Parametric monadsand enriched adjunctions

Paul-André Mellies

Laboratoire Preuves, Programmes, SystèmesCNRS — Université Paris Diderot

June 17, 2012

Abstract

Starting from the particular case of the double negation monad indialogue categories, we investigate what additional structure is re-quired of an adjunction in order to give rise to a strong monad. Theanalysis leads to a purely combinatorial description of enriched func-tors, enriched natural transformations and enriched adjunctions be-tween categorical modules.

1 IntroductionOne basic principle of categorical algebra is that every monad T on a cat-egory A factors as T = R ◦ L where the functor L is left adjoint to thefunctor R in an adjunction of the form

A

L

""⊥R

cc B

This decomposition T = R ◦ L is not unique in general, and some decompo-sitions are more illuminating than others. A typical illustration is providedby the double negation monad T in a dialogue category A . Recall that a dia-logue category C is a monoidal category equipped with an object ⊥ together

1

with two functors

C op −→ C C op −→ CA 7→ A( ⊥ A 7→ ⊥� A

and two families of bijections

C (B,A( ⊥) � C (A⊗B,⊥) � C (A,⊥� B)

natural in A and B. As such, dialogue categories offer a suitable frameworkin order to interpret languages with linear continuations. Every dialoguecategory is equipped with the double negation monad defined as

T : A 7→ ⊥� (A( ⊥).

It is not difficult to see that the monad T decomposes as the adjunction

C

L

""⊥R

cc C op

where L and R are defined as the two negation functors

L : A 7→ A( ⊥, R : A 7→ ⊥� A.

Now, the double negation functor T happens to be strong on the right. Thismeans that there exists a family of morphisms

σA,m : T (A)⊗m −→ T (A⊗m)

natural in the objects A and m of the category C , and making the expectedseries of coherence diagrams commute. Such a natural transformationis called a right strength. At this point, an important question is thusto understand what specific structure of the two functors L and R and ofthe adjunction L a R between them induces the existence of such a rightstrength σ.

First decomposition. One useful observation in our work on dialoguecategories [5] is that the existence of the strength σ follows from the exis-tence of a family of adjunctions

L(−⊗m) a R(m⊗−)

2

parametrized by the objects m of the category C . The point is that each ofthese adjunction amounts to the existence of a pair of families

axm : L(A) −→ m⊗ L(A⊗m)

cutm : R(m⊗ A)⊗m −→ R(m)natural in the object A and making the expected coherence diagrams

RL(A)⊗m axm // R(m⊗ L(A⊗m))⊗m

cutm

��A⊗m ηA⊗m //

ηA⊗m

OO

RL(A⊗m)

A⊗ L(R(m⊗ A)⊗m) cutm //m⊗ LR(A)

m⊗εA

��LR(m⊗ A) εm⊗A //

axm

OO

m⊗ A

commute. By a parametric family of such adjunctions, we mean in additionthat the two families axm and cutm are dinatural in m, and that they aremonoidal in the sense that the diagrams below

DIAGRAM

commute. Here, the two natural transformations η and ε denote the unitand counit of the adjunction L a R. Now, it appears that the strength of thedouble negation monad T factors as:

σA,m : RL(A)⊗m axm // R(m⊗ L(A⊗m))⊗m cutm // RL(A⊗m)

This decomposition may be alternatively formulated by thinking of the ten-sor product of C as an action (on the right) noted

∗ : C × C −→ C

of the monoidal category C on itself, and of the tensor product of C op as anaction (on the right again) noted

∗− : C op × C op −→ C op

3

of the monoidal category C op on itself. See the definition of action in thenext section. The two natural transformations “axiom” and “cut” are thenreformulated accordingly as

axm : L(A) −→ L(A ∗m)∗−m

cutm : R(A∗−m) ∗m −→ R(A)together with the decomposition σ of the strength of the double negationmonad T = R ◦ L:

σA,m : RL(A) ∗m axm // R(L(A ∗m)∗−m) ∗m cutB // RL(A ∗m)

Second decomposition. This decomposition of the strength σ departsquite radically (at least apparently) from another decomposition of the samestrength, based this time on the decomposition of the double negation monadT = U ◦ F as the canonical adjunction

C

F

""⊥U

cc CT

where CT is the Kleisli category induced by the monad T . Observe that, inthat case, the strength σ induces an action on the right

⊗ : CT × C −→ CT

of the monoidal category C on the Kleisli category CT . The situation alsoincludes two families of functorial strengths

