Theory and Applications of Categories, Vol. 28, No. 4, 2013, pp. 123–149.

SEMIUNITAL SEMIMONOIDAL CATEGORIES (APPLICATIONS TO SEMIRINGS AND SEMICORINGS)

JAWAD ABUHLAIL

Abstract. The category _{A}𝕊_{A} of bisemimodules over a semialgebra A, with the so called Takahashi’s tensor-like product − ⊠_{A} −, is semimonoidal but not monoidal. Although not a unit in _{A}𝕊_{A}, the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories (V, •, I) as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call 𝕁-monads (𝕁-comonads) with respect to the endo-functor 𝕁 := I•− ≃ −•I : V ⟶ V. This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endo-functors. Applications to the semiunital semimonoidal variety (_{A}𝕊_{A}, ⊠_{A}, A) provide us with examples of semiunital A-semirings (semicounital A-semicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules).

1. Introduction

A semiring is, roughly speaking, a ring not necessarily with subtraction. The first natural example of a semiring is the set ℕ_{0} of non-negative integers. Other examples include the set Ideal(R) of (two-sided) ideals of every associative ring R and distributive complete lattices. A semimodule is, roughly speaking, a module not necessarily with subtraction. The category of Abelian groups is nothing but the category of modules over ℤ; similarly, the category of commutative monoids is nothing but the category of semimodules over ℕ_{0}.

Semirings were studied by many algebraists beginning with Dedekind [Ded1984]. Since the sixties of the last century, they were shown to have significant applications in several areas as Automata Theory, Optimization Theory, Tropical Geometry and Idempotent Analysis (for more, see [Gol1999], Gal2002). Recently, Durov [Dur2007] demonstrated that semirings are in one-to-one correspondence with the algebraic additive monads on the category Set of sets. The theory of semimodules over semirings was developed by

The author would like to acknowledge the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through project No. IN100008.

Received by the editors 2012-09-13 and, in revised form, 2013-02-24. Transmitted by Ross Street. Published on 2013-02-26. 2010 Mathematics Subject Classification: 18C15, 18D10, 16W30. Key words and phrases: Semimonoidal Categories, Semiunits, Monads, Comonads, Semirings, Semi-

modules, Semicorings, Semicomodules.

\begin{matrix}c \\ ◯ \end{matrix} Jawad Abuhlail, 2013. Permission to copy for private use granted.

many authors including Takahashi, Patchkoria and Katsov (e.g. [Tak1982], [Tak1982], [Pat2006], [Kat1997]).

A strong connection between corings [Swe1975] over a ring A (coalgebras in the monoidal category _{A}Mod_{A} of bimodules over A) and their comodules on one side and comonads induced by the tensor product − ⊗_{A} − and their comodules on the other side has been realized by several authors (e.g. [BW2003]). Moreover, the theory of monads and comonads in (autonomous) monoidal categories received increasing attention in the last decade and extensions to arbitrary categories were carried out in several recent papers (e.g. [BW2009]).

Using the so called Takahashi’s tensor-like product − ⊠_{A} − of semimodules over an associative semiring A [Tak1982], notions of semiunital semirings and semicounital semicorings were introduced by the author in 2008. However, these could not be realized as monoids (comonoids) in the category _{A}𝕊_{A} of (A, A)-bisemimodules. This is mainly due to the fact that the category (_{A}𝕊_{A}, ⊠_{A}, A) is not monoidal in general (an alternative tensor product −⊗_{A}− was recalled by Katsov in [Kat1997]; in fact (_{A}𝕊_{A}, ⊗_{A}, A) is monoidal. For the relation between − ⊗_{A} − and − ⊠_{A} −, see [Abu]). Motivated by the desire to fix this defect, we introduce and investigate a notion of semiunital semimonoidal categories with prototype (_{A}𝕊_{A}, ⊠_{A}, A) and investigate semimonoids (semicomonoids) in such categories as well as their categories of semimodules (semicomodules). In particular, we realize our semiunital A-semirings (semicounital A-semicorings) as semimonoids (semicomonoids) in (_{A}𝕊_{A}, ⊠_{A}, A). Moreover, we introduce and study 𝕁-monads (𝕁-comonads) in an arbitrary category 𝔄, where 𝕁 : 𝔄 ⟶ 𝔄 is an endo-functor, and apply them to semiunital semimonoidal categories in general and to_{A}𝕊_{A} in particular. Our results extend recent ones on monoids (comonoids) in monoidal categories as well as monads (comonads) in arbitrary categories to semimonoids (semicomonoids) in semiunital semimonoidal categories as well as 𝕁-monads (𝕁-comonads) in arbitrary categories.

Throughout, 𝕀 denotes the identity endo-functor on the category under consideration. The paper is organized as follows. After this introduction, we present in Section 2 our (generalized) notion of 𝕁-monads and 𝕁-comonads in arbitrary categories. In Section 3, we introduce and investigate semiunits in semimonoidal categories. In Section 4, we introduce semimonoids (semicomonoids) in semiunital semimonoidal categories as well as their categories of semimodules (semicomodules). Moreover, we present two reconstruction results, namely Theorems 4.8 and 4.17. In Section 5, we consider the semiunital semimonoidal category (variety) of bisemimodules _{A}𝕊_{A} over a semialgebra A which provides us with a rich source of concrete examples for applying our results. As mentioned above, these concrete examples were the main motivation behind introducing all the abstract notions in this paper. Further investigations of 𝕁-bimonads and Hopf 𝕁-monads as well as bisemimonoids and Hopf semimonoids in semiunital semimonoidal categories will be the subject of a forthcoming paper.

2. Monads and Comonads

Recall first the so called Godement product of natural transformations between functors:

2.1. Let 𝔄, 𝔅, ℭ be arbitrary categories. Any natural transformations ψ : F ⟶ G and ψ′ : F′ ⟶ G′ of functors 𝔄 ⟶^{F,G} 𝔅 ⟶^{F′,G′} ℭ can be multiplied using the Godement product to yield a natural transformation ψ′ψ : F′F ⟶ G′G, where

ψ′_{G(X)} ∘ F′(ψ_{X}) = (ψ′ψ)_{X} = G′(ψ_{X}) ∘ ψ′_{F (X)} for every X ∈ 𝔄.  (1)

Moreover, if 𝔄 ⟶^{H} 𝔅 ⟶^{H′} ℭ are functors and ϕ : G ⟶ H, ϕ′ : G′ ⟶ H′ are natural transformations, then the following interchange law holds

(ϕ ∘ ψ)(ϕ′ ∘ ψ′) = (ϕ′ϕ) ∘ (ψ′ψ).  (2)

2.2. Let 𝔄 and 𝔅 be categories, L : 𝔄 ⟶ 𝔅, R : 𝔅 ⟶ 𝔄 be functors and 𝕁 : 𝔄 ⟶ 𝔄, 𝕂 : 𝔅 ⟶ 𝔅 be endo-functors such that R𝕂 ≃ 𝕁R and L𝕁 ≃ 𝕂L. We say that (L, R) is a (𝕁, 𝕂)-adjoint pair iff we have natural isomorphisms in X ∈ 𝔄 and Y ∈ 𝔅 :

𝔅(L𝕁(X), 𝕂(Y)) ≃ 𝔄(𝕁(X), R𝕂(Y)).

For the special case 𝕁 = 𝕀_{𝔄} and 𝕂 = 𝕀_{𝔅}, we recover the classical notion of adjoint pairs. Till the end of this section, 𝔄 is an arbitrary category.

2.3. Let 𝕋 : 𝔄 ⟶ 𝔄 be an endo-functor. An object X ∈ Obj(𝔄) is said to have a 𝕋-action or to be a 𝕋-act iff there is a morphism ϱ_{X} : 𝕋(X) ⟶ X in 𝔄. For two objects X, X′ with 𝕋-actions, we say that a morphism φ : X ⟶ X′ in 𝔄 is a morphism of 𝕋-acts iff the following diagram is commutative

\xymatrix{𝕋(X) \ar[d]_{𝕋(φ)} \ar[r]^{ϱ_{X}} & X \ar[d]^{φ} \\ 𝕋(X′) \ar[r]_{ϱ_{X′}} & X′}

The category of 𝕋-acts is denoted by Act_{𝕋}. Dually, one can define the category Coact^{𝕋} of 𝕋-coacts.

2.4. Remark. The objects of Coact^{𝔽}, where 𝔽 : Set ⟶ Set is an endo-functor, play an important role in logic and theoretical computer science. They are called 𝔽-systems (e.g. [Rut2000]). Some references call these 𝔽-coalgebras (e.g. [Gum1999]). For us, coalgebras are always coassociative and counital unless something else is explicitly specified.

𝕁-Monads.

2.5. Let 𝕁 : 𝔄 ⟶ 𝔄 be an endo-functor. With a 𝕁-monad on 𝔄 we mean a datum (𝕄, μ, ω, ν; 𝕁) consisting of an endo-functor 𝕄 : 𝔄 ⟶ 𝔄 associated with natural transformations

μ : 𝕄𝕄 ⟶ 𝕄, ω : 𝕀 ⟶ 𝕁 and ν : 𝕁 ⟶ 𝕄

such that the following diagrams are commutative

[figure: 𝕄𝕄𝕄; μ𝕄; 𝕄μ; 𝕄𝕄; μ; 𝕄𝕄; μ; μ; 𝕄; 𝕄𝕄; μ; 𝕀𝕄𝕄; ω; 𝕁𝕄; ν𝕄; 𝕁𝕄; 𝕄𝕄; μ; 𝕄𝕀; 𝕄ω; 𝕄𝕁; 𝕄ν; 𝕄𝕁]

i.e. for every X ∈ 𝔄 we have

μ_{X} ∘𝕄(μ_{X}) = μ_{X} ∘μ_{𝕄(X)}, ν_{𝕄(X)}∘ω_{𝕄(X)}∘μ_{X} = 𝕀_{𝕄𝕄(X)} and 𝕄(ν_{X})∘𝕄(ω_{X})∘μ_{X} = 𝕀_{𝕄𝕄(X)}.

