= a · T a · T λ_{T A} · κ_{E,T A} · (1_{T E} ⊗ η_{T A}) · (μ_{E} ⊗ 1_{T A})

= a · μ_{A} · T (T λ_{A} · κ_{E,A} · (η_{E} ⊗ 1_{T A})) · κ_{E,T A} · (1_{T E} ⊗ η_{T A}) · (μ_{E} ⊗ 1_{T A})

= a · T λ_{A} · μ_{E⊗A} · T κ_{E,A} · κ_{T E,T A} · (T η_{E} ⊗ η_{T A}) · (μ_{E} ⊗ 1_{T A})
= a · T λ_{A} · κ_{E,A} · (μ_{E} ⊗ μ_{A}) · (T η_{E} ⊗ η_{T A}) · (μ_{E} ⊗ 1_{T A})
= a · T λ_{A} · κ_{E,A} · (μ_{E} ⊗ 1_{T A}) .

By Lemma 2.3.3, there is therefore a unique C^{𝕋}-morphism \overline{λ}_{A} that makes the diagram

\xymatrix{T (T E ⊗ A) \ar[d]_{q} \ar[r]^{T (a·T λ·κ·(1⊗η))} & T A \ar[d]^{a} \\ T E A \ar[r]^{\overline{λ}} & A}

commute. Its inverse is the C^{𝕋}-morphism induced by l := q_{T E,A} · T (η_{E} ⊗ 1_{A}) · T λ_{A}^{−1} : T A → T E A. Indeed, since a is a coequalizer in C^{𝕋} of (T a, μ_{A}), and

(q_{T E,A} · T (η_{E} ⊗ 1_{A}) · T λ_{A}^{−1}) · T a
= q_{T E,A} · T (1_{T E} ⊗ a) · T (η_{E} ⊗ 1_{T A}) · T λ_{T A}^{−1}
= q_{T E,A} · T (μ_{E} ⊗ a) · T (T η_{E} ⊗ 1_{T A}) · T (η_{E} ⊗ 1_{T A}) · T λ_{T A}^{−1}

= q_{T E,A} · μ_{T E⊗A} · T κ_{T E,A} · T (T η_{E} ⊗ 1_{T A}) · T (η_{E} ⊗ 1_{T A}) · T λ_{T A}^{−1}

= q_{T E,A} · μ_{T E⊗A} · T T (η_{E} ⊗ 1_{A}) · T κ_{E,A} · T (η_{E} ⊗ 1_{T A}) · T λ_{T A}^{−1}
= q_{T E,A} · μ_{T E⊗A} · T T (η_{E} ⊗ 1_{A}) · T T λ_{A}^{−1}
= (q_{T E,A} · T (η_{E} ⊗ 1_{T A}) · T λ_{T A}^{−1}) · μ_{A} ,

there is a unique induced C^{𝕋}-morphism A → T E A (given by l · η_{A}). Using this computation, we can also compose l with T (a · T λ_{A} · κ_{T E,A} · (1_{T E} ⊗ η_{A})) to obtain

q_{T E,A} · T (η_{E} ⊗ 1_{A}) · T λ_{A}^{−1} · T (a · T λ_{A} · κ_{T E,A} · (1_{T E} ⊗ η_{A}))

= q_{T E,A} · T (η_{E} ⊗ 1_{A}) · T λ_{A}^{−1} · T λ_{A} · μ_{T E⊗A} · T κ_{T E,A} · T (1_{T E} ⊗ η_{A})

= q_{T E,A} · μ_{T E⊗A} · T κ_{T E,A} · T (T η_{E} ⊗ η_{A})
= q_{T E,A} · T (μ_{E} ⊗ a) · T (T η_{E} ⊗ η_{A})
= q_{T E,A} .

Since q_{T E,A} is epic, we have l · η_{A} · \overline{λ}_{A} = 1_{T E A}. With

a · (T (a · T λ_{A} · κ_{T E,A} · (1_{T E} ⊗ η_{A}))) · (T (η_{E} ⊗ 1_{A}) · T λ_{A}^{−1}) = a ,

we obtain similarly \overline{λ}_{A} · l · η_{A} = 1_{A}. Hence, \overline{λ}_{A} is an isomorphism in C^{𝕋}. Naturality of \overline{λ} follows from a standard diagram chase involving the defining diagrams of \overline{λ}_{A}, \overline{λ}_{B}, the facts that 1_{T E} f and f are C^{𝕋}-morphisms, and that q_{T E,A} is epic. The natural isomorphism \overline{ρ}_{A} : A T E → A is obtained symmetrically via the defining diagram

\xymatrix{T (A ⊗ T E) \ar[d]_{q} \ar[r]^{T (a·T ρ·κ·(η⊗1))} & T A \ar[d]^{a} \\ A T E \ar[r]^{\overline{ρ}} & A .}

Commutativity of

\xymatrix{(A T E) B \ar[dr]_{\overline{ρ} 1} \ar[rr]^{\overline{α}} &  & A (T E B) \ar[dl]^{1 \overline{λ}} \\  & A B}

follows again by a standard diagram chase involving the defining diagrams of the three given morphisms, and the fact that q_{A,T E} b is epic.

Symmetry. By using symmetry of the monoidal monad (𝕋, κ), one obtains the existence of a family of C^{𝕋}-morphisms \overline{σ}_{A,B} : A B → B A via the diagram

\xymatrix{T (A ⊗ B) \ar[r]^{T σ} \ar[d]_{q} & T (B ⊗ A) \ar[d]^{q} \\ A B \ar[r]^{\overline{σ}} & B A .}

Naturality of \overline{σ}, as well as commutativity of the symmetry diagrams for a symmetric monoidal category then follows by straightforward diagrammatic arguments.

The following result is a more memorable version of Theorem 2.5.5.

2.5.6. Corollary. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors. If A (−) and (−) B preserve reflexive coequalizer diagrams in C^{𝕋} (for all 𝕋-algebras A, B), then (C^{𝕋}, , T E) is a monoidal category whose structure morphisms are induced by those of (C, ⊗, E).

Moreover, if (C, ⊗, E) and (𝕋, κ) are symmetric monoidal, then so is (C^{𝕋}, , T E).

Proof. The statement follows directly from Theorem 2.5.5. Indeed, all q_{A,B} and 𝕋algebras structures are reflexive coequalizers, so Proposition 2.4.1 yields that q_{A,B} c, a q_{B,C} and q_{A,B} q_{C,D} are coequalizers of their respective diagrams.

