2.1 Distributive laws for monads and extension to Kleisli categories

If $\underline {M} = (M, \eta, \mu )$ is a monad on $\mathscr {C}$ , then its Kleisli category $\mathscr {C}_{\underline {M}}$ is the category with the same objects as $\mathscr {C}$ and such that $\mathscr {C}_{\underline {M}}(X, Y) = \mathscr {C}(X, M Y)$ . The identity is given by $\eta _X \in \mathscr {C}_{\underline {M}}(X, X)$ and the composition of $f \in \mathscr {C}_{\underline {M}}(X, Y)$ with $g \in \mathscr {C}_{\underline {M}}(Y, Z)$ is defined as

(5) \begin{equation} g \circ _{\underline {M}} f = \mu _Y \circ M g \circ f. \end{equation}

There is a functor $K_{\underline {M}} : \mathscr {C} \rightarrow \mathscr {C}_{\underline {M}}$ such that $K_{\underline {M}} X = X$ and for any $f \in \mathscr {C}(X, Y)$ , $K_{\underline {M}} f = \eta \circ f \in \mathscr {C}_{\underline {M}}(X, Y)$ . Then for any $f \in \mathscr {C}_{\underline {M}}(X', X)$ , $h \in \mathscr {C}(X, Y)$ and $g \in \mathscr {C}_{\underline {M}}(Y, Y')$ ,

(6) \begin{equation} K_{\underline {M}} h \circ _{\underline {M}} f = M h \circ f \quad \quad g \circ _{\underline {M}} K_{\underline {M}} h = g \circ h. \end{equation}

These equations are obtained by a straightforward computation using naturality and the triangle identities of the monad.

Throughout this section, we assume that $\underline {M}_1 = (M_1, \eta _1, \mu _1)$ is a monad on a category $\mathscr {C}$ , $\underline {M}_2 = (M_2, \eta _2, \mu _2)$ is a monad on a category $\mathscr {D}$ , and $\underline {M}_3 = (M_3, \eta _3, \mu _3)$ is a monad on a category $\mathscr {E}$ . All the following definitions and result can be found in section 5 of Power and Watanabe (Reference Power and Watanabe2002).

Definition 2. A functor $\widehat {F} : \mathscr {C}_{\underline {M}_1} \rightarrow \mathscr {D}_{\underline {M}_2}$ is an extension of the functor $F : \mathscr {C} \rightarrow \mathscr {D}$ if the following diagram commutes.

These extensions are related to the notion of distributive laws.

Definition 5. A natural transformation $\lambda : F \underline {M}_1 \Rightarrow \underline {M}_2 F$ is a distributive law between $F$ and two monads $\underline {M}_1$ and $\underline {M}_2$ if the following two diagrams commute.

Definition 7. Let $\lambda : F \underline {M}_1 \Rightarrow \underline {M}_2 F$ and $\lambda ' : G \underline {M}_1 \Rightarrow \underline {M}_2 G$ two distributive laws. A morphism between $\lambda$ and $\lambda '$ is a natural transformation $\alpha : F \Rightarrow G$ such that the diagram below commutes.

Theorem 8. (part one of Theorem 4.8 of Power and Watanabe (Reference Power and Watanabe2002)) Any extension $\widehat {F} : \mathscr {C}_{\underline {M}_1} \rightarrow \mathscr {D}_{\underline {M}_2}$ induces a distributive law $\lambda : F \underline {M}_1 \Rightarrow \underline {M}_2 F$ . This law is given by the image of $\mathsf {id}_{M_1X} \in \mathscr {C}_{\underline {M}_1}(M_1 X, X)$ by $\widehat {F}$

\begin{equation*} \lambda _X = \widehat {F}(\mathsf {id}_{M_1 X}) \in \mathscr {D}_{\underline {M}_2}(F M_1 X, F X) = \mathscr {D}(F M_1 X, M_2 F X). \end{equation*}

Conversely, any distributive law $\lambda : F \underline {M}_1 \Rightarrow \underline {M}_2 F$ induces an extension $\widehat {F} : \mathscr {C}_{\underline {M}_1} \rightarrow \mathscr {D}_{\underline {M}_2}$ that maps an object $X$ to $\widehat {F} X = F X$ and a morphism $f \in \mathscr {C}_{\underline {M}_1}(X, Y) = \mathscr {C}(X, M_1 Y)$ to

Those two constructions are inverse of each other, so there is a bijection between extensions and distributive laws.