2.6. With JMonad_{𝔄} we denote the category whose objects are 𝕁-monads, where 𝕁 runs over the class of endo-functors on 𝔄. A morphism (φ; ξ) : (𝕄, μ, ω, ν; 𝕁) ⟶ (𝕄′, μ′, ω′, ν′; 𝕁′) in this category consists of natural transformations φ : 𝕄 ⟶ 𝕄′ and ξ : 𝕁 ⟶ 𝕁′ such that the following diagrams are commutative

[figure: 𝕄𝕄; φφ; μ; 𝕄; φ; 𝕄′𝕄′; μ′; 𝕄′; 𝕀; ω; 𝕁; ξ; ν; 𝕄; φ; 𝕀; ω′; 𝕁′; ν′; 𝕄′]

i.e. for every X ∈ 𝔄 we have

φ_{X} ∘ μ_{X} = μ′_{X} ∘ φ_{𝕄′(X)} ∘ 𝕄(φ_{X}), ξ_{X} ∘ ω_{X} = ω′_{X} and φ_{X} ∘ ν_{X} = ν′_{X} ∘ ξ_{X}.

For a fixed endo-functor 𝕁 : 𝔄 ⟶ 𝔄, we denote by 𝕁-Monad_{𝔄} the subcategory of JMonad_{𝔄} of 𝕁-monads on 𝔄 with ω the identity natural transformation. In the special case 𝕁 = 𝕀_{𝔄} and ω is the identity natural transformation, we drop these from our notation and recover the classical notion of monads on 𝔄.

2.7. Remark. As we saw above, a 𝕁-monad (𝕄, μ, ω, ν; 𝕁) is a generalized notion of a monad. However, it can also be seen as just a monad (𝕄, μ, η) whose unit η := 𝕀 ⟶^{ω} 𝕁 ⟶^{ν} 𝕄 factorizes through 𝕁. Having this in mind, a morphism (φ; ξ) : (𝕄, μ, ω, ν; 𝕁) ⟶ (𝕄′, μ′, ω′, ν′; 𝕁′) in JMonad_{𝔄} is just a morphism of monads which is compatible with the factorizations of the units through 𝕁 and 𝕁′.

2.8. Let (𝕄, μ, ω, ν; 𝕁) ∈ JMonad_{𝔄}. An (𝕄; 𝕁)-module is an object X ∈ Obj(𝔄) with a morphism ϱ_{X} : 𝕄(X) ⟶ X in 𝔄 such that the following diagrams are commutative

[figure: 𝕄𝕄(X); μ_{X}; 𝕄(ϱ_{X}); 𝕄(X); ϱ_{X}; 𝕄(X); ϱ_{X}; X; 𝕄(X); ϱ_{X}; X; ω_{X}; 𝕁(X); ν_{X}; 𝕁(X)]

The category of (𝕄; 𝕁)-modules and morphisms those of 𝕄-acts is denoted by 𝔄_{(𝕄;𝕁)}. In case 𝕁 ≃ 𝕀_{𝔄} and ω is the identity natural transformation, we recover the category of 𝕄-modules of the monad 𝕄.

2.9. Let (𝕄, μ, ω, ν; 𝕁) ∈ JMonad_{𝔄}. For every X ∈ Obj(𝔸), 𝕄(X) is an (𝕄; 𝕁)-module through

ϱ_{𝕄(X)} : 𝕄(𝕄(X)) ⟶^{μ_{X}} 𝕄(X).

Such object are called free (𝕄; 𝕁)-modules and we have the so called free functor

F_{(𝕄;𝕁)} : 𝔄 ⟶ 𝔄_{(𝕄;𝕁)}, X ↦ 𝕄(X).

The full subcategory of free (𝕄; 𝕁)-modules is called the Kleisli category and is denoted by 𝔄̃_{(𝕄;𝕁)}.

2.10. Remark. Let (𝕄, μ, ω, ν; 𝕁) ∈ JMonad_{𝔄} with 𝕄𝕁 ≃ 𝕁𝕄. If X is an (𝕄; 𝕁)- module, then 𝕁(X) is also an (𝕄; 𝕁)-module through

ϱ_{𝕁(X)} : 𝕄𝕁(X) ≃ 𝕁𝕄(X) ⟶^{𝕁(ϱ_{X})} 𝕁(X).

Moreover, if Y = 𝕄(X) is a free (𝕄; 𝕁)-module, then 𝕁(Y) = 𝕁𝕄(X) ≃ 𝕄𝕁(X) is also a free (𝕄; 𝕁)-module. One can easily see that 𝕁 can be lifted to endo-functors 𝕁′ : 𝔄_{(𝕄;𝕁)} ⟶ 𝔄_{(𝕄;𝕁)} and 𝕁̃: 𝔄̃_{(𝕄;𝕁)} ⟶ 𝔄̃_{(𝕄;𝕁)}.

2.11. Let (𝕄, μ, ω, ν; 𝕁) ∈ JMonad_{𝔄} and assume that 𝕄𝕁 ≃ 𝕁𝕄. We have a natural isomorphism for every X ∈ 𝔄 and Y ∈ 𝔄_{(𝕄;𝕁)} :

𝔄_{(𝕄;𝕁)}(F_{(𝕄;𝕁)}(X), 𝕁(Y)) ≃ 𝔄(X, 𝕁(Y)), f ↦ f ∘ (ν ∘ ω)_{X}

with inverse g ⟼ ϱ_{𝕁(Y)} ∘ F_{(𝕄;𝕁)}(g). Consider the forgetful functor U : 𝔄_{(𝕄;𝕁)} ⟶ 𝔄 and the endo-functor 𝕁′ : 𝔄_{(𝕄;𝕁)} ⟶ 𝔄_{(𝕄;𝕁)} (see Remark 2.10). We have a natural isomorphism

𝔄_{(𝕄;𝕁)}(F_{(𝕄;𝕁)}(𝕁(X)), 𝕁′(Y)) ≃ 𝔄(𝕁(X), U(𝕁′(Y)));

i.e. (F_{(𝕄;𝕁)}(−), U) is a (𝕁, 𝕁′)-adjoint pair.

𝕁-Comonads.

2.12. Let 𝕁 be an endo-functor on 𝔄. With a 𝕁-comonad on 𝔄 we mean a datum (ℂ, Δ, ω, θ) consisting of an endo-functor ℂ : 𝔄 ⟶ 𝔄 associated with natural transformations

Δ : ℂ ⟶ ℂℂ, ω : 𝕀 ⟶ 𝕁 and θ : ℂ ⟶ 𝕁

such that the following diagrams are commutative

[figure: ℂ; Δ; Δ; ℂℂ; Δℂ; ℂℂ; ℂΔ; ℂℂℂ; 𝕀ℂ; ωℂ; Δ; ℂℂ; θℂ; 𝕁ℂ; 𝕁ℂ; ℂ𝕀; ℂω; Δ; ℂℂ; ℂθ; ℂ𝕁; ℂ𝕁]

i.e. for every X ∈ 𝔄 we have

Δ_{ℂ(X)} ∘ Δ_{X} = ℂ(Δ_{X}) ∘ Δ_{X}, θ_{ℂ(X)} ∘ Δ_{X} = ω_{ℂ(X)} and ℂ(θ_{X}) ∘ Δ_{X} = ℂ(ω_{X}).

2.13. By JComonad_{𝔄} we denote the category whose objects are 𝕁-comonads, where 𝕁 runs over the class of endo-functors on 𝔄. A morphism (ψ; ξ) : (ℂ, Δ, ω, θ; 𝕁) ⟶ (ℂ′, Δ′, ω′, θ′; 𝕁′) in this category consists of natural transformations ψ : ℂ ⟶ ℂ′ and ξ : 𝕁 ⟶ 𝕁′ such that the following diagrams are commutative

[figure: ℂ; ψ; Δ; ℂℂ; ψψ; ℂ′; Δ′; ℂ′ℂ′; ℂ; θ; ψ; 𝕁; ξ; 𝕀; ω; ℂ′; θ′; 𝕁′; 𝕀; ω′]

i.e. for every X ∈ 𝔄 we have

ψ_{ℂ′(X)} ∘ ℂ(ψ_{X}) ∘ Δ_{X} = Δ′_{X} ∘ ψ_{X}, ξ_{X} ∘ θ_{X} = θ′_{X} ∘ ψ_{X} and ξ_{X} ∘ ω_{X} = ω′_{X}.

For a fixed endo-functor 𝕁 : 𝔄 ⟶ 𝔄, we denote by 𝕁-Comonad_{𝔄} the subcategory of 𝕁-comonads on 𝔄 with ω the identity transformation. In the special case 𝕁 = 𝕀_{𝔄} and ω is the identity natural transformation, we drop these from our notation and recover the notion of comonads on 𝔄.