In the case where (C, ⊗, E) is closed symmetric monoidal, the closed structure can be lifted to C^{𝕋} provided that C has equalizers and reflexive coequalizers.

2.5.7. Corollary. Let (𝕋, κ) be a monoidal monad on a closed symmetric monoidal category (C, ⊗, E) with equalizers and such that C^{𝕋} has tensors. Then (C^{𝕋}, , T E) is a monoidal category whose structure morphisms are induced by those of (C, ⊗, E).

Proof. The equalizer hypothesis allows us to apply the construction of an internal hom in C^{𝕋} described in [9, Theorem 2.2] that makes A (−) left adjoint for any A ∈ C^{𝕋} (see also the proof of [8, Proposition 1.3]). In particular, A (−) preserves reflexive coequalizers, and Corollary 2.5.6 applies.

2.5.8. Examples.

(1) In the case where (𝕋, κ) = (𝕀, 1) is the identity monoidal monad on a monoidal category (C, ⊗, E), Theorem 2.5.5 reduces to a tautology, namely that (C, ⊗, E) is monoidal.

(2) For the free abelian group cartesian monad (𝔸, κ) on Set (Example 1.2.1(2)), Corollary 2.5.7 describes the usual tensor product over ℤ of the category Set^{𝔸} ≅ AbGrp of abelian groups.

(3) For the monoidal powerset monad (ℙ, ι) on Set (Example 1.2.1(3)), Corollary 2.5.7 yields the classical tensor product on Set^{ℙ} ≅ Sup (see [16]).

2.6. Alternative hypotheses. In the case where (C, ⊗, E) is not closed symmetric monoidal (see Corollary 2.5.7), it might be delicate to verify the more general hypotheses of Theorem 2.5.5 or even of Corollary 2.5.6 since each of these involve the induced tensor (− −). Theorem 2.6.4 below presents a situation where the hypotheses can be directly tested on the original data 𝕋 and (C, ⊗, E). The proof hinges on how certain coequalizers can be tensored.

2.6.1. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E), and consider coequalizer diagrams

\xymatrix{T Z \ar[r]^{μ·T r} \ar[r]_{T s} & T Y \ar[r]^{p} & X & and & T Z′ \ar[r]^{μ·T r′} \ar[r]_{T s′} & T Y′ \ar[r]^{p′} & X′}

in C^{𝕋} (here, (X, x), (X′, x′) are 𝕋-algebras, and the objects of the form T A are equipped with their free structure μ_{A}). Suppose that T (− ⊗ −) : C^{𝕋} × C^{𝕋} → C^{𝕋} preserves the coequalizer diagram of (p, p′), that T (T p⊗T p′) is an epimorphism, and that the coequalizer q_{X,X′} : T (X ⊗ X′) → X X exists.

If the coequalizer of μ_{Y ⊗Y′} · T (κ_{Y,Y′} · (r ⊗ r′)) and T (s ⊗ s′) exists C^{𝕋}, then it can be given by q_{X,X′} · T (p · η_{Y} ⊗ p′ · η_{Y′}):

\xymatrix{T (Z ⊗ Z′) \ar[r]^{μ·T (κ·(r⊗r′))} \ar[r]_{T (s⊗s′)} & T (Y ⊗ Y′) \ar[r]^{q·T (p·η⊗p′·η)} & X X′ .}

Proof. By hypothesis, there are coequalizer diagrams

\xymatrix{T (T Z ⊗ T Z′) \ar[r]^{T ((μ·T r)⊗(μ·T r′))} \ar[r]_{T (T s⊗T s′)} & T (T Y ⊗ T Y′) \ar[r]^{T (p⊗p′)} & T (X ⊗ X′) .}

and

\xymatrix{T (Z ⊗ Z′) \ar[r]^{μ·T (κ·(r⊗r′))} \ar[r]_{T (s⊗s′)} & T (Y ⊗ Y′) \ar[r]^{t} & X ⊛ X′}

in C^{𝕋}. We proceed to show that X ⊛ X′ and X X′ are isomorphic. Since the diagram

\xymatrix{T (T Z ⊗ T Z′) \ar[r]^{T ((μ·T r)⊗(μ·T r′))} \ar[r]_{T (T s⊗T s′)} & T (T Y ⊗ T Y′) \ar[r]^{t·μ·T κ} & X ⊛ X′}

commutes, there exists a unique C^{𝕋}-morphism \overline{t} : T (X ⊗ X′) → X ⊛ X′ such that

\overline{t} · T (p ⊗ p′) = t · μ_{Y ⊗Y′} · T κ_{Y,Y′} .  (2.6.ii)

We can therefore consider the diagram

\xymatrix{T (T T Y ⊗ T T Y′) \ar[r]^{T (T p⊗T p′)} \ar[d]^{μ·T κ} \ar[d]_{T (μ⊗μ)} & T (T X ⊗ T X′) \ar[d]_{μ·T κ} \ar[d]^{T (x⊗x′)} \\ T (T Y ⊗ T Y′) \ar[r]^{T (p⊗p′)} \ar[d]_{μ·T κ} & T (X ⊗ X′) \ar[d]^{\overline{t}} \\ T (Y ⊗ Y′) \ar[r]^{t} & X ⊛ X′}

in which the inner- and outer-upper squares commute, as does the lower one; Proposition 2.5.2 moreover implies that the two left vertical arrows of the large square can be identified. By hypothesis, T (T p ⊗ T p′) is an epimorphism, so the vertical C^{𝕋}-morphism T (T X ⊗ T X′) → X ⊛ X′ that makes the large square commute is unique, that is,

\xymatrix{T (T X ⊗ T X′) \ar[r]^{μ·T κ} \ar[r]_{T (x⊗x′)} & T (X ⊗ X′) \ar[r]^{\overline{t}} & X ⊛ X′}

commutes. Thus, the universal property of q_{X,X′} yields a unique C^{𝕋}-morphism \overline{\overline{t}} : X X′ → X ⊛ X′ such that

\overline{\overline{t}} · q_{X,X′} = \overline{t} .

Let us verify now that \overline{\overline{t}} is an isomorphism. With (2.6.ii), one obtains

\overline{\overline{t}} · q_{X,X′} · T (p ⊗ p′) = t · μ_{Y ⊗Y′} · T κ_{Y,Y′} ,  (2.6.iii)

and therefore

t = \overline{\overline{t}} · q_{X,X′} · T (p ⊗ p′) · T (η_{Y} ⊗ η_{Y′}) = \overline{\overline{t}}_{X,X′} · q_{X,X′} · T (p · η_{Y} ⊗ p′ · η_{Y′}) .