σF : F (A)⊗m −→ F (A⊗m)

σU : U(A)⊗m −→ U(A⊗m)where the strength σU is moreover invertible. This leads to an alternativedecomposition of the strength σ of the double negation monad T = U ◦ F :

σA,m : UF (A)⊗m σU // U(F (A)⊗m) σF // UF (A⊗m)

Using the notation of actions rather than of tensor products, one may rewritethe functorial strength as

σF : F (A) ∗m −→ F (A ∗m)

4

σU : U(A) ∗m −→ U(A ∗m)

together with the decomposition as:

σA,m : UF (A) ∗m σU // U(F (A) ∗m) σF // UF (A ∗m) .

where ∗ denotes the two actions on the right

∗ : C × C −→ C ∗ : CT × C −→ CT

of the monoidal category C on the category C and on its Kleisli category CT

respectively.

The two decompositions of the strength σ are apparently very different.In particular, the first decomposition seems to be of an entirely new andunknown nature, since it requires a combination of actions of the monoidalcategory C as well as of its opposite category C op. The second decomposi-tion is more familiar, since it only requires a pair of actions of the monoidalcategory C on the category C and on its Kleisli category CT . For a very longtime, we believed that the two constructions were intrinsically different.We were simply wrong... Indeed, the purpose of this note is precisely tounify the two decompositions of T by recasting the two adjunctions L a Rand F a U in the same enriched framework. In particular, we show that thefirst decomposition of the double negation monad is based on an enrichedadjunction – an observation which one may find interesting for its own sake.Also, we will see that our purpose is not to translate everything in the lan-guage of enriched category theory. Quite on the contrary: our main purposeis to clarify what combinatorial structures underlie the notions of “enrich-ment” in a number of examples of interest where the enriched categoriesare coming from an action of a monoidal category M .

Plan of the paper. Given monoidal category M , we start by introducingin §2 the notion of positive M -category. We explain in §3 how every suchpositive M -category may be seen as an enriched category over the presheafcategory [M op, Set]. We then introduce in §4 the dual notion of negative M -category which may be also seen as an enriched category over the presheafcategory [M op, Set], but in a different way. In each positive and negativecase, we characterize the enriched functors and natural transformationsbetween these M -categories. We complete the pictureby characterizing in§5 the enriched functors and natural transformations between a positive

5

and a negative M -category. This leads to the definition in §6 of the 2-category M – Mod which embeds fully and faithfully in the 2-category ofenriched categories, enriched functors and enriched natural transforma-tions. We illustrate these ideas with a series of examples provided in §7and conclude the paper in §8.

2 Positive M -categoriesThe starting point of our analysis is provided by the notion of M -category,defined as a categorical module A over a monoidal category M . Morespecifically, given a monoidal category M , we construct a 2-category M + – Modwith

• positive M -categories as 0-cells,

• M -functors as 1-cells,

• natural M -transformations as 2-cells.

We will see in the next section that these data are tightly connected to thenotions of enriched functors and enriched natural transformations.

Definition 1 (positive action) A positive action of a monoidal category(M ,⊗, I) on a category A is defined as a functor

∗ : M ×A −→ A

which transports every pair of objects (m,A) to an object m ∗ A of the cate-gory A , together with a pair of natural transformations:

δm,n,A : (m⊗ n) ∗ A −→ n ∗ (m ∗ A) δA : I ∗ A −→ A

satisfying the three expected coherence laws:

(m⊗ n⊗ p) ∗ A

δ

��

δ //m ∗ ((n⊗ p) ∗ A)

δ

��(m⊗ n) ∗ (p ∗ A) δ // (m ∗ (n ∗ (p ∗ A)))

6

m ∗ A id

��

m ∗ A

(m⊗ I) ∗ A δ //m ∗ (I ∗ A)

δ

OO

m ∗ A id

��

m ∗ A

(I ⊗m) ∗ A δ // I ∗ (m ∗ A)

δ

OO

An action is called pseudo when the natural transformations are isomor-phisms, and strict when the monoidal category M is strict and the naturaltransformations are equalities.

Note that the notion of positive action generalizes the notion of comonad:indeed, a comonad in a category A is the same thing as positive action ofthe trivial monoidal category M = 1 on the category A .

Definition 2 (positive categories) A positive M , or M +-category, is acategory A equipped with a positive action of the monoidal category M .