2.14. Remark. 𝕁-Comonads are not fully dual to 𝕁-monads. Recall from Remark 2.7 that a 𝕁-monad can be seen as a monad whose unit factorizes through 𝕁. On the other hand, 𝕁-comonads cannot be seen as a special type of comonads. The lack of duality is because not all arrows are reversed; the arrow ω : 𝕀 ⟶ 𝕁 is assumed for both. Notice that keeping this arrow is suggested by the concrete example in Section 5.

2.15. Let (ℂ, Δ, ω, θ; 𝕁) ∈ JComonad_{𝔄}. A (ℂ; 𝕁)-comodule is an object X ∈ Obj(𝔄) along with a morphism ϱ^{X} : X ⟶ ℂ(X) in 𝔄 such that the following diagrams are commutative

[figure: X; ϱ^{X}; ϱ^{X}; ℂ(X); ℂ(ϱ^{X}); ℂ(X); Δ_{X}; ℂℂ(X); X; ω_{X}; ϱ^{X}; ℂ(X); θ_{X}; 𝕁(X); 𝕁(X)]

The category of (ℂ; 𝕁)-comodules and morphisms those of ℂ-coacts is denoted by 𝔄^{(ℂ;𝕁)}. In case 𝕁 = 𝕀_{𝔄} and ω is the identity natural transformation, we recover the category of ℂ-comodules for the comonad ℂ.

2.16. Let (ℂ, Δ, ε; 𝕁) ∈ JComonad_{𝔄}. For every X ∈ Obj(𝔄), ℂ(X) has a canonical structure of a (ℂ; 𝕁)-comodule through

ϱ^{ℂ(X)} : ℂ(X) ⟶^{Δ_{X}} ℂℂ(X).

Such object are called cofree (ℂ; 𝕁)-comodules and we have the so called cofree functor

F^{ℂ} : 𝔄 ⟶ 𝔄^{(ℂ;𝕁)}, X ↦ ℂ(X).

The full subcategory of cofree (ℂ; 𝕁)-comodules is called the Kleisli category of ℂ and is denoted by 𝔄̃^{(ℂ;𝕁)}.

2.17. Remark. Let (ℂ, Δ, ω, θ; 𝕁) ∈ JComonad_{𝔄} with 𝕁ℂ ≃ ℂ𝕁. If X is a (ℂ; 𝕁)- comodule, then 𝕁(X) is also a (ℂ; 𝕁)-comodule through

ϱ^{𝕁(X)} : 𝕁(X) ⟶^{𝕁(ϱ^{X})} 𝕁ℂ(X) ≃ ℂ(𝕁(X)).

If Y = ℂ(X) is a cofree (ℂ; 𝕁)-comodule, then 𝕁(Y) = 𝕁ℂ(X) ≃ ℂ𝕁(X) is also a cofree (ℂ, 𝕁)-comodule. One case easily see that 𝕁 lifts to endo-functors 𝕁′ : 𝔄^{(ℂ;𝕁)} ⟶ 𝔄^{(ℂ;𝕁)} and 𝕁̃: 𝔄̃^{(ℂ;𝕁)} ⟶ 𝔄̃^{(ℂ;𝕁)}.

2.18. Let (ℂ, Δ, ω, θ; 𝕁) ∈ JComonad_{𝔄} with 𝕁 idempotent and 𝕁ℂ ≃ ℂ𝕁. Consider the forgetful functor U : 𝔄^{(ℂ;𝕁)} ⟶ 𝔄 and the endo-functor 𝕁′ : 𝔄^{(ℂ;𝕁)} ⟶ 𝔄^{(ℂ;𝕁)}. We have a natural isomorphism for X ∈ 𝔄 and Y ∈ 𝔄^{(ℂ;𝕁)} :

𝔄^{(ℂ;𝕁)}(𝕁′(Y), F^{ℂ}(𝕁(X))) ≃ 𝔄(U(𝕁′(Y)), 𝕁(X)), f ↦ θ_{𝕁(X)} ∘ f

with inverse g ⟼ F^{ℂ}(g) ∘ ϱ^{𝕁(Y)}; i.e. (U, F^{(ℂ;𝕁)}(−)) is a (𝕁′, 𝕁)-adjoint pair.

2.19. Proposition. Let 𝔄 and 𝔅 be categories, L : 𝔄 ⟶ 𝔅, R : 𝔅 ⟶ 𝔄 be functors and 𝕁 : 𝔄 ⟶ 𝔄, 𝕂 : 𝔅 ⟶ 𝔅 endo-functors such that L𝕁 ≃ 𝕂L, 𝕁R ≃ R𝕂 and (L, R) is a (𝕁, 𝕂)-adjoint pair.

1. (L, R) is an adjoint pair where L : 𝕁(𝔄) ⟶^{L} 𝕂(𝔅) and R : 𝕂(𝔅) ⟶^{R} 𝕁(𝔄) with unit and counit of adjunction given by

η : 𝕁 ⟶ RL𝕁 and ε : LR𝕂 ⟶ 𝕂.

2. RL is a monad on 𝕁(𝔄) with

μ_{RL} : (RL)(RL)𝕁 ≃ R(LR𝕂)L ⟶^{RεL} R𝕂L ≃ (R)L𝕁 and η_{RL} := η.

3. LR is a comonad on 𝕂(𝔅) with

Δ_{LR} : (LR)𝕂 ≃ L𝕁R ⟶^{LηR} L(RL𝕁)R ≃ (LR)(LR)𝕂 and ε_{LR} := ε.

4. L is a monad on 𝕁(𝔄) if and only if R is a comonad on 𝕂(𝔅). In this case, 𝕁(𝔄)_{L} ≃ 𝕂(𝔅)^{R}.

5. L is a comonad on 𝕁(𝔄) if and only if R is a monad on 𝕂(𝔅). In this case, 𝕁̃(𝔄)^{L} ≃

{ \atop 𝕂(𝔅)}_{R}.

Proof. By assumption, L𝕁 ≃ 𝕂L whence L(𝕁(𝔄)) := L𝕁(𝔄) = 𝕂L(𝔄) ⊆ 𝕂(𝔅) and 𝕁R ≃ R𝕂 whence R(𝕂(𝔅)) := R(𝕂(𝔅)) = 𝕁R(𝔅) ⊆ 𝕁(𝔄). The assumptions imply that (L, R) is an adjoint pair. The result follows now from the classical result on right adjoint pairs (e.g. [EM1965, Proposition 3.1], [BW2009, 2.5, 2.6]).

3. Semiunital Semimonoidal Categories

A semimonoidal category is roughly speaking a monoidal category not necessarily with a unit object. The reader might consult the literature for the precise definitions and for the notions of (op)-semimonoidal functors between such categories. In this section, we introduce a notion of semiunital semimonoidal categories and semiunital (op-)semimonoidal functors.

Semiunits.

3.1. Let (V, •) be a semimonoidal category with natural isomorphisms γ_{X,Y,Z} : (X • Y) • Z ⟶ X • (Y • Z) for all X, Y, Z ∈ V. We say that I ∈ V is a semiunit iff

1. there is a natural transformation ω : 𝕀 ⟶ (I • −);

2. there exists an isomorphisms of functors I • − ≃ − • I, i.e. there is a natural isomorphism I • X ≃^{ℓ_{X}} X • I in V with inverse ℘_{X}, for each object X of V, such that ℓ_{I} = ℘_{I} and the following diagram is commutative for all X, Y ∈ V :

\xymatrix{(I • X) • Y \ar[r]^{γ_{I,X,Y}} \ar[d]_{ℓ_{X}•Y} & I • (X • Y) \ar[r]^{ℓ_{X•Y}} & (X • Y) • I \ar[d]^{γ_{X,Y,I}} \\ (X • I) • Y \ar[r]_{γ_{X,I,Y}} & X • (I • Y) \ar[r]_{X•ℓ_{Y}} & X • (Y • I)}

3. the following diagram is commutative for all X, Y ∈ V :

\xymatrix{(I • X) • Y \ar[dr]_{≃} & X • Y \ar[l]_{ω_{X}•Y} \ar[d]_{ω_{X•Y}} \ar[r]^{X•ω_{Y}} & X • (I • Y) \ar[dl]^{≃} \\  & I • (X • Y)}

3.2. If X ≃^{ω_{X}} I•X (≃^{ℓ_{X}} X •I), then we say that X is firm and set λ_{X} := ω_{X}^{−1} : I•X ⟶ X

and κ_{X} : X • I ≃^{℘_{X}} I • X ⟶^{ω_{X}^{−1}} X. With V^{firm} we denote the full subcategory of firm objects in V. If I is firm (called also pseudo-idempotent) and ω_{I}^{−1} • I = I • ω_{I}^{−1}, then one says that I is idempotent [Koc2008].

3.3. Remark. Let (V, •; γ) be a semimonoidal category. One says that V is monoidal [Mac1998] iff V has a unit (or an LR unit), i.e. a distinguished object I ∈ V with natural

isomorphisms I • X ≃^{λ_{X}} X and X • I ≃^{κ_{X}} X such that X • λ_{Y} = κ_{X} • Y for all X, Y ∈ V (equivalently, λ_{I} = κ_{I}, λ_{X•Y} = λ_{X} • Y and κ_{X•Y} = X • κ_{Y} for all X, Y ∈ V). Kock [Koc2008] called an object I ∈ V a Saavedra unit – called also a reduced unit – iff it is pseudo-idempotent and cancellable in the sense that the endo-functors I • − and − • I are full and faithful (equivalently, I is idempotent and the endo-functors I • − and − • I are equivalences of categories). Moreover, he showed that I is a unit if and only if I is a Saavedra unit. Indeed, every unit is a semiunit, whence our notion of semiunital semimonoidal categories generalizes the classical notion of monoidal categories.