The equality

q_{X,X′} · T (p · η_{Y} ⊗ p′ · η_{Y′}) · μ_{Y ⊗Y′} · T κ_{Y,Y′} = q_{X,X′} · T (p ⊗ p′) ,  (2.6.iv)

with Lemma 2.5.1 shows that

\xymatrix{T (Z ⊗ Z′) \ar[r]^{μ·T (κ·(r⊗r′))} \ar[r]_{T (s⊗s′)} & T (Y ⊗ Y′) \ar[r]^{q·T (p·η⊗p′·η)} & X X′}

commutes. The universal property of t then yields a unique C^{𝕋}-morphism u : X ⊛ X′ → X X′ such that

q_{X,X′} · T (p · η_{Y} ⊗ p′ · η_{Y′}) = u · t ,

and we can consider the following commutative diagram:

\xymatrix{T (T Y ⊗ T Y′) \ar[r]^{T (p⊗p′)} \ar[d]_{μ·T κ} & T (X ⊗ X′) \ar[r]^{q} \ar@{-->}[dr]^{\overline{t}} & X X′ \ar[d]^{\overline{\overline{t}}} \\ T (Y ⊗ Y′) \ar[rr]^{t} \ar[d]_{T (η⊗η)} &  & X ⊛ X′ \ar[d]^{u} \\ T (T Y ⊗ T Y′) \ar[r]^{T (p⊗p′)} & T (X ⊗ X′) \ar[r]^{q} & X X′ .}

By using (2.6.iv), one then observes

q_{X,X′} · T (p ⊗ p′) = u · \overline{\overline{t}} · q_{X,X′} · T (p ⊗ p′) ,

so u · \overline{\overline{t}} = 1_{X X′} because both q_{X,X′} and T (p ⊗ p′) are epimorphisms. By exchanging the displayed upper and lower diagrams, one obtains similarly that \overline{\overline{t}} · u = 1_{X⊛X′}, and can conclude that \overline{\overline{t}} is an isomorphism with inverse u.

2.6.2. Corollary. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) with reflexive coequalizers. If T (X ⊗ −) and T (− ⊗ Y) preserve reflexive coequalizers in C (for all X, Y ∈ ob C), then

\xymatrix{T ((T A ⊗ T B) ⊗ T C) \ar[r]^{μ·T (κ·(κ⊗1))} \ar[r]_{T ((a⊗b)⊗c)} & T ((A ⊗ B) ⊗ C) \ar[r]^{q·T (q·η⊗1)} & (A B) C}

and

\xymatrix{T (T A ⊗ (T B ⊗ T C)) \ar[r]^{μ·T (κ·(1⊗κ))} \ar[r]_{T (a⊗(b⊗c))} & T (A ⊗ (B ⊗ C)) \ar[r]^{q·T (1⊗q·η)} & A (B C)}

are coequalizer diagrams in C^{𝕋} (for all 𝕋-algebras (A, a), (B, b) and (C, c)).

Proof. We prove the statement for the first diagram, the proof for the second following similarly. For this, we only need to verify that the hypotheses of Proposition 2.6.1 are verified for the reflexive coequalizer diagrams

\xymatrix{T (T A ⊗ T B) \ar[r]^{μ·T κ} \ar[r]_{T (a⊗b)} & T (A ⊗ B) \ar[r]^{q} & A B & and & T T C \ar[r]^{μ} \ar[r]_{T c} & T C \ar[r]^{c} & C .}

Since T ≅ T (E ⊗ −) preserves reflexive coequalizers, these lift to C^{𝕋} by Proposition 2.4.2, and T (q_{A,B} ⊗ 1_{C}) is therefore a coequalizer in C^{𝕋}; thus, T (q_{A,B} ⊗ c) is one too by Proposition 2.4.1. Similarly, T (T q_{A,B} ⊗ T c) is a reflexive coequalizer in C^{𝕋} and consequently an epimorphism. Finally, the coequalizer of

μ_{(A⊗B)⊗C} · T (κ_{A⊗B,C} · (κ_{A⊗B} ⊗ 1_{C})) and T (a ⊗ b) ⊗ T c

exists in C^{𝕋} because it is a reflexive pair (split by T ((η_{A} ⊗ η_{B}) ⊗ η_{C})).

2.6.3. Corollary. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) with reflexive coequalizers. If T (X ⊗ −) and T (− ⊗ Y) preserve reflexive coequalizers in C (for all X, Y ∈ ob C), then

\xymatrix{T ((A ⊗ B) ⊗ (C ⊗ D)) \ar[r]^{q·T (q·η⊗q·η)} & (A B) (C D)}

is an epimorphism in C^{𝕋} (for all 𝕋-algebras (A, a), (B, b), (C, c), and (D, d)).

Proof. The given morphism is in fact a reflexive coequalizer in C^{𝕋}, obtained by applying Proposition 2.6.1 to the coequalizer diagrams of q_{A,B} and q_{C,D}.

2.6.4. Theorem. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) with reflexive coequalizers. If T (X ⊗ −) and T (− ⊗ Y) preserve reflexive coequalizers in C (for all X, Y ∈ ob C), then (C^{𝕋}, , T E) is a monoidal category whose structure morphisms are induced by those of (C, ⊗, E).

Moreover, if (C, ⊗, E) and (𝕋, κ) are symmetric monoidal, then so is (C^{𝕋}, , T E).