Definition 3 (positive functors) An M +-functor

(F, σ) : (A , ∗) −→ (B, ∗)

between two M +-categories is defined as a functor

F : A −→ B

between the underlying categories, together with a natural transformation

σm,A : m ∗ F (A) −→ F (m ∗ A)

making the two coherence diagrams

F ((m⊗ n) ∗ A)δ

))(m⊗ n) ∗ FA

�

σm⊗n55

F (m ∗ (n ∗ A))

m ∗ (n ∗ FA) σn //m ∗ F (n ∗ A)

σm

OO(1)

7

I ∗ FA

σI

##

// FA

F (I ∗ A)

<<

(2)

commute.

Definition 4 (natural transformation) A natural M +-transformation

θ : (F, σ) ⇒ (G, τ) : (A , ∗) −→ (B, ∗)

is a natural transformation between the underlying functors

θ : F ⇒ G : A −→ B

such that the coherence diagram

m ∗ FA σm //

m ∗ θA

��

F (m ∗ A)

θm∗A

��m ∗GA τm // G(m ∗ A)

commutes for all objects A and m.

Recall from the work of Anders Kock [3] that

Definition 5 A strong monad on a M +-category (A , ∗) is a pair consistingof a monad T in the category A together with a natural transformation

σ : m ∗ T (A) −→ T (m ∗ A)

making the familiar diagrams commute.

The following statement is essentially straightforward.

Proposition 1 A strong monad on a M +-category (A , ∗) is the same thingas a monad

(T, µ, η) : (A , ∗) −→ (A , ∗)

in the 2-category M + – Mod.

8

3 EnrichmentWe suppose from now on that the monoidal category M is small. At thispoint, we are ready to establish that

Proposition 2 Every M +-category A may be seen as an enriched cate-gory ι+A over the presheaf category [M op, Set] equipped with the Day tensorproduct on M . The hom-space

A [A,B]

from an object A to an object B in the enriched category A is defined as thefollowing presheaf:

A [A,B] : M op −→ Setm 7→ A (m ∗ A,B)

where A (A,B) denotes the hom-set from A to B in the category A .

Recall that the Day tensor product of two presheaves is defined using thecoend formula:

ϕ⊗Day ψ : m 7→∫ m1,m2∈M

M (m,m1 ⊗m2)× ϕ(m1)× ψ(m2)

This tensor product equips the presheaf category [M op, Set] with the struc-ture of a monoidal category, whose tensorial unit is provided by the repre-sentable presheaf

yI : m 7→ M (m, I)associated to the unit of the original monoidal category M . Note that thisconstruction of the enriched category generalizes the familar constructionof the Kleisli category associated to a comonad in the category A . Indeed,in the case of a trivial category M = 1, the category ι+A is enriched over[M op, Set] which coincides in that case with Set. This category ι+A coin-cides with the Kleisli category associated to the comonad A 7→ I ∗ A whereI denotes the unique object of the monoidal category M = 1.

Proposition 3 The construction of Proposition 2 induces a 2-functor

ι+ : M + – Mod −→ [M op, Set] –Cat

which is moreover full and faithful in the sense that

M + – Mod((A , ∗), (B, ∗)) � [M op, Set] –Cat(ι+(A , ∗), ι+(B, ∗))

where � means isomorphism of categories.

9

In other words, an M +-functor and a natural M +-transformation betweentwo positive M -categories (A , ∗) and (B, ∗) are the same things as an en-riched functor and an enriched natural transformation between the associ-ated enriched categories.

4 Negative M -categoriesWe introduce below the notion of negative M -category, or M−-category,which is dual to the notion of positive M -category.

Definition 6 (negative action) A negative action of a monoidal categoryM = (M ,⊗, I) on a category A is defined as a positive action of the oppositemonoidal category M op on the opposite category A op.

Now, recall that the opposite monoidal category M op(0,1) = (M op,⊗ op, I) isobtained from M by

• reversing the direction of the morphisms of dimension 1, this justify-ing the index 1,

• reversing the orientation of the tensor product of dimension 0, thisjustifying the index 0.

This leads to the definition of negative M -category.

Definition 7 (negative categories) A negative M -category, or M−-category,is a category A equipped with a negative action of the opposite monoidalcategory M op(0,1).

The definition of negative M -category is fine, but a bit concise, and we thusexpand it in the following statement.