3.4. Let (V, •, I_{V}; ω_{V}) and (W, ⊗, I_{W}; ω_{W}) be semiunital semimonoidal categories. A semimonoidal functor F : V ⟶ W, with a natural transformation ϕ : F (−) ⊗ F (−) ⟶ F (− • −), is said to be semiunital semimonoidal iff there exists a coherence morphism ϕ̃: I_{W} ⟶ F (I_{V}) in W such that the following diagram is commutative

\xymatrix{I_{W} ⊗ F (X) \ar[r]^{ℓ_{F (X)}} \ar[d]_{ϕ̃⊗F (X)} & F (X) ⊗ I_{W} \ar[d]^{F (X)⊗ϕ̃} \\ F (I_{V}) ⊗ F (X) \ar[d]_{ϕ_{I_{V} ,X}} & F (X) ⊗ F (I_{V}) \ar[d]^{ϕ_{X,I_{V}}} \\ F (I_{V} • X) \ar[r]_{F (ℓ_{X})} & F (X • I_{V})}

Moreover, we say that F is a strong (strict) semiunital semimonoidal functor iff F is strong (strict) as a semimonoidal functor and ϕ is an isomorphism (identity). For two semimonoidal functors F, F′ : V ⟶ W, we say that a semimonoidal natural transformation ς : F ⟶ F′ is semiunital semimonoidal iff the following diagram is commutative

\xymatrix{ & I_{W} \ar[dl]_{ϕ̃} \ar[dr]^{ϕ′} \\ F (I_{V}) \ar[rr]_{ς_{I_{V}}} &  & F′(I_{V})}

One can dually define semiunital (strong, strict) op-semimonoidal functors and semiunital natural transformations between them.

3.5. Remark. Let (V, •, I; ω) be a semiunital semimonoidal category and consider the functor

𝕁 := I • − : V ⟶ V.

1. We have natural isomorphisms

𝕁(I) • X ≃ 𝕁(𝕁(X)) ≃ X • 𝕁(I) and 𝕁(X) • Y ≃ X • 𝕁(Y)

for all X, Y ∈ V.

2. 𝕁 is op-semimonoidal with natural transformation ψ_{X,Y} : 𝕁(− • −) ⟶ 𝕁(−) • 𝕁(−) is given by the composition of morphisms

ψ_{X,Y} : I • (X • Y) ⟶^{ω_{I}•(X•Y)} (I • I) • (X • Y) ≃ (I • X) • (I • Y)

for all X, Y ∈ V.

3. Assume that I is firm.

(a) 𝕁 is strong semiunital semimonoidal with

ϕ_{X,Y} : (I•X)•(I•Y) ≃ (I•I)•(X•Y) ≃^{ω_{I}^{−1}•(X•Y)} I•(X•Y) and ϕ̃:= ω_{I} : I ⟶ I•I

for all X, Y ∈ V.

(b) 𝕁 is strong semiunital op-semimonoidal with

ψ̃ := ω_{I}^{−1} : I • I ⟶ I.

(c) the full subcategory (V^{firm}, •, I) is monoidal.

(d) (𝕁(V), •, I) is a monoidal full subcategory of (V^{firm}, •, I) with

λ_{I•X} : I • (I • X) ≃^{γ_{I,I,X}^{−1}} (I • I) • X ≃^{ω_{I}^{−1}} I • X;
κ_{I•X} : (I • X) • I ≃^{γ_{I,X,I}} I • (X • I) ≃^{I•℘_{X}} I • (I • X) ≃^{λ_{I•X}} I • X

for every X ∈ V.

3.6. Definition. Let (V, •, I; ω) be a semiunital semimonoidal category. We say that V ∈ V has a left dual iff there exists V_{l}^{⊛} ∈ V along with morphisms f : I ⟶ I • V • V_{l}^{⊛} and g : I • V_{l}^{⊛} • V ⟶ I in V such that

(V • g) ∘ (ℓ_{V} • V_{l}^{⊛} • V) ∘ (f • V) = ℓ_{V} and (g • V_{l}^{⊛}) ∘ (℘_{V^{⊛}} • V • V_{l}^{⊛}) ∘ (V_{l}^{⊛} • f) = ℘_{V^{⊛}}.

A right dual V_{r}^{⊛} of V is defined symmetrically. We say that V is left ( right) autonomous, or left ( right) rigid iff every object in V has a left (right) dual.

3.7. Definition. Let (V, •, I; ω) be a semiunital semimonoidal category. We say that V is right (left) closed iff for every V ∈ V, the functor − • V : 𝕁(V) ⟶ 𝕁(V) (V • − : 𝕁(V) ⟶ 𝕁(V)) has a right-adjoint, i.e. there exists a functor G : 𝕁(V) ⟶ 𝕁(V) and a natural isomorphism for every pair of objects X, Y ∈ V :

V(X • I • V, Y • I) ≃ V(X • I, G(Y • I)) (resp. V(V • I • X, Y • I) ≃ V(X • I, G(Y • I))).

Moreover, V is said to be closed iff V is left and right closed.

3.8. Lemma. Let (V, •, I; ω) be a semiunital semimonoidal category. If V ∈ V has a left (right) dual V^{⊛}, then (− • V, − • V^{⊛}) ((V • −, V^{⊛} • −)) is a (𝕁, 𝕁)-adjoint pair. In particular, if V is left (right) autonomous, then V is right (left) closed.

Proof. Assume that V ∈ V has a left dual V^{⊛}. For all X, Y ∈ V we have a natural isomorphism

V(X • I • V, Y • I) ≃ V(X • I, Y • I • V^{⊛}), f ⟼ (f • V^{⊛}) ∘ (X • υ)  (3)

with inverse g ⟼ (Y • ϖ) ∘ (g • V).

4. Semimonoids and Semicomonoids

In this section, we introduce notions of semimonoids and semicomonoids in semiunital semimonoidal categories. Throughout, (V, •, I; ω) is a semiunital semimonoidal category, where I is a semiunit, ω : 𝕀 ⟶ 𝕁 is a natural transformation between the identity functor and the endo-functor 𝕁 := I•− ≃ −•I : V ⟶ V and γ_{X,Y,Z} : (X•Y)•Z ⟶ X•(Y •Z) are natural isomorphisms for all X, Y, Z ∈ V (we assume the existence of natural isomorphisms

I • X ≃^{ℓ_{X}} X • I with inverse X • I ⟶^{℘_{X}:=ℓ_{X}^{−1}} I • X for every X ∈ V).

Semimonoids.

4.1. A V-semimonoid consists of a datum (A, ζ, ϖ), where A ∈ V and ζ : A • A ⟶ A, ϖ : I ⟶ A are morphisms in V such that the following diagrams are commutative

[figure: A • A • A; ζ•A; A•ζ; A • A; ζ; A • A; ζ; A; A • A; ζ; A; ω_{A}; A • A; ζ; I • A; ϖ•A; I • A; ℓ_{A}; A • I; A•ϖ]

If A ≃^{ω_{A}} I•A, then we say that A is a unital V-semimonoid. A morphism of V-semimonoids f : (A, ζ, ϖ) ⟶ (A′, ζ′, ϖ′) is a morphism in V such that the following diagrams are commutative

[figure: A • A; ζ; f•f; A; f; A′ • A′; ζ′; A′; I; ϖ; A; f; I; ϖ′; A′]

The category of V-semimonoids is denoted by SMonoid(V); the full subcategory of unital V-semimonoids is denoted by USMonoid(V).

4.2. Let (A, ζ, ϖ) be a V-semimonoid. A right A-semimodule is a datum (M, ρ_{M}) where M ∈ V and ρ_{M} : M • A ⟶ M is a morphism in V such that the following diagrams are commutative

\xymatrix{M • A • A \ar[r]^{ρ_{M} •A} \ar[d]_{M•ζ} & M • A \ar[d]^{ρ_{M}} & M • A \ar[r]^{ρ_{M}} & M \ar[d]^{ω_{M}} \\ M • A \ar[r]_{ρ_{M}} & M & M • I \ar[u]^{M•ϖ} & I • M \ar[l]^{ℓ_{M}}}

If M ≃^{ω_{M}} I • M, then we say that M is a unitary right A-semimodule. A morphism of right A-semimodules is a morphism f : M ⟶ M′ in V such that the following diagram is commutative

\xymatrix{M • A \ar[r]^{ρ_{M}} \ar[d]_{f•A} & M \ar[d]^{f} \\ M′ • A \ar[r]_{ρ_{M′}} & M′}

The category of right A-semimodules is denoted by S_{A}; the full subcategory of unitary right A-semimodules is denoted by US_{A}. Analogously, one can define the category _{A}S of left A-semimodules and its full subcategory _{A}US of unitary left A-semimodules.

4.3. Example. If I ≃^{ω_{I}} I • I, then I is a unital V-semimonoid with

ζ_{I} : I • I ⟶^{ω_{I}^{−1}} I and ϖ_{I} : I ⟶^{id} I.