Proof. We verify the hypotheses of Theorem 2.5.5. By Proposition 2.4.2, q_{A,B} exists and is therefore is a reflexive coequalizer of (μ_{A⊗B} · T κ_{A,B}, T (a ⊗ b)) in C^{𝕋}. With the convention of 2.5.3, the first coequalizer diagram in Corollary 2.6.2 is in fact

\xymatrix{T (T A ⊗ T B) T T C \ar[r]^{μ·T κ μ} \ar[r]_{T (a⊗b) T c} & T (A ⊗ B) T C \ar[r]^{q c} & (A B) C .}

Indeed, we already know that T ((a ⊗ b) ⊗ c) = T (a ⊗ b) T c; moreover, with q_{T A,T B} = μ_{A⊗B} · T κ_{A,B} in the diagram

\xymatrix{T (T (T A ⊗ T B) ⊗ T T C) \ar[d]_{μ·T κ} \ar[r]^{T (μ·T κ⊗μ)} & T (T (A ⊗ B) ⊗ T C) \ar[d]^{μ·T κ} \\ T ((T A ⊗ T B) ⊗ T C) \ar[r]^{μ·T (κ·(κ⊗1))} & T ((A ⊗ B) ⊗ C) ,}

we see that μ_{(A⊗B)⊗C} · T (κ_{A⊗B,C} · (κ_{A,B} ⊗ 1_{T C})) = μ_{A⊗B} · T κ_{A,B} μ_{C}; finally, since q_{A,B} is a C^{𝕋}-morphism and q_{A B,C} = (a ▷◁ b) c, we have by functoriality of (− −):

q_{A B,C} · T (q_{A,B} · η_{A⊗B} ⊗ 1_{C}) = ((a ▷◁ b) c) · (T (q_{A,B} · η_{A⊗B}) 1_{T C})
= (q_{A,B} c) · ((μ_{A⊗B} · T η_{A⊗B}) 1_{T C}) = q_{A,B} c .

By symmetry, the second coequalizer diagram of Corollary 2.6.2 shows that a q_{B,C} is the coequalizer of (μ_{A} μ_{B⊗C} · T κ_{B,C}, T a T (b ⊗ c)). By using that q_{A,B} = a b and q_{C,D} = c d are C^{𝕋}-morphisms in Corollary 2.6.3, we obtain that

q_{A,B} q_{C,D} = q_{A B,C D} · T (q_{A,B} · η_{A⊗B} ⊗ q_{C,D} · η_{C⊗D})

is an epimorphism. Hence, Theorem 2.5.5 applies.

2.6.5. Example. For the monoidal category (C, +, ∅) of Example 1.2.1(4), we note that the functor X + (−) preserves coequalizers for all X ∈ ob C. In general, C is not monoidal closed (for example, if C = Set and X is a non-empty set, the functor X + (−) is not left adjoint, as it does not preserve coproducts), so Corollary 2.5.7 does not apply. Nevertheless, if C^{𝕋} has tensors, then (− −) turns out to be the binary coproduct in C^{𝕋} (see [11, Proposition 2]). In this case, (C^{𝕋}, , T ∅) is monoidal because of the universal property of coproducts; alternatively, (− −) is left adjoint to the diagonal functor, so Corollary 2.5.6 applies.

In the case where C has reflexive coequalizers and the monad functor T preserves them (so that C^{𝕋} has reflexive coequalizers by Proposition 2.4.2), Theorem 2.6.4 states that (C^{𝕋}, , T ∅) is monoidal, a result that essentially summarizes the previous discussion.

2.7. Monoidal monad morphisms and Eilenberg–Moore categories. Once a monoidal structure on the Eilenberg–Moore category has been established, it is not surprising that monoidal monad morphisms induce monoidal functors, although these need not be strict as in Proposition 1.3.2.

2.7.1. Proposition. Let (C, ⊗, E) be a monoidal category, and (𝕊, ι), (𝕋, κ) monoidal monads on C. Suppose that C^{𝕋} and C^{𝕊} both have tensors that make (C^{𝕋}, , T E) and (C^{𝕊}, ⊛, SE) monoidal categories with structures induced by those of (C, ⊗, E). If ϕ : (𝕊, ι) → (𝕋, κ) is a monoidal monad morphism, then the induced functor

C^{ϕ} : C^{𝕋} → C^{𝕊}

(that commutes with the forgetful functors to C) is itself monoidal.

Proof. Set 𝕊 = (S, ν, δ) and 𝕋 = (T, μ, η). Recall that the algebraic functor C^{ϕ} sends a 𝕋-algebra (A, a) to the 𝕊-algebra (A, a · ϕ_{A}) and is identical on morphisms. For 𝕋algebras (A, a) and (B, b), one has ϕ_{A⊗B} · ν_{A⊗B} = μ_{A⊗B} · ϕ_{T (A⊗B)} · Sϕ_{A⊗B}, that is, ϕ_{A⊗B} : S(A ⊗ B) → T (A ⊗ B) is a C^{𝕊}-morphism. Hence, the inner- and outer-left squares in the diagram

\xymatrix{S(SA ⊗ SB) \ar[r]^{ν·Sι} \ar[r]_{S((a·ϕ)⊗(b·ϕ))} \ar[d]_{ϕ·S(ϕ⊗ϕ)} & S(A ⊗ B) \ar[r]^{p} \ar[d]^{ϕ} & A ⊛ B \ar@{.>}[d]^{\overline{ϕ}} \\ T (T A ⊗ T B) \ar[r]_{μ·T κ} \ar[r]^{T (a⊗b)} & T (A ⊗ B) \ar[r]^{q} & A B}

commute in C^{𝕊}. If p_{A,B} denotes the coequalizer of the upper row, there is consequently a unique C^{𝕊}-morphism \overline{ϕ}_{A,B} : (A ⊛ B, (a · ϕ_{A}) ▷◁ (b · ϕ_{B})) → (A B, (a ▷◁ b) · ϕ_{A B}) that makes the square on the right commute. These \overline{ϕ}_{A,B} (for A, B ∈ ob C^{𝕋}) are easily seen to be natural in A and B.

A standard diagram chase involving defining diagrams of the induced structures yields commutativity of

\xymatrix{(A ⊛ B) ⊛ C \ar[r]^{\overline{ϕ}⊛1} \ar[d]_{\overline{α}} & (A B) ⊛ C \ar[r]^{\overline{ϕ}} & (A B) C \ar[d]^{\overline{α}} \\ A ⊛ (B ⊛ C) \ar[r]^{1⊛\overline{ϕ}} & A ⊛ (B C) \ar[r]^{\overline{ϕ}} & A (B C) .}

Similarly, commutativity of

[figure: SE ⊛ A; ϕ⊛1; \overline{λ}; T E ⊛ A; \overline{ϕ}; T E A; \overline{λ}; A; and; A ⊛ SE; 1⊛ϕ; \overline{ρ}; A ⊛ T E; \overline{ϕ}; A T E; \overline{ρ}; A]

is routinely verified. Hence, the functor C^{ϕ} with \overline{ϕ} : C^{ϕ}(−) ⊛ C^{ϕ}(−) → C^{ϕ}(− −) and ϕ_{E} : SE → T E is monoidal.