Proposition 4 A negative M -category is the same thing as a category Aequipped with a functor

−∗ : M op ×A −→ A

which transports every pair of objects (m,A) to an object m−∗A of the cate-gory A , together with a pair of natural transformations:

µm,n,A : n−∗(m−∗A) −→ (m⊗ n)−∗A µA : A −→ I−∗A

10

satisfying the three coherence laws:

p−∗(n−∗(m−∗A))

µ

��

µ // (n⊗ p)−∗(m−∗A)

µ

��p−∗((m⊗ n)−∗A) µ // (m⊗ n⊗ p)−∗A

m−∗A id

µ

��

m−∗A

��I−∗(m−∗A) µ // (m⊗ I)−∗A

m−∗A id

µ

��

m−∗A

��m−∗(I−∗A) µ // (I ⊗m)−∗A

Definition 8 (negative functors) An M−-functor

(F, σ) : (A ,−∗) −→ (B,−∗)

between two negative M -categories is defined as a functor

F : A −→ B

between the underlying categories, together with a natural transformation

σm,A : F (m−∗A) −→ m−∗F (A)

making the two coherence diagrams

11

n−∗F (m−∗A)σn

))F (n−∗(m−∗A))

µ��

σn55

n−∗(m−∗FA)µ��

F ((m⊗ n)−∗A) σm⊗n // (m⊗ n)−∗FA

F (I−∗A)

σI$$

µ // FA

I−∗FAµ

==

commute.

Definition 9 (natural transformation) A natural M -transformation

θ : (F, σ) ⇒ (G, τ) : (A ,−∗) −→ (B,−∗)

is a natural transformation between the underlying functors

θ : F ⇒ G : A −→ B

such that the coherence diagram

F (m−∗A)

θm−∗A

��

σm //m−∗FA

m−∗θA

��G(m−∗A) τm //m−∗GA

commutes for all objects A and m.

One reason for introducing the notion of negative M -category is that itprovides us with an alternative way to see a category A as an enrichedcategory over the presheaf category [M op, Set]. The idea is to let the ob-jects m of the monoidal category M act on the target object B rather thanon the source object A of a morphism A→ B of the category A . Indeed, weestablish that:

12

Proposition 5 Every negative M -category (A ,−∗) may be seen as an en-riched category over the presheaf category [M op, Set] equipped with the Daytensor product. The presheaf [A,B] over the category M is defined as follows:

A [A,B] : m 7→ A (A,m−∗B)

where m−∗B denotes the result of letting the object m in M op act on theobject B of the category A .

This very observation leads us to introduce the 2-category M− – Mod with

• negative M -categories as 0-cells,

• M−-functors as 1-cells,

• natural M−-transformations as 2-cells.

Proposition 6 The construction of Proposition 5 induces a 2-functor

ι− : M− – Mod −→ [M op, Set] –Cat

which is moreover full and faithful in the strong sense that

M− – Mod((A , ∗), (B, ∗)) � [M op, Set] –Cat(ι−(A , ∗), ι−(B, ∗))

where � means isomorphism of categories.

In other words, an M−-functor and a natural M−-transformation betweennegative M -categories (A ,−∗) and (B,−∗) are the same thing as an en-riched functor and an enriched natural transformation between the associ-ated enriched categories.

5 Transverse functorsAt this point, we have constructed two full and faithful functors

ι+ : M + – Mod −→ [M op, Set] –Cat ι− : M− – Mod −→ [M op, Set] –Cat

each of them characterizing the enriched functors and natural transforma-tions between positive M -categories and negative M op(0,1)-categories, seenas enriched categories. This leads to the question of characterizing the en-riched functors and natural transformations from an object of M + – Mod toan object of M− – Mod, and conversely, from an object of M− – Mod to anobject of M + – Mod. As we will see, the answer is illuminating, and clari-fies the algebraic status of the adjunction L a R between the two negationsof a dialogue category mentioned in the introduction. So, let us consider

13

• a positive M -module (A , ∗),

• a negative M -module (B,−∗).

Definition 10 (transverse functor) A transverse M +−-functor

(F, ax) : (A , ∗) −→ (B,−∗)

is a functor between the underlying categories

F : A −→ B

together with a family of morphisms

axm,A : FA −→ m−∗F (m ∗ A)

natural in A and dinatural in m, such that the coherence diagrams

F (A) axm⊗n //

axn

��

(m⊗ n)−∗F ((m⊗ n) ∗ A)

δ

��

n−∗F (n ∗ A)

axn

��n−∗(m−∗F (m ∗ (n ∗ A))) µ // (m⊗ n)−∗F (m ∗ (n ∗ A))

By dinaturality in m, we mean that the diagram

FAaxm //

axn

��

m−∗F (m ∗ A)

m−∗F (h∗A)

��n−∗F (n ∗ A) h−∗F (n∗A) //m−∗F (n ∗ A)

commutes for every morphism h : m→ n in the category M .