Moreover, every M ∈ V^{firm} is a unitary (I, I)-bisemimodule with ρ_{M}^{l} : I • M ⟶^{ω_{M}^{−1}} M and

ρ_{M}^{r} : M • I ⟶^{ω_{M}^{−1}∘℘_{M}} M.

4.4. Let A be a V-semimonoid and M a right A-semimodule. We have a functor

− • M : V ⟶ S_{A},

where for all X ∈ V we have a structure of a right A-semimodule on X • M given by

ρ_{X•M} : (X • M) • A ⟶^{γ_{X,M,A}} X • (M • A) ⟶ X • M.

Similarly, if M is a left A-semimodule, then we have a functor M • − : V ⟶ _{A}S.

4.5. Let A and B be V-semimonoids. Let M be a left B-semimodule as well as a right A-semimodule and consider B • M ∈ S_{A} and M • A ∈ _{B}M. We say that M is a (B, A)- bisemimodule iff ρ_{(M;B)} : B • M ⟶ M is a morphism in S_{A}, or equivalently iff ρ_{(M;A)} : M • A ⟶ M is a morphism in _{B}S. The category of (unitary) (B, A)-bisemimodules with morphisms being in _{B}S ∩ S_{A} is denoted by _{B}S_{A} (_{B}US_{A}). Indeed, every (unital) V-semimonoid A is a (unitary) (A, A)-bisemimodule in a canonical way.

4.6. Proposition. Every semiunital semimonoidal functor F : (V, •, I_{V}) ⟶ (W, ⊗, I_{W}) lifts to a functor

F̃ : SMonoid(V) ⟶ SMonoid(W), A ⟼ F (A)

that commutes with the forgetful functors

U_{V} : SMon(V) ⟶ V and U_{W} : SMon(W) ⟶ W.

Proof. Let (A, ζ_{A}, ϖ_{A}) be a semimonoid in V and consider B := F (A). Define

ζ_{B} : F (A) ⊗ F (A) ⟶^{ϕ_{A,A}} F (A • A) ⟶^{F (ζ_{A})} F (A);
ϖ_{B} : I_{W} ⟶^{ϕ̃} F (I_{V}) ⟶^{F (ϖ_{A})} F (A).

One checks easily that (B, ζ_{B}, ϖ_{B}) is a semimonoid in W. If f : A ⟶ A′ is a morphism of V-semimonoids, then examining the involved diagrams shows that F (f) : F (A) ⟶ F (A′) is a morphism of W-semimonoids. Finally, it is clear that U_{W} ∘ F̃ = F ∘ U_{V}.

4.7. Proposition. Let (A, ζ, ϖ) be a V-semimonoid.

1. We have 𝕁-monads

− • A : V ⟶ V and A • − : V ⟶ V

and isomorphisms of categories

S_{A} ≃ V_{(−•A;𝕁)} and _{A}S ≃ V_{(A•−;𝕁)}.

2. If B is a V-semimonoid, then we have 𝕁-monads

− • A : _{B}S ⟶ _{B}S and B • − : S_{A} ⟶ S_{A}

and isomorphisms of categories

(_{B}S)_{(−•A;𝕁)} ≃ _{B}S_{A} ≃ (S_{A})_{(B•−;𝕁)}.

Proof. Consider the natural transformations

μ : (− • A) • A ⟶ − • A, μ_{X} : (X • A) • A ≃^{γ_{X,A,A}} X • (A • A) ⟶^{X•μ} X • A,
ν : 𝕁 ⟶ − • A, ν_{X} : I • X ≃^{ℓ_{X}} X • I ⟶^{X•ϖ} X • A.

One can easily check that (− • A, μ, ω, ν) is a 𝕁-monad. The isomorphism S_{A} ≃ V_{(−•A;𝕁)} follows immediately from comparing the corresponding diagrams. The other assertions can also be checked easily.

An object G in a cocomplete category 𝔄 is said to be a (regular) generator iff for every X ∈ 𝔄, there exists a canonical (regular) epimorphism f_{X} : ⨆_{f∈𝔄(G,X)} G ⟶ X

[BW2005, p. 199] (see also [Kel2005], [Ver]); recall that an arrow in 𝔄 is said to be a regular epimorphism iff it is a coequalizer (of its kernel pair).

4.8. Theorem. Let V be cocomplete, I and A ∈ V be firm and assume that I is a regular generator in V and that both A • − and − • A preserve colimits in V. There is a bijective correspondence between the structures of unital semimonoids on A, the structures of 𝕁monads on − • A and the structures of 𝕁-monads on A • −.

Proof. Assume that (− • A, μ, ω, ν) is a 𝕁-monad and consider

μ : A • A ⟶^{ω_{A}•A} I • A • A ⟶^{μ_{I}} I • A ≃^{λ_{A}} A;
ϖ : I ⟶^{ω_{I}} I • I ⟶^{ν_{I}} I • A ≃^{λ_{A}} A.

Clearly, (A, μ, ϖ) is a (unital) semimonoid. The converse follow by Proposition 4.7. The proof of the bijective correspondence is similar to that in the proof of [Ver, Theorem 3.9]. The statement corresponding to the endo-functor A • − can be proved analogously.

Semicomonoids.

4.9. A V-semicomonoid is a datum (C, δ, ϵ) where C ∈ V, δ : C ⟶ C • C, ϵ : C ⟶ I are morphisms in V such that the following diagrams are commutative

[figure: C; δ; δ; C • C; δ•C; C • C; C•δ; C • C • C; C • C; ϵ•C; C; ω_{C}; δ; δ; C • C; C•ϵ; I • C; I • C; ℓ_{C}; C • I]

If C ≃^{ω_{C}} I • C, then we say that C is a counital V-semicomonoid. A morphism of Vsemicomonoids f : (C, δ, ϵ) ⟶ (C′, δ′, ϵ′) is a morphism in V such that the following diagrams are commutative

[figure: C; δ; f; C • C; f•f; C′; δ′; C′ • C′; C; f; ϵ; I; C′; ϵ′; I]

The category of V-semicomonoids is denoted by SComonoid(V); the full subcategory of counital V-semicomonoids is denoted by CSComonoid(V).

4.10. Let (C, δ, ϵ) be a V-semicomonoid. A right C-semicomodule is a datum (M, ρ^{M}) where M ∈ V and ρ^{M} : M ⟶ M • C is a morphisms in V such that the following diagrams are commutative

\xymatrix{M \ar[r]^{ρ^{M}} \ar[d]_{ρ^{M}} & M • C \ar[d]^{ρ^{M} •C} & M \ar[d]_{ω_{M}} \ar[r]^{ρ^{M}} & M • C \ar[d]^{M•ϵ} \\ M • C \ar[r]_{M•δ_{C}} & M • C • C & I • M & M • I \ar[l]^{℘_{M}}}

A morphism of right C-semicomodules is a morphism f : M ⟶ M′ in V such that the following diagram is commutative

\xymatrix{M \ar[r]^{ρ^{M}} \ar[d]_{f} & M • C \ar[d]^{f•C} \\ M′ \ar[r]_{ρ^{M′}} & M′ • C}

The category of right C-semicomodules is denoted by S^{C}; the category of counitary right C-semicomodules is denoted by CS^{C}. Analogously, one can define the category ^{C}S of left C-semicomodules and its full subcategory ^{C}CS of counitary left C-semicomodules.

4.11. Remark. We prefer to use the terminology unital semimonoids (counital semicomonoids) to distinguish them from monoids (comonoids) which we reserve for monoidal categories. For example, the category of unital semimonoids in the monoidal category Set of sets is the category Monoid of usual monoids of the sense of Abstract Algebra. The same applies for unitary semimodules (counitary semicomodules). This is also consistent with the classical terminology of semirings and semimodules used in Section 5.

4.12. Let C be a V-semicomonoid and M a right C-semicomodule. We have a functor

− • M : V ⟶ S^{C},

where for every X ∈ V we have a structure of a right C-semicomodule on X • M given by

ρ^{X•M} : X • M ⟶^{X•ρ^{M}} X • (M • C) ⟶^{γ_{X,M,C}^{−1}} (X • M) • C.

Similarly, if M is a left C-semicomodule, then we have a functor M • − : V ⟶ ^{C}S.

4.13. Let C and D be V-semicomonoids. Let M be a left D-semicomodule and a right C-semicomodule and consider D • M ∈ S^{C} and M • C ∈ ^{D}S. We say that M is a (D, C)-bisemicomodule iff ρ^{(M;D)} : M ⟶ D • M is a morphism in S^{C} or equivalently iff ρ^{(M;C)} : M ⟶ M • C is a morphism in ^{D}S. The category of (D, C)-bisemicomodules with morphisms in ^{D}S ∩ S^{C} is denoted by ^{D}S^{C}. The full subcategory of counitary (D, C)- bisemicomodules is denoted by ^{D}CS^{C}. Indeed, every (counital) V-semicomonoid C is a (counitary) (C, C)-bisemicomodule in a canonical way.

4.14. Example. I is V-semicomonoid with

δ_{I} : I ⟶^{ω_{I}} I • I and ϵ_{I} : I ⟶^{id} I.

Moreover, every (firm) M ∈ V is a (counitary) (I, I)-bisemicomodule with (ρ^{M})^{l} : M ⟶^{ω_{M}}

I • M and (ρ^{M})^{r} : M ⟶^{ℓ_{M} ∘ω_{M}} M • I.