2.7.2. Example. The unit monad morphism η : 𝕀 → 𝕋 of a monoidal monad 𝕋 (Example 1.3.1) induces the forgetful functor C^{η} : C^{𝕋} → C^{𝕀} ≅ C that is therefore monoidal (whenever (C^{𝕋}, ⊗, T E) is monoidal with structures morphisms induced by (C, ⊗, E)).

2.7.3. Remark. Example 2.7.2 shows that in the context of Theorem 2.5.5, the forgetful functor G^{𝕋} : C^{𝕋} → C is monoidal. Standard diagram chases also show that the left adjoint F^{𝕋} : C → C^{𝕋} is strong monoidal with respect to \overline{κ} : F^{𝕋}(−) F^{𝕋}(−) → F^{𝕋}(− ⊗ −) and 1_{T E} : T E → T E (where \overline{κ}_{X,Y} : (T X T Y, μ_{X} ▷◁ μ_{Y}) → (T (X ⊗Y), μ_{X⊗Y}) denotes the C^{𝕋}- morphism induced by the bimorphism κ_{X,Y} for all X, Y ∈ ob C). Hence, the adjunction F^{𝕋} ⊣ G^{𝕋} : C^{𝕋} → C is monoidal, mirroring the closed case studied in [9] in relation with commutative monads.

3. Actions

3.1. Monoids. Let (C, ⊗, E) be a monoidal category. The category of monoids

(M, m : M ⊗ M → M, e : E → M)

in C with their homomorphisms is denoted by Mon(C) (see [13, Section VII.3]).

3.1.1. Examples.

(1) For Set with its cartesian structure, Mon(Set) = Mon, the usual category of monoids with their homomorphisms.

(2) A unital ring R is an abelian group that is also a monoid in which the distributive laws hold, that is, the multiplication R×R → R is ℤ-bilinear and is therefore equivalently described as a group homomorphism R⊗_{ℤ}R → R. Hence, unital rings are precisely the monoids in AbGrp (with its usual tensor product), and Mon(AbGrp) = Rng is the category of unital rings and their homomorphisms.

(3) A quantale V is a complete lattice with a monoid operation (−⊗−) : V ×V → V that preserves suprema in each variable; with the tensor product in Sup, the category of complete lattices and sup-preserving maps, the monoid operation may equivalently be considered a morphism V ⊗ V → V in Sup. Hence, one has Mon(Sup) = Qnt, the category of quantales and their homomorphisms.

3.2. Actions in a monoidal category. Let (C, ⊗, E) be a monoidal category, and M = (M, m, e) a monoid in C. The category of M-actions

a : M ⊗ A → A

in C (that is, of left actions of M on objects of C, see [13, Section VII.4]) with their equivariant C-morphisms is denoted by C^{M} (indeed, a monoid M in a monoidal category (C, ⊗, E) gives rise to a monad on C whose Eilenberg–Moore category is the category of M-actions and equivariant C-morphisms, in generalization of the monadic group actions in Set described in [13, Section VI.2]).

3.2.1. Examples.

(1) If C = Set, and M is a monoid, the category Set^{M} is the usual category of M-actions and equivariant maps.

(2) The monoidal structure of AbGrp is given by the tensor product over ℤ, and a monoid R in AbGrp is a ring. Hence, AbGrp^{R} is the usual category R-Mod of left R-modules.

(3) Given a quantale V = (V, ⊗, k), that is, a monoid is Sup, the category Sup^{V} is described as follows. A V -action X in Sup is a complete lattice X together with a bimorphism (−) · (−) : V × X → X in Sup such that

(u ⊗ v) · x = u · (v · x) , k · x = x ,

for all v ∈ V, x ∈ X, and a sup-map f : X → Y is equivariant whenever

f(v · x) = v · f(x)

for all v ∈ V, x ∈ X.

3.3. Monadic actions. In general, monadic functors do not compose. In the case of actions in C^{𝕋} however, they do (Theorem 3.3.3 below).

3.3.1. Proposition. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors, and (C^{𝕋}, , T E) a monoidal category whose structure morphisms are induced by those of (C, ⊗, E).

A monoid (M, ξ) in (C^{𝕋}, , T E) induces a monad M 𝕋 on C whose functor is M T (−) : C → C, and whose multiplication and unit are given by their components at X ∈ ob C as follows:

μ̃_{X} = (m 1_{T X}) · \overline{α}_{M,M,T X}^{−1} · (1_{M} (ξ ▷◁ μ_{X})) ,
η̃_{X} = (e 1_{T X}) · \overline{λ}_{T X}^{−1} · η_{X} .

Proof. The composite of the adjunctions

[figure: (C^{𝕋})^{M}; ⊥; C^{𝕋}; ⊥; M (−); C; T]

yields an adjunction [figure: (C^{𝕋})^{M}; ⊥; C; M T (−)] that induces the described monad.

3.3.2. Lemma. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors, and (C^{𝕋}, , T E) a monoidal category whose structure morphisms are induced by those of (C, ⊗, E).

The C-morphisms (e 1_{T X}) · \overline{λ}_{T X}^{−1} : T X → M T X are the components of a monad morphism τ : 𝕋 → M 𝕋.

Proof. Naturality of the components τ_{X} := (e 1_{T X}) · \overline{λ}_{T X}^{−1} (for X ∈ ob C) is immediate. Agreement on the units is also easily verified:

η̃_{X} = (e 1_{T X}) · \overline{λ}_{T X}^{−1} · η_{X} = τ_{X} · η_{X} .

For the multiplications, one has

μ̃_{X}·(1_{M} T τ_{X}) · τ_{T X}
= (m 1_{T X}) · \overline{α}_{M,M,T X}^{−1} · (1_{M} ((e 1_{T X}) · \overline{λ}_{T X}^{−1} · μ_{X})) · τ_{T X}
= (\overline{λ}_{M} 1_{T X}) · ((1_{T E} e) 1_{T X}) · \overline{α}_{T E,T E,T X}^{−1} · (1_{T E} (\overline{λ}_{T X}^{−1} · μ_{X})) · \overline{λ}_{T X}^{−1}
= (e 1_{T X}) · (\overline{λ}_{T E} 1_{T X}) · \overline{α}_{T E,T E,T X}^{−1} · \overline{λ}_{T E T X}^{−1} · \overline{λ}_{T X}^{−1} · μ_{X}
= (e 1_{T X}) · \overline{λ}_{T X}^{−1} · μ_{X} = τ_{X} · μ_{X}

(we use [6] for the penultimate equality), so τ is indeed a monad morphism.