Definition 11 (natural transformations) A natural M +−-transformation

θ : (F, ax) ⇒ (G, ax) : (A , ∗) −→ (B,−∗)

14

is a natural transformation between the underlying functors

θ : F ⇒ G : A −→ B

making the diagram

F (A)

θA

��

axm //m−∗F (m ∗ A)

m−∗ θm∗A

��G(A) axm //m−∗G(m ∗ A)

commute for all objects A and m.

The definition of transverse functor and natural transformations are justi-fied by the following proposition.

Proposition 7 A transverse M +−-functor

(A , ∗) −→ (B,−∗)

is the same thing as a functor

ι+(A , ∗) −→ ι−(B,−∗)

enriched over the presheaf category [M op, Set]. Similarly, a natural M +−-transformation

θ : (F, σ) ⇒ (G, τ) : (A , ∗) −→ (B,−∗)

is the same thing as an enriched natural transformation between the under-lying enriched functors.

Similarly, the enriched functors and natural transformations from a nega-tive M -category to a positive M -category are characterized by the followingdefinitions. Suppose given

• a negative M -module (A ,−∗),

• a positive M -module (B, ∗).

15

Definition 12 (transverse functor) A transverse M−+-functor

(F, cut) : (A ,−∗) −→ (B, ∗)

is a functor between the underlying categories

F : A −→ B

together with a family of morphisms

cutm,A : m ∗ F (m−∗A) −→ FA

natural in A and dinatural in m, such that the coherence diagrams

(m⊗ n) ∗ F (n−∗(m−∗A)) δ //

µ

��

m ∗ (n ∗ F (n−∗(m−∗A)))

cutn

��m ∗ F (m−∗A)))

cutm

��(m⊗ n) ∗ F ((m⊗ n)−∗A) cutm⊗n // FA

By dinaturality in m, we mean that the diagram

m ∗ F (n−∗A) m∗F (h−∗A) //

h∗F (n−∗A)

��

m ∗ F (m−∗A)

cutm

��n ∗ F (n−∗A) cutn // FA

commutes for every morphism h : m→ n in the category M .

Definition 13 (natural transformations) A natural M−+-transformation

θ : (F, cut) ⇒ (G, cut) : (A ,−∗) −→ (B, ∗)

is a natural transformation between the underlying functors

θ : F ⇒ G : A −→ B

16

making the diagram

m ∗ F (m−∗A)

m ∗ θm−∗A

��

cutm // FA

θA

��m ∗ G(m−∗A) cutm // GA

commute for all objects m and A.

6 General M -categoriesAt this point, we are ready to construct the 2-category M – Mod with

• the positive as well as the negative M -categories as 0-cells,

• the positive, negative or transverse M -functors as 1-cells,

• the positive, negative or transverse natural M -transformations as 2-cells.

As we explained in the introduction, the 2-category M – Mod provides apurely combinatorial description of the 2-category of enriched categoriesover [M op, Set] “restricted” to the positive and negative M -categories. Thisinformal statement is formalized as follows.

Proposition 8 Put together, the constructions ι+ and ι− induce a 2-functor

M – Mod −→ [M op, Set] –Cat

which is full and faithful in the sense that

M – Mod(A ,B) � [M op, Set] –Cat(ιA , ιB)

for every positive or negative M -categories A and B, where � means iso-morphism of categories.

A technical benefit of this combinatorial reformulation of enriched functorsand natural transformations is that one may relax the condition that themonoidal category M is small.

17

7 IllustrationsIn this section, we review a series of examples of strong monads T = R ◦ Lon M -categories arising from an enriched adjunction L a R. Our startingpoint is provided by the two different decompositions of the double negationmonad in a dialogue category.