Dual to Proposition 4.6, we have

4.15. Proposition. Every semiunital op-semimonoidal functor F : (V, •, I_{V}) ⟶ (W, ⊗, I_{W}) lifts to a functor

F̃ : SCMonoid(V) ⟶ SCMonoid(W), C ⟼ F (C)

which commutes with the forgetful functors

U_{V} : SCMon(V) ⟶ V and U_{W} : SCMon(W) ⟶ W.

Dual to Proposition 4.7, we obtain

4.16. Proposition. Let (C, δ, ϵ) be a V-semicomonoid.

1. We have 𝕁-comonads

− • C : V ⟶ V and C • − : V ⟶ V

and isomorphisms of categories

S^{C} ≃ V^{(−•C;𝕁)} and ^{C}S ≃ V^{(C•−;𝕁)}.

2. If D is a V-semicomonoid, then we have 𝕁-comonads

− • C : ^{D}S ⟶ ^{D}S and D • − : S^{C} ⟶ S^{C}

and isomorphisms of categories

(^{D}M)^{(−•C;𝕁)} ≃ ^{D}S^{C} ≃ (S^{C})^{(D•−;𝕁)}.

Our second reconstruction result is obtained in a way similar to that of Theorem 4.8:

4.17. Theorem. Let V be cocomplete, I and C ∈ V be firm and assume that I is a regular generator and that both C • − and − • C respect colimits in V. There is a bijective correspondence between the structures of counital semicomonoids on C, the structures of 𝕁-comonads on (− • C, Δ, ω, ε; 𝕁) and the structures of 𝕁-comonads on (C • −, Δ̃, ω, ε̃; 𝕁).

4.18. Proposition. If (C, δ, ϵ) is a semicomonoid and (A, ζ, ϖ) is a unital semimonoid, then (V(C, A), ∗, 𝔢) is a monoid in Set with multiplication and neutral element given by

f ∗ g := ζ ∘ (f • g) ∘ δ and 𝔢 := ϖ ∘ ϵ.

Proof. For every f, g, h ∈ V(C, A), we have

((f ∗ g) ∗ h) = ζ ∘ (ζ • A) ∘ ((f • g) • h) ∘ (δ • C) ∘ δ
= ζ ∘ (A • ζ) ∘ (f • (g • h)) ∘ (C • δ) ∘ δ
= f ∗ (g ∗ h),

whence ∗ is associative. On the other hand, we have for every f ∈ V(C, A)

ℓ_{A} ∘ ω_{A} ∘ (f ∗ 𝔢) = ℓ_{A} ∘ ω_{A} ∘ ζ ∘ (f • ϖ) ∘ (C • ϵ) ∘ δ
= (ℓ_{A} ∘ ω_{A} ∘ ζ ∘ (A • ϖ)) ∘ (f • I) ∘ (C • ϵ) ∘ δ
= (A • I) ∘ (f • I) ∘ (C • ϵ) ∘ δ
= (f • I) ∘ (C • ϵ) ∘ δ
= (f • I) ∘ (ℓ_{C} ∘ ω_{C})
= (ℓ_{A} ∘ ω_{A}) ∘ f

Since A ≃^{ℓ_{A}∘ω_{A}} A • I is an isomorphism (in particular a monomorphism), we conclude that f ∗ 𝔢 = f. One can conclude similarly that 𝔢 ∗ f for all f ∈ V(C, A).

4.19. Proposition. If φ : (C, δ_{C}, ϵ_{C}) ⟶ (D, δ_{D}, ϵ_{D}) is a morphism of semicomonoids and σ : (A, ζ_{A}, ϖ_{A}) ⟶ (B, ζ_{B}, ϖ_{B}) is a morphism of unital semimonoids, then

V(D, A) ⟶^{<−,A>} V(C, A), f ⟼ f ∘ φ and V(C, A) ⟶^{<C,−>} V(C, B), g ⟼ σ ∘ g

are morphisms of monoids in Set. In particular, we have functors

V(C, −) : SMonoid_{V} ⟶ Monoid and V(−, A) : SCMonoid_{V}^{op} ⟶ Monoid.

5. A concrete example

In this section we give applications to the category of bisemimodules over a semialgebra. For the convenience of the reader and to make the manuscript self-contained, we begin this section by recalling some basic definitions and results on semirings and their semimodules.

Semirings and Semimodules.

5.1. Definition. A semiring is an algebraic structure (S, +, ·, 0, 1) consisting of a nonempty set S with two binary operations “+” (addition) and “·” (multiplication) satisfying the following axioms:

1. (S, +, 0) is a commutative monoid with neutral element 0_{S};

2. (S, ·, 1) is a monoid with neutral element 1;

3. x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x for all x, y, z ∈ S;

4. 0 · s = 0 = s · 0 for every s ∈ S ( i.e. 0 is absorbing).

5.2. Let S, S′ be semirings. A map f : S ⟶ S′ is said to be a morphism of semirings iff for all s_{1}, s_{2} ∈ S :

f(s_{1} + s_{2}) = f(s_{1}) + f(s_{2}), f(s_{1}s_{2}) = f(s_{1})f(s_{2}), f(0_{S}) = 0_{S′} and f(1_{S}) = 1_{S′}.

The category of semirings is denoted by SRng.

5.3. Let (S, +, ·) be a semiring. We say that S is

cancellative iff the additive semigroup (S, +) is cancellative, i.e. whenever s, s′, s′′ ∈ S we have

s + s′ = s + s′′ ⇒ s′ = s′′;

commutative iff the multiplicative semigroup (S, ·) is commutative;

semifield iff (S\{0}, ·, 1) is a commutative group.

5.4. Examples. Rings are indeed semirings. The first natural example of a ( commutative) semiring which is not a ring is (ℕ_{0}, +, ·), the set of non-negative integers. The semirings (ℝ_{0}^{+}, +, ·) and (ℚ_{0}^{+}, +, ·) are indeed semifields. For every associative ring R we have a semiring structure (Ideal(R), +, ·) on the set Ideal(R) of (two-sided) ideals of R. Every distributive complete lattice (L, ∧, ∨, 0, 1) is a semiring. For more examples, the reader may refer to [Gol1999]. In the sequel, we assume that 0_{S} ≠ 1_{S}.

5.5. Definition. Let S be a semiring. A right S-semimodule is an algebraic structure (M, +, 0_{M}) consisting of a non-empty set M, a binary operation “+” along with a right S-action

M × S ⟶ M, (m, s) ↦ ms,

such that:

1. (M, +, 0_{M}) is a commutative monoid with neutral element 0_{M};

2. (ms)s′ = m(ss′), (m + m′)s = ms + m′s and m(s + s′) = ms + ms′ for all s, s′ ∈ S and m, m′ ∈ M;

3. m1_{S} = m and m0_{S} = 0_{M} = 0_{M}s for all m ∈ M and s ∈ S.

5.6. Let M, M′ be right S-semimodules. A map f : M ⟶ M′ is said to be a morphism of S-semimodules (or S-linear) iff for all m_{1}, m_{2} ∈ M and s ∈ S :

f(m_{1} + m_{2}) = f(m_{1}) + f(m_{2}) and f(ms) = f(m)s.

The set Hom_{S}(M, M′) of S-linear maps from M to M′ is clearly a commutative monoid under addition. The category of right S-semimodules is denoted by 𝕊_{S}. Analogously, one can define the category _{S}𝕊 of left S-semimodules. A right (left) S-semimodule is said to be cancellative iff the semigroup (M, +) is cancellative. With ℂ𝕊_{S} ⊆ 𝕊_{S} (resp. _{S}ℂ𝕊 ⊆ S𝕊) we denote the full subcategory of cancellative right (left) S-semimodules. For two semirings S and T, an (S, T)-bisemimodule M has a structure of a left S-semimodule and a right T -semimodule such that (sm)t = s(mt) for all m ∈ M, s ∈ S and t ∈ T. The category of (S, T)-bisemimodules and S-linear T -linear maps is denoted by _{S}𝕊_{T}; the full subcategory of cancellative (S, T)-bisemimodules is denoted by _{S}ℂ𝕊_{T} .

5.7. Let M be a right S-semimodule. An S-congruence on M is an equivalence relation ≡ such that

m_{1} ≡ m_{2} ⇒ m_{1}s + m ≡ m_{2}s + m for all m_{1}, m_{2}, m ∈ M and s ∈ S.

In particular, we have an S-congruence relation ≡_{[0]} on M defined by

m ≡_{[0]} m′ ⟺ m + m′′ = m′ + m′′ for some m′′ ∈ M.

The quotient S-semimodule M/ ≡_{[0]} is indeed cancellative and we have a canonical surjection 𝔠_{M} : M ⟶ 𝔠(M), where 𝔠(M) := M/ ≡_{[0]}, with

Ker(𝔠_{M}) = {m ∈ M | m + m′′ = m′′ for some m′′ ∈ M}.

The class of cancellative right S-semimodules is a reflective subcategory of 𝕊_{S} in the sense that the functor 𝔠 : 𝕊_{S} ⟶ ℂ𝕊_{S} is left adjoint to the embedding functor ℂ𝕊_{S} ↪ 𝕊_{S}, i.e. for every S-semimodule M and every cancellative S-semimodule N we have a natural isomorphism of commutative monoids Hom_{S}(𝔠(M), N) ≃ Hom_{S}(M, N) [Tak1982, p.517].

Takahashi’s Tensor-like Product.