Remark 9. This bijection is compositional. The natural transformation $\mathsf {id}_{M} : M \Rightarrow M$ is a distributive law between the identity functor $\mathsf {Id}^{\mathscr {C}}$ and the monad $\underline {M}$ , associated to the extension $\mathsf {Id}^{\mathscr {C}_{\underline {M}}}$ . We can check that if $\widehat {F}$ is an extension of $F : \mathscr {C} \rightarrow \mathscr {D}$ whose associated distributive law is $\lambda$ , and if $\widehat {G}$ is an extension of $G : \mathscr {D} \rightarrow \mathscr {E}$ whose associated distributive law is $\lambda '$ , then $\widehat {G} \widehat {F}$ is an extension of $G F$ whose associated distributive law is defined as the following pasting diagram.

(7)

which corresponds to the following composition: .

The result below is also proved in Power and Watanabe (Reference Power and Watanabe2002). The proof is simple, but this result is crucial in the development of this article.

2.2 Distributive law for comonads and extension to coKleisli categories

The coKleisli category of a comonad $\overline {D}$ is defined in a dual way to the Kleisli category of a monad. If $\overline {D} = (D, \epsilon, \delta )$ is a comonad on $\mathscr {C}$ , then its coKleisli category $\mathscr {C}_{\overline {D}}$ is the category with the same objects as $\mathscr {C}$ and such that $\mathscr {C}_{\overline {D}}(X, Y) = \mathscr {C}(D X, Y)$ . The identity is given by $\epsilon _X \in \mathscr {C}_{\overline {D}}(X, X)$ and the composition of $f \in \mathscr {C}_{\overline {D}}(X, Y)$ with $g \in \mathscr {C}_{\overline {D}}(Y, Z)$ is defined as

(8) \begin{equation} g \circ _{\overline {D}} f = g \circ D f \circ \delta. \end{equation}

There is a functor $K_{\overline {D}} : \mathscr {C} \rightarrow \mathscr {C}_{\overline {D}}$ such that $K_{\overline {D}} X = X$ and for any $f \in \mathscr {C}(X, Y)$ , $K_{\overline {D}} f = f \circ \epsilon \in \mathscr {C}_{\overline {D}}(X, Y)$ .

The notion of structure extension to the coKleisli category of a comonad is the same as the one of Definitions 2 and 3 for Kleisli categories. Extensions to the coKleisli categories are also related to distributive law, except that this time the natural transformation goes in the opposite direction and has type $\overline {D}_2 F \Rightarrow F \overline {D}_1$ .

Definition 12. A natural transformation $\lambda : \overline {D}_2 F \Rightarrow F \overline {D}_1$ is a distributive law between $F$ and two comonads $\overline {D}_1$ and $\overline {D}_2$ if the following two diagrams commute.

Definition 13. Let $\lambda : \overline {D}_2 F \Rightarrow F \overline {D}_1$ and $\lambda ' : \overline {D}_2 G \Rightarrow G \overline {D}_1$ two distributive laws. A morphism between $\lambda$ and $\lambda '$ is a natural transformation $\alpha : F \Rightarrow G$ such that the diagram below commutes.

Again, this bijection is compositional in a similar way to Remark 9 and provides an isomorphism of $2$ -categories.

2.3 Distributive laws and lifting to (co)Eilenberg-moore category

There are two other notions of distributive law, by taking the dual direction of Definitions 5 and 12.