First decomposition. The adjunction

C

L

""⊥R

cc C op

where L and R are defined as the two negation functors

L : A 7→ A( ⊥, R : A 7→ ⊥� A.

happens to define an adjunction in the 2-category M – Mod where M isdefined as C op(1) = (C ,⊗ op, I). The fact that one needs to take the oppositetensor product is just a matter of convention in the definition of our M -modules. The important point to observe is that the monoidal category Macts positively on the category C by the tensor product:

∗ : M × C −→ C(m,A) 7→ A⊗m

and that its opposite monoidal category M op(0,1) = C op acts negatively onthe category C op by the same tensor product:

−∗ : M op(0,1) × C op −→ C op

(m,A) 7→ m⊗ A.

considered this time in the opposite category C op. This means that thecategory C should be considered as a positive M -category (C , ∗) whereasits opposite category C op defines a negative M -category (C op,−∗). Fromthis, it is not difficult to see that the adjunction L a R defines an adjunctionin the 2-category M – Mod. The axiom and cut are given by the naturaltransformations

axm : L(X) −→ m⊗ L(X ⊗m)

18

cutm : R(m⊗X)⊗m −→ R(X).From this follows by Proposition 8 that the canonical adjunction L a Rof dialogue categories is enriched over the category [C op(0,1), Set] equippedwith its Day tensor product. More generally, one can show that

Proposition 9 An adjunction

(A , ∗)L

""⊥R

cc (B,−∗)

between a positive M -category A and a negative M -category B is the samething as a family of categorical adjunctions

L(m ∗ −) a R(m−∗−)

natural in the object m of the category M .

Second decomposition. The following statement clarifies the algebraicstatus of our second decomposition of the double negation monad as T =U ◦ F .

Proposition 10 Every monad T in the 2-category M + – Mod induces anadjunction

C

F

""⊥U

cc CT

living in the 2-category M + – Mod, where CT is the Kleisli category of themonad T equipped with the action

~ : M × CT −→ CT

defined on two morphisms

f : A −→ T (B) h : m −→ n

as follows:

h~ f : m ∗ A h∗f // n ∗ T (B) σn // T (n ∗B) .

This is precisely the way the double negation monad T factors as T = U ◦Fin the introduction.

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The state monad. Another example of adjunction in M + – Mod where M =C is provided by the state monad

T : A 7→ S ( (S ⊗ A)

in any monoidal closed category C , where S is a specific object of C , describ-ing a space of states for the system. In that case, the monad T factors asthe adjunction

C

L

""⊥R

cc C

where L : A 7→ S ⊗ A and R : A 7→ S ( A. It is not difficult to see that thetwo functors L and R are M -functors between the positive M -module Cand itself – as testified by the existence of the natural transformations

L(A)⊗m = (S ⊗ A)⊗m � S ⊗ (A⊗m) = L(A⊗m)

R(A)⊗m = (S ( A)⊗m −→ S ( (A⊗m) = R(A⊗m)

As a matter of fact, one easily checks that the adjunction lives in the cat-egory M + – Mod. This provides a conceptual way to understand why thestate monad is strong.

The Eilenberg-Moore adjunction. Suppose given a strong monad Ton a monoidal closed category C . By monoidal closed, we mean that thefunctor

A 7→ m⊗ A : C −→ C

has a right adjoint denoted as

A 7→ m( A : C −→ C

for every object m of the monoidal category M = C . In that case, thecategory of algebras C T defines a negative M -module (C T ,(). The actionof an object m of C on a T -algebra

a : TA −→ A

is defined as the T -algebra

m( a : T (m( A) σ−→ m( T (A) −→ m( A.

20

where the natural transformation

σm,A : T (m( A) −→ m( T (A)

follows from the strength of the monad T . Hence, the category C defines apositive M -category whereas its category of T -algebras C T defines a nega-tive category. It is not difficult to check then that the adjunction

C

F

""⊥U

cc C T

is an adjunction living inside the 2-category M – Mod of positive and neg-ative M -categories. In other words, it is an adjunction F a U together witha family of adjunctions

F (m ∗ −) a U(m−∗−)

parametrized by the object m of the category C . This enables to decomposethe strength σ of the monad T in the same way as in the case of the doublenegation monad:

σA,m : m⊗ UF (A) axm // m⊗ U(m( F (m⊗ A)) cutm // UF (m⊗ A)

using the two natural transformations

axm : F (A) −→ m( F (m⊗ A)

cutm : m⊗ U(m( A) −→ U(A).

Note that there is another way to decompose the strength of the monad T ,which is to think of the original category C as a negative M -category Cthanks to the action

( : M op × C −→ C .

In that acception, the strength of T is decomposed in a more expected way:

σm,A : UF (m( A) σF−→ U(m( FA) σU−→ m( UFA.

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The effect calculus. In a series of recent papers, Jeff Egger, RasmusMøgelberg, Alex Simpson introduce a basic effect-calculus (EC) togetherwith a semantic notion of model for the calculus, see [1] and [2]. Here,we follow [2] and consider an EC model based on a cartesian category Mrather than on a cartesian closed category as in the preliminary version ofthe paper [1]. We recall the definition of EC model below.