3.3.3. Theorem. Let (𝕋, κ) be a monoidal monad on a monoidal category (C, ⊗, E) such that C^{𝕋} has tensors, and (C^{𝕋}, , T E) a monoidal category whose structure morphisms are induced by those of (C, ⊗, E).

For a monoid (M, ξ) in (C^{𝕋}, , T E), there is an isomorphism between the category of algebras of the monad M 𝕋 and the category of M-actions in C^{𝕋}:

C^{M 𝕋} ≅ (C^{𝕋})^{M} .

In particular, the forgetful functor (C^{𝕋})^{M} → C is strictly monadic.

Proof. The comparison functor K : (C^{𝕋})^{M} → C^{M 𝕋} sends a 𝕋-algebra (A, a_{1}) with an action a_{2} : M A → A in C^{𝕋} to the (M 𝕋)-algebra (A, a_{2} · (1_{M} a_{1})). We proceed to verify that K is an isomorphism.

The monad morphism τ : 𝕋 → M 𝕋 of Lemma 3.3.2 induces a functor C^{τ} : C^{M 𝕋} → C^{𝕋} that sends a (M 𝕋)-algebra (A, a) to the 𝕋-algebra (A, a · τ_{A}) and commutes with the forgetful functors to C. Set a_{1} := a · τ_{A}, so that

\xymatrix{T (T M ⊗ T A) \ar[r]^{μ·T κ} \ar[r]_{T (ξ⊗a_{1})} & T (M ⊗ A) \ar[r]^{q} & M A}

is a coequalizer diagram. There is then a unique C^{𝕋}-morphism a_{2} : M A → A such that

\xymatrix{T (M ⊗ A) \ar[rr]^{q} \ar[d]_{T (1⊗η)} &  & M A \ar[d]^{a_{2}} \\ T (M ⊗ T A) \ar[r]^{q} & M T A \ar[r]^{a} & A}

commutes. To see this, we use the universal property of q_{M,A}. Indeed, τ_{A} = (e 1_{T A})·\overline{λ}_{T A}^{−1} : T A → M T A is a C^{𝕋}-morphism, so the diagram

\xymatrix{T (T M ⊗ T A) \ar[r]^{T (ξ⊗η)} \ar@{-->}[dr]_{T (ξ⊗1)} & T (M ⊗ T T A) \ar[r]^{T (1⊗T ((e 1)·\overline{λ}^{−1}))} \ar[d]^{T (1⊗μ)} & T (M ⊗ T (M T A)) \ar[d]^{T (1⊗(ξ▷◁μ))} \\  & T (M ⊗ T A) \ar[r]^{T (1⊗((e 1)·\overline{λ}^{−1}))} & T (M ⊗ (M T A))}

commutes; by definition of a (M 𝕋)-algebra (A, a), the diagram

\xymatrix{T (M ⊗ T (M T A)) \ar[d]_{T (1⊗(ξ▷◁μ))} \ar[r]^{q} & M T (M T A) \ar[d]_{1 (ξ▷◁μ)} \ar[r]^{1 T a} & M T A \ar[dr]^{a} \\ T (M ⊗ (M T A)) \ar[r]^{q} & M (M T A)) \ar[r]^{(m 1)·\overline{α}^{−1}} & M T A \ar[r]^{a} & A}

also commutes. By glueing these diagrams together along T (1_{M} ⊗(ξ ▷◁ μ_{A})), one obtains, with a_{1} = a · (e 1_{T A}) · \overline{λ}_{T A}^{−1} and q_{M,T A} · T (1 ⊗ T a) = (1_{M} T a) · q_{M,T (M T A)},

a · q_{M,T A} · T (1_{M} ⊗ η_{A}) · T (ξ ⊗ a_{1})
= a · q_{M,T A} · T (1_{M} ⊗ T a_{1}) · T (ξ ⊗ η_{T A})
= a · (1_{M} T a) · q_{M,T (M T A)} · T (1_{M} ⊗ T ((e 1_{T A}) · \overline{λ}_{T A}^{−1})) · T (ξ ⊗ η_{T A})

= a · (m 1_{T A}) · \overline{α}_{M,M,T A}^{−1} · q_{M,M T A} · T (1_{M} ⊗ ((e 1_{T A}) · \overline{λ}_{T A}^{−1})) · T (ξ ⊗ 1_{T A})

= a · (m 1_{T A}) · \overline{α}_{M,M,T A}^{−1} · (1_{M} ((e 1_{T A}) · \overline{λ}_{T A}^{−1})) · q_{M,T A} · T (ξ ⊗ 1_{T A})
= a · (\overline{ρ}_{M} 1_{T A}) · \overline{α}_{M,T E,T A}^{−1} · (1_{M} \overline{λ}_{T A}^{−1}) · q_{M,T A} · T (ξ ⊗ 1_{T A})
= a · q_{M,T A} · T (ξ ⊗ 1_{T A}) .

Since one also has

a · q_{M,T A} · T (1_{M} ⊗ η_{A}) · μ_{M⊗A} · T κ_{M,A}
= a · q_{M,T A} · μ_{M⊗T A} · T κ_{M,T A} · T (1_{T M} ⊗ T η_{A})
= a · q_{M,T A} · T (ξ ⊗ 1_{T A}) ,

the existence and unicity of the required C^{𝕋}-morphism a_{2} follows. Moreover,

a · q_{M,T A} = a · q_{M,T A} · T (1_{M} ⊗ η_{A}) · T (1_{M} ⊗ a_{1})
= a_{2} · q_{M,A} · T (1_{M} ⊗ a_{1})
= a_{2} · (1_{M} a_{1}) · q_{M,T A} ,

so a = a_{2} · (1_{M} a_{1}) because q_{M,T A} is epic. Let us verify that a_{2} : M A → A defines an action. For this, we use

a · (1_{M} T a) = a_{2} · (1_{M} a_{1}) · (1_{M} T a)
= a_{2} · (1_{M} a) · (1_{M} (ξ ▷◁ μ_{A}))
= a_{2} · (1_{M} a_{2}) · (1_{M} (1_{M} a_{1})) · (1_{M} (ξ ▷◁ μ_{A}))

and

a · μ̃_{A} = a_{2} · (m 1_{A}) · (1_{M M} a_{1}) · \overline{α}_{M,M,T A}^{−1} · (1_{M} (ξ ▷◁ μ_{A}))
= a_{2} · (m 1_{A}) · \overline{α}_{M,M,A}^{−1} · (1_{M} (1_{M} a_{1})) · (1_{M} (ξ ▷◁ μ_{A})) .