Those notions of distributive law are tied to the notion of lifting of a functor and natural transformations to the Eilenberg-Moore categories $\mathscr {C}^{\underline {M}_1}$ and $\mathscr {C}^{\underline {M}_2}$ of the monads (and similarly the coEilenberg-Moore categories $\mathscr {C}^{\overline {D}_1}$ and $\mathscr {C}^{\overline {D}_2}$ of the comonads). A lifting of a functor $\mathscr {C} \rightarrow \mathscr {D}$ is a functor $\overline {F} : \mathscr {C}^{\underline {M}_1} \rightarrow \mathscr {C}^{\underline {M}_2}$ such that the diagram below commutes (here, $\mathscr {U}$ is the forgetful functor), see Power and Watanabe (Reference Power and Watanabe2002) for more details.

We summarize the different configurations for a distributive law, depending on the direction of the natural transformation and the choice between monads and comonads in Figure 2.

Figure 2.

Configurations for distributive laws.

2.4 Distributive laws between monads and comonads

The central notion of this article, Taylor expansion, is given by a distributive law $\partial : ! \mathsf {S} \Rightarrow \mathsf {S} !$ between a functor $\mathsf {S}$ that represents partial summability (see section 4) and the resource comonad $!\_$ of LL (see Definition 98). It turns out that $\mathsf {S}$ is a monad, and that $\partial : ! \mathsf {S} \Rightarrow \mathsf {S} !$ is also a distributive law between the functor $!\_$ , and the monad $\mathsf {S}$ . As such, the following notion of distributive law of a comonad over a monad is central to this article.

Similarly, there are distributive laws between two monads, as well as distributive laws between two comonads, with similar definitions as above. In fact, distributive laws were first introduced in Beck (Reference Beck and Eckmann1969) between two monads. The notion of distributive law of a comonad over another comonad is also of interest in the article, because the functor $\mathsf {S}$ admits a comonadic structure and $\partial : ! \mathsf {S} \Rightarrow \mathsf {S} !$ is a distributive law of the comonad $!\_$ over the comonad $\mathsf {S}$ .

Observe that the following assertions are equivalent, by definition.

1. (1) $\lambda$ is a distributive law of the comonad $\overline {D}$ over the monad $\underline {M}$ ,

2. (2) $\lambda$ is a distributive law $D \underline {M} \Rightarrow \underline {M} D$ and $\epsilon$ and $\delta$ are morphisms of distributive laws,

3. (3) $\lambda$ is a distributive law $\overline {D} M \Rightarrow M \overline {D}$ and $\eta$ and $\mu$ are morphisms of distributive laws,

where the distributive laws $D^2 \underline {M} \Rightarrow \underline {M} D^2$ , $\overline {D} M^2 \Rightarrow M^2 \overline {D}$ , $\underline {M} \Rightarrow \underline {M}$ and $\overline {D} \Rightarrow \overline {D}$ involved above are given by compositionality of the distributive laws, see Remark 9.

By Theorem 14, a distributive law of a comonad over a monad provides a monad $\widehat {\underline {M}}$ on $\mathscr {C}_{\overline {D}}$ that extends $\underline {M}$ . By Theorems 8 and 10, such distributive law also provides a comonad $\widehat {\overline {D}}$ on $\mathscr {C}_{\underline {M}}$ that extends $\overline {D}$ . We can also check that $(\mathscr {C}_{\overline {D}})_{\widehat {\underline {M}}} = (\mathscr {C}_{\underline {M}})_{\widehat {\overline {D}}}$ , so a distributive law of a comonad over a monad allows to combine them in arbitrary ways. This result was first proved in Van Osdol (Reference Van Osdol1971).

There is also another notion of distributive laws of the monad $\underline {M}$ over the comonad $\overline {D}$ obtained by reversing the arrows. Such distributive laws are associated to lifting of the monad $\underline {M}$ to the coEilenberg-Moore category of $\overline {D}$ , and lifting of $\overline {D}$ to the Eilenberg-Moore category of $\underline {M}$ .

Those notions of distributive laws between monads and comonads allow in particular to define the notion of bimonad as a functor equipped with a structure of monad and a structure of comonad together with a distributive law expressing that these two structures commute in some sense. Bimonads are formally similar to the notion of Hopf bialgebra in a symmetric monoidal category, apart that the role of the symmetry is played here by the distributive law.