Definition 14 ([2]) An effect-calculus model is an [M op, Set]-enriched ad-junction

M

F

##⊥ A

U

bb

between a cartesian category M and a [M op, Set]-enriched category A . Onerequires moreover that the category A has [M op, Set]-cotensors indexed byrepresentables, and finite [M op, Set]-enriched products.

We relate this definition of effect-calculus model to the notion of M -categoryusing the following elementary observation. Suppose given a monoidal cat-egory M . In that case:

Proposition 11 A category A enriched over [M op, Set] which is moreoverequipped with [M op, Set]-cotensors indexed by representables ym = M (−,−)is the same thing as a negative M -category (A,−∗) with invertible coercions

µm,n,A : n−∗(m−∗A) −→ (m⊗ n)−∗A, µA : A −→ I−∗A.

This correspondence enables to reformulate the effect-calculus models in apurely combinatorial way, where the underlying notion of enrichment is notmentioned anymore.

Proposition 12 An effect-calculus model on a cartesian category M is thesame thing as a categorical adjunction

M

F

##⊥ A

U

bb

22

together with a pseudo-action

⇒ : M op ×A −→ A

of the cartesian category M on the category A , and a family of adjunctions

F (m×−) ` U(m⇒ −)

parametrized by m. Finally, one requires that the category A has finiteproducts, and that the canonical morphisms

m⇒ (A×B) −→ (m⇒ A)× (m⇒ B) m⇒ I −→ I

are invertible for all objects m of the category M and all objects A of thecategory A .

Here, it should be noticed that the family of adjunctions

F (m×−) ` U(m⇒ −)

may be understood as a form of closure property, providing a natural familyof bijections

ϕm,A,B : M (m× A,UB) � M (A,U(m⇒ B)).

Note that this situation is precisely what is called an exponential ideal

U : A −→ M

on the cartesian category M . In a preliminary version of their journalpaper [2], the cartesian category M is also required to be closed, see [1] fordetails. In that case, another formulation based on implication in M ratherthan product becomes possible.

Proposition 13 An effect-calculus model on a cartesian closed category Mis the same thing as a categorical adjunction

M

F

##⊥ A

U

bb

23

together with a pseudo-action

⇒ : M op ×A −→ A

of the cartesian category M on the category A , and a family of isomorphisms

σm,A : U(m⇒ A) −→ m⇒ U(A)

natural in m and A, making the two coherence diagrams

n⇒ U(m⇒ A)σn

))U(n⇒ (m⇒ A))

µ��

σn44

n⇒ (m⇒ UA)µ��

U((m⊗ n)⇒ A) σm⊗n // (m⊗ n)⇒ U(A)

U(I ⇒ A)

σI%%

µ // U(A)

I ⇒ U(A)µ

;;

commute for all object m of the category M and all objects A of the cate-gory A . Finally, one requires that the category A has finite products, andthat the canonical morphisms

m⇒ (A×B) −→ (m⇒ A)× (m⇒ B) m⇒ I −→ I

are invertible for all objects m of the category M and all objects A of thecategory A .

Note that a enriched calculus model where the category A has finite sumssuch that the

(m−∗A) + (m−∗B) −→ m−∗(A+B) 0 −→ m−∗0

is equivalent to the model of Paul Blain Levy’s call-by-push-value in [?].

Linear effect systems. This leads to a tentative definition of linear ef-fect system.

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Definition 15 A linear effect system is an adjunction

L

L

""⊥R

cc A

between a monoidal category L and a L -category A (either positive or neg-ative). The adjunction is moreover required to be enriched over the presheafcategory [L op, Set].

It is possible to give a purely combinatorial definition of linear effect sys-tems based on our reformulation of enriched adjunctions in the 2-categoryL – Mod. Quite obviously, there are two cases to treat, depending whetherthe L -category A is positive or negative. When the L -category (A , ∗) ispositive, the enrichment requirement on L a R boils down to the existenceof a family of isomorphisms

σ`,A : `⊗ L(A) −→ L(` ∗ A)

natural in the objects ` and A of the category L , and satisfying the coher-ence diagrams (1) and (2) of a positive functor. When the L -category (A ,−∗)is negative, the enrichment requirement on L a R boils down to the exis-tence of a family of adjunctions

L(` ∗ −) a R(`−∗−)

parametrized by the objects ` of the category L . Suppose now that wehave a resource modality on the category L , in the sense of linear logic.In its modern formulation, such a modality is described as a lax monoidaladjunction