Moreover, for any 𝕋-algebra structure b : T B → B, the C^{𝕋}-morphisms T (1_{M} ⊗ b) is an epimorphism, so that q_{M,A} · T (1_{M} ⊗ b) = (1_{M} b) · q_{M,T A} implies that (1_{M} b) is epic. Since a · (1_{M} T a) = a · μ̃_{A}, and (1_{M} (1_{M} a_{1})), (1_{M} (ξ ▷◁ μ_{A})) are both epic, we obtain

a_{2} · (1_{M} a_{2}) = a_{2} · (m 1_{A}) · \overline{α}_{M,M,A}^{−1} ,

the first condition for a_{2} to be an action. The second condition comes from

a · η̃_{A} = a_{2} · (1_{M} a_{1}) · (e 1_{T A}) · \overline{λ}_{T A}^{−1} · η_{A} = a_{2} · (e 1_{A}) · \overline{λ}_{A}^{−1} · a_{1} · η_{A} = a_{2} · (e 1_{A}) · \overline{λ}_{A}^{−1}

with a · η̃_{A} = 1_{A}. Finally, any (M 𝕋)-algebra homomorphism f : (A, a) → (B, b) yields a 𝕋-algebra homomorphism f : (A, a · τ_{A}) → (B, b · τ_{B}) that is equivariant: one has

f ·a_{2} ·(1_{M} a_{1}) = f ·a = b·(1_{M} T f) = b_{2} ·(1_{M} b_{1})·(1_{M} T f) = b_{2} ·(1_{M} f)·(1_{M} a_{1}) ,

so that f · a_{2} = b_{2} · (1_{M} f) because (1_{M} a_{1}) is epic.

Hence, a (M 𝕋)-algebra (A, a) yields a 𝕋-algebra (A, a_{1}) with an action a_{2} : M A → A, and K((A, a_{1}), a_{2}) returns the original (M 𝕋)-algebra (A, a), since a = a_{2} · (1_{M} a_{1}).

Conversely, the K-image of a 𝕋-algebra (A, a_{1}) with an action a_{2} : M A → A is a (M 𝕋)-algebra (A, a_{2} · (1_{M} a_{1})). Since

a_{2} · (1_{M} a_{1}) · τ_{A} = a_{2} · (e 1_{A}) · \overline{λ}_{A}^{−1} · a_{1} = a_{1}

and the diagram

\xymatrix{T (M ⊗ A) \ar[r]^{1} \ar[d]_{T (1⊗η)} & T (M ⊗ A) \ar[rr]^{q} &  & M A \ar[d]^{a_{2}} \\ T (M ⊗ T A) \ar[rr]^{q} \ar[ur]_{T (1⊗a_{1})} &  & M T A \ar[ur]^{1 a_{1}} \ar[r]^{a} & A}

commutes, one recuperates the original triplet ((A, a_{1}), a_{2}) from (A, a_{2} · (1_{M} a_{1})); that is, K is an isomorphism.

3.3.4. Examples.

(1) Any monoid M in a monoidal category (C, ⊗, E) yields an isomorphism (C^{𝕀})^{M} ≅ C^{M} ≅ C^{M⊗𝕀}. This immediate result is also the 𝕋 = 𝕀 case of Theorem 3.3.3.

(2) If 𝕋 = 𝔸 is the free abelian group monad and M = R is a ring, the isomorphisms

R-Mod ≅ AbGrp^{R} ≅ (Set^{𝔸})^{R} ≅ Set^{R⊗_{ℤ}𝔸}

recall the classical monadicity of R-modules over Set, and describe the free R-module over a set X as R ⊗_{ℤ} AbX.

(3) For a quantale V, the category Sup^{V} of V -actions (Example 3.2.1(3)) is isomorphic to the category Set^{V ⊗ℙ}. The classical description of the tensor in Sup (see for example [2]) yields isomorphisms

V ⊗ P X ≅ Sup(Sup(V, Sup(P X, 2)), 2) ≅ Set(X, V) .

The case where V is integral was treated in [15], where it is proved that Sup^{V} is monadic over Set.

3.4. Monad morphisms. Suppose that 𝕊 and 𝕋 are monoidal monads on (C, ⊗, E) that respectively induce monoidal categories (C^{𝕊}, ⊛, SE) and (C^{𝕋}, , T E). Proposition 3.4.2 below shows that every pair of monoid homomorphism f : N → M and monoidal monad morphism ϕ : 𝕊 → 𝕋 induces a monad morphism (f, ϕ) : N⊛𝕊 → M 𝕋, and thus a functor (C^{𝕋})^{M} → (C^{𝕊})^{N} between the respective categories of actions. The usual restriction-of-scalars functor between categories of modules then appears as the 𝕊 = 𝕋 = 𝔸 instance of this result (Corollary 3.4.3).

3.4.1. Lemma. Let (C, ⊗, E) be a monoidal category, and (𝕊, ι), (𝕋, κ) monoidal monads on C. Suppose that C^{𝕋} and C^{𝕊} both have tensors that make (C^{𝕋}, , T E) and (C^{𝕊}, ⊛, SE) monoidal categories with structures induced by those of (C, ⊗, E).

If ϕ : 𝕊 → 𝕋 is a monoidal monad morphism and ((N, ζ), n, d) is a monoid in C^{𝕋}, then (N, ζ · ϕ_{N}) with multiplication n · \overline{ϕ}_{N,N} and unit d · ϕ_{E} is a monoid in C^{𝕊}.

Proof. Since (N, ζ) is a 𝕋-algebra, (N, ζ · ϕ_{N}) is an 𝕊-algebra. The structures d · ϕ_{E} : SE → N and n · \overline{ϕ}_{N,N} : N ⊛ N → N are C^{𝕊}-morphisms (see the proof of Proposition 2.7.1). Commutativity of the corresponding monoid diagrams in (C^{𝕊}, ⊛, SE) follows from commutativity of the monoid diagrams of (N, n, d) in (C^{𝕋}, , T E) combined with commutativity of the diagrams showing that the functor C^{ϕ} : C^{𝕋} → C^{𝕊} is monoidal with respect to \overline{ϕ} and ϕ_{E} (Proposition 2.7.1).