Definition 17 (6.2 of Mesablishvili and Wisbauer (Reference Mesablishvili and Wisbauer2011)). Assume that $\overline {H} = (H, \eta, \mu )$ is a monad on $\mathscr {C}$ and $\underline {H} = (H, \epsilon, \delta )$ is a comonad on $\mathscr {C}$ . Then $(H, \eta, \mu, \epsilon, \delta )$ is a $\tau$ -bimonad if $\tau$ is a distributive law $\overline {H} \underline {H} \Rightarrow \underline {H} \overline {H}$ of the monad $\overline {H}$ over the comonad $\underline {H}$ , and a distributive law $\underline {H} \overline {H} \Rightarrow \overline {H} \underline {H}$ of the comonad $\underline {H}$ over the monad $\overline {H}$ , and if the diagrams below commute.

Recall that $\ast$ is defined as the horizontal composition of natural transformations: if $F, G : \mathscr {C} \rightarrow \mathscr {D}$ , $F', G' : \mathscr {D} \rightarrow \mathscr {E}$ , $\alpha : F \Rightarrow G$ and $\beta : F' \Rightarrow G'$ , then $\beta \ast \alpha : GF \Rightarrow G' F'$ is defined as

(9) \begin{equation} (\beta \ast \alpha )_X = \beta _{F' X} \circ G \alpha _X = G' \alpha _X \circ \beta _{F X}. \end{equation}

3.1 Distributive laws on product categories

Let us recall first some fact and notation about product categories. The category $\mathbf {Cat}$ whose objects are the (small) categories and whose morphisms are the functors is a cartesian category, with terminal object the category $\mathbf {1}$ which contains one object and one morphism, and whose categorical product is defined as follows.

Given two categories $\mathscr {C}_1$ and $\mathscr {C}_2$ , the product category $\mathscr {C}_1 \times \mathscr {C}_2$ is the category whose objects are the pairs $(X_1, X_2)$ with $X_1 \in \mathbf {Obj}(\mathscr {C}_1)$ and $X_2 \in \mathbf {Obj}(\mathscr {C}_2)$ and whose morphisms are the pairs $(f_1, f_2)$ with $f_1 \in \mathscr {C}_1(X_1, Y_1)$ and $f_2 \in \mathscr {C}_2(X_2, Y_2)$ .

For any functors $F_1 : \mathscr {C}_1 \rightarrow \mathscr {D}_1$ and $F_2 : \mathscr {C}_2 \rightarrow \mathscr {D}_2$ , we can define the functor $F_1 \times F_2 : \mathscr {C}_1 \times \mathscr {C}_2 \rightarrow \mathscr {D}_1 \times \mathscr {D}_2$ by $(F_1 \times F_2) (X_1, X_2) = (F_1 X_1, F_2 X_2)$ and $(F \times F_2) (f_1, f_2) = (F_1 f_1, F_2 f_2)$ .

Given $F_1 : \mathscr {C} \rightarrow \mathscr {D}_1$ and $F_2 : \mathscr {C} \rightarrow \mathscr {D}_2$ , we can define $\langle F_1, F_2 \rangle : \mathscr {C} \rightarrow \mathscr {D}_1 \times \mathscr {D}_2$ by $\langle F_1, F_2 \rangle X = (F_1 X, F_2 X)$ and $\langle F_1, F_2 \rangle f = (F_1 f, F_2 f)$ . For any category $\mathscr {C}$ , we can define the functor $\Delta ^{\mathscr {C}} : \mathscr {C} \rightarrow \mathscr {C} \times \mathscr {C}$ by $\Delta ^{\mathscr {C}}(X) = (X, X)$ and $\Delta ^{\mathscr {C}} f = (f, f)$ . That is, $\Delta ^{\mathscr {C}} = \langle \mathsf {Id}, \mathsf {Id} \rangle$ .