5.8. ([Gol1999, page 187]) Let M_{S} be a right S-semimodule, _{S}N a left S-semimodule and consider the commutative monoid U := S^{(M×N)}×S^{(M×N)}. Let U′ ⊆ S^{(M×N)}×S^{(M×N)} be the symmetric S-subsemimodule generated by the set of elements of the form

\begin{matrix}(f_{(m_{1}+m_{2},n)}, f_{(m_{1},n)} + f_{(m_{2},n)}), & (f_{(m_{1},n)} + f_{(m_{2},n)}, f_{(m_{1}+m_{2},n)}), \\ (f_{(m,n_{1}+n_{2})}, f_{(m,n_{1})} + f_{(m,n_{2})}), & (f_{(m,n_{1})} + f_{(m,n_{2})}, f_{(m,n_{1}+n_{2})}), \\ (f_{(ms,n)}, f_{(m,sn)}), & (f_{(m,sn)}, f_{(ms,n)}), \end{matrix}

where

f_{(m,n)}(m′, n′) = \{ \begin{matrix}1_{S}, & (m, n) = (m′, n′) \\ 0, & (m, n) ≠ (m′, n′). \end{matrix}

Let ≡ be the S-congruence relation on S^{(M×N)} defined by

f ≡ f′ ⟺ f + g = f′ + g′ for some (g, g′) ∈ U′.

Takahashi’s tensor-like product of M and N is defined as M ⊠_{S} N := U/ ≡ . Notice that there is an S-balanced map

τ̃: M × N ⟶ M ⊠_{S} N, (m, n) ↦ m ⊠_{S} n := (m, n)/ ≡

with the following universal property [Tak1982]: for every commutative monoid G and every S-bilinear S-balanced map β : M × N ⟶ G there exists a unique morphism of monoids γ : M ⊠_{S} N ⟶ 𝔠(G) such that we have a commutative diagram

\xymatrix{M × N \ar[d]_{τ} \ar[r]^{β} & G \ar[d]^{𝔠_{G}} \\ M ⊠_{S} N \ar@{.>}[r]_{γ} & 𝔠(G)}  (4)

The following result collects some properties of − ⊠_{S} − (compare with [Abu] and [Gol1999, Proposition 16.15, 16.16]):

5.9. Proposition. Let S and T be semirings, M be a right S-semimodule and N a left S-semimodule.

1. M ⊠_{S} N is a cancellative commutative monoid.

2. M_{S} (_{S}N) is cancellative if and only if 𝔠(M) ≃ M (𝔠(N) ≃ N).

3. We have natural isomorphisms of functors

− ⊠_{S} S ≃ 𝔠(−) : 𝕊_{S} ⟶ 𝕊_{S} and S ⊠_{S} − ≃ 𝔠(−) : _{S}𝕊 ⟶ _{S}𝕊.

Moreover, we have isomorphisms of functors

− ⊠_{S} S ≃ 𝔠(−) ≃ S ⊠_{S} − : _{S}𝕊_{S} ⟶ _{S}𝕊_{S}.

We set

M ⊠_{S} S ≃^{ϑ_{M}^{r}} M and S ⊠_{S} N ≃^{ϑ_{M}^{l}} N.

4. We have idempotent functors

𝕁 : S ⊠_{S} − : _{S}𝕊 ⟶ _{S}𝕊 and 𝕂 := − ⊠_{T} T : 𝕊_{T} ⟶ 𝕊_{T} .  (5)

In particular, 𝔠(𝔠(M)) ≃ 𝔠(M) and 𝔠(𝔠(N)) ≃ 𝔠(N).

5. We have natural isomorphisms of commutative monoids

𝔠(M) ⊠_{S} N ≃ 𝔠(M) ⊠_{S} 𝔠(N) ≃ M ⊠_{S} 𝔠(N) ≃ M ⊠_{S} N ≃ 𝔠(M ⊠_{S} N). (6)

5.10. Proposition. Let S and T be semirings, M a right S-semimodule and N an (S, T)-bisemimodule. Consider the functors

− ⊠_{S} N : 𝕊_{S} ⟶ 𝕊_{T} , N ⊠_{T} − : _{T} 𝕊 ⟶ _{S}𝕊

and the endo-functors 𝕁 and 𝕂 in (5).

1. (− ⊠_{S} N, Hom_{−T} (N, −)) is a (𝕁, 𝕂)-adjoint pair.

2. (N ⊠_{T} −, Hom_{S−}(N, −)) is a (𝕂, 𝕁)-adjoint pair.

Proof. For every right T -semimodule G we have natural isomorphisms of commutative monoids

Hom_{−T} (𝕁(M) ⊠_{S} N), 𝕂(G)) ≃ Hom_{−T} (𝔠(M) ⊠_{S} N), 𝔠(G))
≃ Hom_{−T} (M ⊠_{S} N, 𝔠(G))
≃ Hom_{−S}(M, Hom_{−T} (N, 𝔠(G)))  ([Gol1999, 16.15])
≃ Hom_{−S}(𝔠(M), Hom_{−T} (N, 𝔠(G))) ([Tak1982, p. 517])
≃ Hom_{−S}(𝕁(M), Hom_{−T} (N, 𝕂(G))).

The second statement can be proved symmetrically.

In what follows, S denotes a commutative semiring with 1_{S} ≠ 0_{S}, A is an S-semialgebra (i.e. a semiring with a morphism of semirings ι_{A} : S ⟶ A), _{A}𝕊_{A} is the category of (A, A)- bisemimodules and _{A}ℂ𝕊_{A} is its full subcategory of cancellative (A, A)-bisemimodules. Moreover, we fix the idempotent endo-functor 𝕁 given by

𝔠(−) ≃ A ⊠_{A} − ≃ − ⊠_{A} A : _{A}𝕊_{A} ⟶ _{A}𝕊_{A}.

5.11. Theorem.

1. (_{A}𝕊_{A}, ⊠_{A}, A) is a closed semiunital semimonoidal category.

2. (_{A}ℂ𝕊_{A}, ⊠_{A}, 𝔠(A)) is a closed monoidal category.

5.12. By a semiunital A-semiring we mean an (A, A)-bisemimodule A associated with (A, A)-bilinear maps ζ_{A} : A ⊠_{A} A ⟶ A and ϖ_{A} : A ⟶ A such that the following diagrams are commutative

\xymatrix{A ⊠_{A} A ⊠_{A} A \ar[r]^{ζ_{A}⊠_{A}A} \ar[d]_{A⊠_{A}ζ_{A}} & A ⊠_{A} A \ar[d]^{ζ_{A}} & A ⊠_{A} A \ar[r]^{ζ_{A}} & A \ar[d]_{𝔠_{A}} & A ⊠_{A} A \ar[l]_{ζ_{A}} \\ A ⊠_{A} A \ar[r]_{ζ_{A}} & A & A ⊠_{A} A \ar[u]^{ϖ_{A}⊠_{A}A} \ar[r]_{ϑ_{A}^{l}} & 𝔠(A) & A ⊠_{A} A \ar[u]_{A⊠_{A}ϖ_{A}} \ar[l]^{ϑ_{A}^{r}}}

Let A and A′ be semiunital A-semirings. An (A, A)-bilinear map f : A ⟶ A′ is called a morphism of semiunital A-semirings iff

f ∘ ζ_{A} = ζ_{A′} ∘ (f ⊠_{A} f) and f ∘ ϖ_{A} = ϖ_{A′}.

The set of morphisms of semiunital A-semirings form A to A′ is denoted by SSRng_{A}(A, A′). The category of semiunital A-semirings will be denoted by SSRng_{A}. Indeed, we have an isomorphism of categories SSRng_{A} ≃ SMonoid(_{A}𝕊_{A}).

5.13. Let A be a semiunital A-semiring. A semiunitary right A-semimodule is a right Asemimodule along with a right A-linear map ρ_{M} : M ⊠_{A} A ⟶ M such that the following diagrams are commutative

\xymatrix{M ⊠_{A} A ⊠_{A} A \ar[r]^{ρ_{M} ⊠_{A}A} \ar[d]_{M⊠_{A}ζ_{A}} & M ⊠_{A} A \ar[d]^{ρ_{M}} & M ⊠_{A} A \ar[d]_{ϑ_{M}^{r}} \ar[r]^{M⊠_{A}ϖ_{A}} & M ⊠_{A} A \ar[d]^{ρ_{M}} \\ M ⊠_{A} A \ar[r]_{ρ_{M}} & M & 𝔠(M) & M \ar[l]^{𝔠_{M}}}

A morphism of semiunitary right A-semimodules (A-linear) is an A-linear map f : M ⟶ M′ such that the following diagram is commutative

\xymatrix{M ⊠_{A} A \ar[r]^{ρ_{M}} \ar[d]_{f⊠_{A}A} & M \ar[d]^{f} \\ M′ ⊠_{A} A \ar[r]_{ρ_{M′}} & M′}

The category of semiunitary right A-semimodules and A-linear maps is denoted by 𝕊𝕊_{A}; the full subcategory of unitary cancellative right A-semimodules is denoted by ℂ𝕊_{A}. Analogously, one can define the category _{A}𝕊𝕊 of semiunital left A-semimodules and its full subcategory _{A}ℂ𝕊 of unitary cancellative left A-semimodules. For two semiunital Asemirings A and B, one can define the category_{B}𝕊𝕊_{A} of semiunitary (B, A)-bisemimodules and its full subcategory_{B}ℂ𝕊_{A} of unitary cancellative (B, A)-bisemimodules in the obvious way. Considering any semiunital A-semiring as a semimonoid in (_{A}𝕊_{A}, ⊠_{A}, A), we have indeed isomorphisms of categories for any two semiunital A-semirings A and B:

B𝕊𝕊 ≃ (_{A}𝕊𝕊)_{(B⊠_{A}−;𝔠)}, 𝕊𝕊_{A} ≃ (𝕊_{A})_{(−⊠_{A}A;𝔠)}, _{B}𝕊𝕊_{A} ≃ _{B}S_{A}
Bℂ𝕊 ≃ (_{A}ℂ𝕊)_{B⊠_{A}−}, ℂ𝕊_{A} ≃ (ℂ𝕊_{A})_{−⊠_{A}A}, _{B}ℂ𝕊_{A} ≃ _{B}US_{A}

5.14. Remark. We use the terminology semiunital A-semirings to stress that such semimonoids are defined in the semiunital semimonoidal category (_{A}𝕊_{A}, ⊠_{A}, A) and to avoid confusion with (unital) A-semirings which can be defined as monoids in the monoidal category (_{A}𝕊_{A}, ⊗_{A}, A). The same applies for semicounitary A-semicorings below.