3.4.2. Proposition. Let (C, ⊗, E) be a monoidal category, and (𝕊, ι), (𝕋, κ) monoidal monads on C. Suppose that C^{𝕋} and C^{𝕊} both have tensors that make (C^{𝕋}, , T E) and (C^{𝕊}, ⊛, SE) monoidal categories with structures induced by those of (C, ⊗, E).

If ϕ : 𝕊 → 𝕋 is a monoidal monad morphism, and f : N → M a monoid homomorphism in C^{𝕋}, then there is a monad morphism (f, ϕ) : N ⊛ 𝕊 → M 𝕋 whose components at X ∈ ob C are given by the C^{𝕊}-morphism \overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}) : N ⊛ SX → M T X.

Proof. By Lemma 3.4.1, N can be seen as a monoid in (C^{𝕊}, ⊛, SE), thus defining a monad N ⊛ 𝕊. Via the functor C^{ϕ} : C^{𝕋} → C^{𝕊}, the arrows f and ϕ_{X} are C^{𝕊}-morphisms, so f ⊛ ϕ_{X} is defined in C^{𝕊}, and so is \overline{ϕ}_{M,T X} by Proposition 2.7.1. To verify that these components define a monad morphism, we use our usual notations 𝕊 = (S, ν, δ), 𝕋 = (T, μ, η) for the monads, and (N, n, d), (M, m, e) for the monoids. Moreover, we let the context differentiate between the induced structure morphisms \overline{α}, \overline{δ} and \overline{ρ} of C^{𝕋} or C^{𝕊}.

By using the definitions and properties of the involved morphisms, we compute

\overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}) · δ̃_{X} = \overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}) · (d · ϕ_{E} ⊛ 1_{SX}) · \overline{λ}_{SX}^{−1} · δ_{X}
= \overline{ϕ}_{M,T X} · (e ⊛ 1_{T X}) · (ϕ_{E} ⊛ 1_{T X}) · (1_{SE} ⊛ ϕ_{X}) · \overline{λ}_{SX}^{−1} · δ_{X}
= (e 1_{T X}) · \overline{ϕ}_{T E,T X} · (ϕ_{E} ⊛ 1_{T X}) · \overline{λ}_{T X}^{−1} · ϕ_{X} · δ_{X}
= (e 1_{T X}) · \overline{λ}_{T X}^{−1} · η_{X} = η̃_{X} .

For the multiplications, we use that

\overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}) · (n · \overline{ϕ}_{N,N} ⊛ 1_{SX}) · \overline{α}_{N,N,SX}^{−1}

= (m · (f f) 1_{T X}) · \overline{ϕ}_{N N,T X} · (\overline{ϕ}_{N,N} ⊛ 1_{T X}) · \overline{α}_{N,N,T X}^{−1} · (1_{N} ⊛ (1_{N} ⊛ ϕ_{X}))

= (m · (f f) 1_{T X}) · \overline{α}_{N,N,T X}^{−1} · \overline{ϕ}_{N,N T X} · (1_{N} ⊛ \overline{ϕ}_{N,T X}) · (1_{N} ⊛ (1_{N} ⊛ ϕ_{X}))

= (m 1_{T X}) · \overline{α}_{M,M,T X}^{−1} · \overline{ϕ}_{M,M T X} · (1_{M} ⊛ \overline{ϕ}_{M,T X}) · (f ⊛ (f ⊛ ϕ_{X}))

and since \overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}) : N ⊛ SX → M T X is a C^{𝕊}-morphism,

\overline{ϕ}_{M,M T X} · (1_{M} ⊛ \overline{ϕ}_{M,T X}) · (f ⊛ (f ⊛ ϕ_{X})) · (1_{N} ⊛ (ζ · ϕ_{N} ▷◁ ν_{X}))
= \overline{ϕ}_{M,M T X} · (f ⊛ ((ξ ▷◁ μ_{X}) · ϕ_{M⊛T X})) · (1_{N} ⊛ S(\overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X})))
= (1_{M} (ξ ▷◁ μ_{X})) · \overline{ϕ}_{M,T (M T X)} · (f ⊛ ϕ_{M T X}) · (1_{N} ⊛ S(\overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}))) .

Hence,

\overline{ϕ}_{M,T X}·(f ⊛ ϕ_{X}) · ν̃_{X}
= \overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}) · (n · \overline{ϕ}_{N,N} ⊛ 1_{SX}) · \overline{α}_{N,N,SX}^{−1} · (1_{N} ⊛ (ζ · ϕ_{N} ▷◁ ν_{X}))
= μ̃_{X} · \overline{ϕ}_{M,T (M T X)} · (f ⊛ ϕ_{M T X}) · (1_{N} ⊛ S(\overline{ϕ}_{M,T X} · (f ⊛ ϕ_{X}))) ,

so the components \overline{ϕ}_{M,T X}·(f ⊛ϕ_{X}) satisfy the two conditions for being a monad morphism.

3.4.3. Corollary. [Restriction of scalars] Let (C, ⊗, E) be a monoidal category, and (𝕋, κ) a monoidal monad on C. Suppose that C^{𝕋} has tensors that make (C^{𝕋}, , T E) a monoidal category with structure morphisms induced by those of (C, ⊗, E).

If f : N → M is a monoid homomorphism in C^{𝕋}, then the monad morphism (f, 1_{𝕋}) : N 𝕋 → M 𝕋 induces a functor C^{(f,1_{𝕋})} : (C^{𝕋})^{M} → (C^{𝕋})^{N}.

Proof. The monad morphism is the ϕ = 1_{𝕋} case of Proposition 3.4.2, and the result follows from the isomorphisms (C^{𝕊})^{N} ≅ C^{N⊛𝕊} and (C^{𝕋})^{M} ≅ C^{M 𝕋} (Theorem 3.3.3).

Acknowledgement. I would like to heartily thank Fred Linton that inspired this work by his insightful Remark in [11], and followed it with a kind and enlightening e-mail correspondence more than 40 years later [12]. Dirk Hofmann also helped spot an error in an earlier version of this work, and Martin Brandenburg provided useful comments and additional references: thanks to them, too.

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[8] A. Kock. Bilinearity and Cartesian closed monads. Math. Scand., 29:161–174, 1971.

[9] A. Kock. Closed categories generated by commutative monads. J. Austral. Math. Soc., 12:405–424, 1971.

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[11] F.E.J. Linton. Coequalizers in categories of algebras. In Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), pages 75–90. Springer, Berlin, 1969.

[12] F.E.J. Linton. Private communication, 2011.

[13] S. MacLane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer, New York, 1971. Second edition: 1998.

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