M

Lin

""⊥Mult

cc L

between a cartesian category M and the monoidal category L . Recall thatsuch a lax monoidal adjunction Lin a Mult is the same thing as a cate-gorical adjunction where the left adjoint functor Lin is strongly monoidal,this meaning that the monoidal coercions

Lin(A)⊗ Lin(B) −→ Lin(A×B) I −→ Lin(I)

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are isomorphisms. From this follows that the functor

M ×LLin−→ L ×L

⊗−→ L

defines an action of the category M on the category L . Similarly, the func-tor

M ×ALin−→ L ×A −→ A

transfers the positive (resp. negative) action of the L to a positive (resp.negative) action of M on the category A . This enables to see the adjunc-tion L a R as an adjunction in the 2-category L – Mod. Now, the twoadjunctions live in L – Mod and may be composed in the following way:

M

Lin

""⊥Mult

cc L

L

&&⊥ A

R

cc

From this follows that the adjunction defines

M ×ALin−→ L ×A −→ A

transfers the positive (resp. negative) action of the L to a positive (resp.negative) action of M on the category A . From this follows that

Proposition 14 Suppose given a linear effect system between a monoidalcategory L and a category A with finite products commuting to the actionof L , in the sense that the canonical morphisms

`−∗(A×B) −→ (`−∗A)× (`−∗B) `−∗A −→ 1

are invertible. In that case, every resource modality between a cartesiancategory M and the category L induces an adjunction

M

L ◦Lin

""⊥Mult ◦R

cc A

which defines an effect calculus model. The resulting strong monad T on thecartesian category M is equal to

T = Mult ◦ R ◦ L ◦ Lin : M −→ M .

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A typical illustration is provided by the case of a dialogue category L whichalways defines a linear effect system with its opposite category L op seenas a negative L -category. Every resource modality on L thus induces aneffect calculus model between the cartesian category M involved in theresource modality and the category L op seen this time as a negative M -category. In that case, the proposition enables to transfer the double nega-tion monad of the dialogue category L based on linear continuations to adouble negation monad living in the cartesian category M , and thus basedon general continuations.

8 ConclusionThis paper may be seen as a step towards a more combinatorial and uni-fied understanding of side effects in concrete case studies. We started thepaper in that spirit by describing two apparently different ways to recoverthe strength σ of the double negation monad T from its decomposition intoan adjunction L a R or F a U . We then showed that each of the two recipesto recover σ may be understood as a procedure to lift the original adjunc-tion L a R or F a U living in the 2-category Cat to an adjunction living inthe 2-category of enriched categories over the presheaf category [M op, Set].The statement is established by reformulating the traditional notions ofenriched functors and enriched natural transformations in a purely combi-natorial way in the particular case of functors and natural transformationsbetween M -categories. The combinatorial reformulation in itself is basedon functorial strengths, costrengths, axioms and cuts. All this leads to thedefinition of a 2-category M – Mod of positive and negative M -categorieswhere these specific adjunctions decomposing strong monads may be for-mulated and studied. The combinatorial reformulation is then applied inorder to clarify the structure of categories equipped with computational ef-fects. In particular, we take this opportunity to reformulate in that style thenotion of model of effect calculus described by Jeff Egger, Rasmus Møgel-berg and Alex Simpson in a series of recent papers. A natural question is tounderstand in a similar fashion other effect systems considered in the liter-ature, and more specifically the enriched version of the effect calculus [2].We leave that point for future work.

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References[1] Jeff Egger, Rasmus Ejlers Møgelberg, Alex Simpson. Enriching an Effect Cal-

culus with Linear Types. Proceedings of Computer Science Logic 2009.

[2] Jeff Egger, Rasmus Møgelberg and Alex Simpson. The Enriched Effect Calcu-lus: Syntax and Semantics. Submitted June 2011.

[3] Anders Kock. On the double dualization monad. Math. Scand 29 (1971), pp 161- 174.

[4] Paul Blain Levy. Adjunction models for call-by-push-value with stacks. Theoryand Applications of Categories, 14:75?110, 2005.

[5] Paul-André Melliès. Game semantics in string diagrams. To appear in the Pro-ceedings of LICS 2012.

[6] Paul-André Melliès. Braided notions of dialogue categories. Manuscriptsubmitted to a journal and available on the author’s homepage:http://www.pps.jussieu.fr/∼mellies/tensorial-logic.html

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