5.15. Being a variety, in the sense of Universal Algebra, the category _{A}𝕊_{A} of (A, A)- bisemimodules is cocomplete. The class of regular epimorphism in _{A}𝕊_{A} coincides with that of surjective (A, A)-bilinear maps. For every (A, A)-bisemimodule M, there is a surjective (A, A)-bilinear map from a free (A, A)-bisemimodule to M (compare with [Gol1999, Proposition 17.11]); whence, A is a regular generator. Moreover, for every (A, A)-bisemimodule X, both X ⊠_{A} −, − ⊠_{A} X : _{A}𝕊_{A} ⟶ _{A}ℂ𝕊_{A} respect colimits since they are left adjoints [Tak1982, Corollary 4.5].

Applying Theorem 4.8 to _{A}𝕊_{A}, we obtain:

5.16. Corollary. Let A be cancellative and A a cancellative (A, A)-bisemimodule. There is a bijective correspondence between the structures of unital A-semirings on A, the structures of 𝔠-monads on A ⊠_{A} − and the structures of 𝔠-monads on − ⊠_{A} A.

5.17. A semicounital A-semicoring is an (A, A)-bisemimodule associated with (A, A)- bilinear maps δ_{C} : C ⟶ C ⊠_{A} C and ϵ_{C} : C ⟶ A such that the following diagrams are commutative

\xymatrix{C \ar[r]^{δ_{C}} \ar[d]_{δ_{C}} & C ⊠_{A} C \ar[d]^{C⊠_{A}δ_{C}} & C ⊠_{A} C \ar[d]_{ϵ_{C}⊠_{A}C} & C \ar[l]_{δ_{C}} \ar[r]^{δ_{C}} \ar[d]_{𝔠_{C}} & C ⊠_{A} C \ar[d]^{C⊠_{A}ϵ_{C}} \\ C ⊠_{A} C \ar[r]_{δ_{C}⊠_{A}C} & C ⊠_{A} C ⊠_{A} C & A ⊠_{A} C \ar[r]_{ϑ_{C}^{l}} & 𝔠(C) & C ⊠_{A} A \ar[l]^{ϑ_{C}^{r}}}  (7)

The map δ_{C} (ϵ_{C}) is called the comultiplication (counity) of C. Using Sweedler-Heyneman’s notation, we have for every c ∈ C :

∑ c_{11} ⊠_{A} c_{12} ⊠_{A} c_{2} = ∑ c_{1} ⊠_{A} c_{21} ⊠_{A} c_{22};
𝔠_{C}(∑ c_{1}ϵ_{C}(c_{2})) = 𝔠_{C}(c) = 𝔠_{C}(∑ ϵ_{C}(c_{1})c_{2}).

Let (C, δ, ϵ) and (C′, δ′, ϵ′) be semicounital A-semicorings. We call an (A, A)-bilinear map f : C ⟶ C′ a morphism of semicounital A-semicorings iff

(f ⊠_{A} f) ∘ δ_{C} = δ_{C′} ∘ f and ϵ_{C′} ∘ f = ϵ_{C}.

The set of morphisms of semicounital A-semicoring from C to C′ is denoted by SSCog_{A}(C, C′). The category of semicounital A-semicorings is denoted by SSCrng_{A}. Indeed, we have an isomorphism of categories SSCrng_{A} ≃ SCMonoid(_{A}𝕊_{A}).

5.18. Let (C, δ, ϵ) be an A-semicoring. A semicounitary right C-semicomodule is a right A-semimodule M associated with an A-linear map

ρ^{M} : M ⟶ M ⊠_{A} C, m ↦ ∑ m_{<0>} ⊠_{A} m_{<1>},

such that the following diagrams are commutative

\xymatrix{M \ar[r]^{ρ^{M}} \ar[d]_{ρ^{M}} & M ⊠_{A} C \ar[d]^{M⊠_{A}δ_{C}} & M \ar[r]^{ρ^{M}} \ar[d]_{𝔠_{M}} & M ⊠_{A} C \ar[d]^{M⊠_{A}ϵ_{C}} \\ M ⊠_{A} C \ar[r]_{ρ^{M} ⊠_{A}C} & M ⊠_{A} C ⊠_{A} C & 𝔠(M) & M ⊠_{A} A \ar[l]^{ϑ_{M}^{r}}}

Using Sweedler-Heyneman’s notation, we have for every m ∈ M :

∑ m_{<0>} ⊠_{A} m_{<1>1} ⊠_{A} m_{<1>2} = ∑ m_{<0><0>} ⊠_{A} m_{<0><1>} ⊠_{A} m_{<1>};
𝔠(∑ m_{<0>}ϵ_{C}(m_{<1>})) = 𝔠_{M}(m).

For semicounitary right C-comodules M, M′, we call an A-linear map f : M ⟶ M′ a morphism of semicounitary right C-semicomodules (or C-colinear) iff the following diagram is commutative

\xymatrix{M \ar[r]^{f} \ar[d]_{ρ^{M}} & N \ar[d]^{ρ^{N}} \\ M ⊠_{A} C \ar[r]_{f⊠_{A}C} & N ⊠_{A} C}

The category of semicounitary right C-semicomodules and C-colinear maps is denoted by 𝕊𝕊^{C}; the full subcategory of counitary right C-semicomodules is denoted by ℂ𝕊^{C}. Analogously, one can define the category ^{C}𝕊𝕊 of semicounitary left C-semicomodules and its full subcategory^{C}ℂ𝕊 of counitary left C-semicomodules. For two semicounital A-semicorings C and D one can define the category ^{D}𝕊𝕊^{C} of semicounitary (D, C)-bisemicomodules and its full subcategory ^{D}ℂ𝕊^{C} of counitary (D, C)-bisemicomodules in the obvious way. Considering any semicounital A-semicoring as a semicomonoid in (_{A}𝕊_{A}, ⊠_{A}, A), we have indeed isomorphisms of categories for any two semicounital A-semicorings C and D:

D𝕊𝕊 ≃ (_{A}𝕊)^{(D⊠_{A}−;𝔠)}, 𝕊𝕊^{C} ≃ (𝕊_{A})^{(−⊠_{A}C;𝔠)}, ^{D}𝕊𝕊^{C} ≃ ^{D}S^{C}
Dℂ𝕊 ≃ ^{D⊠_{A}−}(_{A}ℂ𝕊), ℂ𝕊^{C} ≃ (ℂ𝕊_{A})^{−⊠_{A}C}, ^{D}ℂ𝕊^{C} ≃ ^{D}CS^{C}

Applying Theorem 4.17 to _{A}𝕊_{A}, we obtain:

5.19. Corollary. Let A be cancellative and C a cancellative (A, A)-bisemimodule. There is a bijective correspondence between the structures of counital A-semicorings on C, the structures of 𝔠-comonads on C ⊠_{A} − and the structures of 𝔠-comonads on − ⊠_{A} C.

Almost all structures of corings over rings (e.g. [Abu2003], [BW2003]) can be transferred to obtain structures of semicorings over semirings.

5.20. Example. Let f : B ⟶ A be an extension of S-semialgebras and consider A as a (B, B)-bisemimodule in the canonical way. One can define Sweedler’s counital Asemicoring C := (A ⊠_{B} A, δ, ϵ) with

δ : A ⊠_{B} A ⟶ (A ⊠_{B} A) ⊠_{A} (A ⊠_{B} A), a ⊠_{B} ã ↦ (a ⊠_{B} 1_{A}) ⊠_{A} (1_{A} ⊠_{B} ã);
ϵ : A ⊠_{B} A ⟶ A, a ⊠_{B} ã ↦ aã.

Acknowledgement: The author thanks the anonymous referee for her/his careful reading of the paper and, in particular, for suggestions to fix some results in the initial version of the paper. He also thanks Prof. Robert Wisbauer for the fruitful discussions on monads and comonads during his visit to the University of Düsseldorf in Summer 2009. He also thanks Deutsche Akademische Austausch Dienst (DAAD) for supporting that visit.

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J. Vercruysse, Hopf algebras – Variant notions and reconstruction theorems, to appear in C. Heunen, M. Sadrzadeh, E. Grefenstette (eds.): Compositional methods in Physics and Linguistics, Oxford University Press, preprint; arXiv:1202.3613.

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