Given the functors $F_1, G_1 : \mathscr {C}_1 \rightarrow \mathscr {D}_1$ and $F_2, G_2 : \mathscr {C}_2 \rightarrow \mathscr {D}_2$ and two natural transformation $\alpha _1 : F_1 \Rightarrow G_1$ and $\alpha _2 : F_2 \Rightarrow G_2$ , we can define the natural transformation $(\alpha _1, \alpha _2) : F_1 \times F_2 \Rightarrow G_1 \times G_2$ by $(\alpha _1, \alpha _2)_{(X_1, X_2)} = (\alpha _{1, X_1}, \alpha _{2, X_2})$ .

We can check that $(\mathscr {C}_1 \times \mathscr {C}_2)_{\underline {M}_1 \times \underline {M}_2} = \mathscr {C}_{\underline {M}_1} \times \mathscr {C}_{\underline {M}_2}$ , so the following lemmas make sense.

3.2 Lax monoidal functor and lax monoidal monads

We recall in this section basic definitions on symmetric monoidal category and monoidal monads, we refer the reader to Melliès (Reference Melliès2009) for more details. Then we show that any symmetric monoidal monad can be interpreted as a distributive law. This explains why a symmetric monoidality structure on a monad allows to extend the symmetric monoidal structure of the category on which the monad is defined to its Kleisli category.

Definition 26. A lax symmetric monoidal functor from a symmetric monoidal category $(\mathscr {L}, \otimes, 1)$ to another symmetric monoidal category $(\mathscr {L}', \bullet, I)$ is a tuple $(F, m^0, m^2)$ where $F : \mathscr {L} \rightarrow \mathscr {L}'$ is a functor, $m^0 \in \mathscr {L}'(I, F 1)$ and $m^2_{X,Y} \in \mathscr {L}'(F X \bullet F Y, F (X \otimes Y))$ is a natural transformation that are compatible with the monoidal structure

and compatible with the symmetry.

A strong symmetric monoidal functor is a lax symmetric monoidal functor such that $m^0$ is an iso, and $m^2$ is a natural isomorphism.

Definition 27. A monoidal natural transformation between two monoidal functors $(F, m^0_F, m^2_F)$ and $(G, m^0_G, m^2_G)$ is a natural transformation $\alpha : F \Rightarrow G$ such that the following diagram commutes.

There is in fact a 2-category in which the objects are the (small) monoidal categories, the morphisms are the lax monoidal functors, and the 2-cells are the monoidal natural transformations. The composition of two lax monoidal functors is given by the composition of the functor and a suitable composition of their associated natural transformations. The compositions of $2$ -cells are the same as the compositions in $\mathbf {Cat}$ .

Definition 28. A lax symmetric monoidal monad on a symmetric monoidal category $(\mathscr {L}, \otimes, 1)$ is the data of $(M, \eta, \mu, m^0, m^2)$ such that $(M, m^0, m^2)$ is a lax symmetric monoidal functor from $(\mathscr {L}, \otimes, 1)$ to itself, $(M, \eta, \mu )$ is a monad on $\mathscr {L}$ , and such that $\eta$ and $\mu$ are monoidal natural transformations. This last assumption corresponds to the fact that $m^0 = \eta _{1} \in \mathscr {L}(1, M 1)$ and to the commutation of the following diagrams.

Lax symmetric monoidal monads are a well-studied notion because they are related to the extension of the symmetric monoidal structure to the Kleisli category.

It turns out that lax monoidal monads are an example of distributive laws. This observation sheds light on Theorem 29 above, and is doubtlessly folklore, but seems to be often overlooked in the literature. We can trace this observation to Guitart (Reference Guitart1980).Footnote ⁴

The diagrams of Definition 28 correspond to the fact that the natural transformation $m^2 : (\_ \otimes \_) (M \times M) \Rightarrow M (\_ \otimes \_)$ is a distributive law between the functor $\_ \otimes \_$ and the monads $\underline {M} \times \underline {M}$ and $\underline {M}$ . By Theorem 8, it means that $\_ \otimes \_$ extends to a functor $\_ \widehat {\otimes } \_ : \mathscr {L}_{\underline {M}} \times \mathscr {L}_{\underline {M}} \rightarrow \mathscr {L}_{\underline {M}}$ . Then the commutations of Definition 26 corresponds to the fact that $\lambda ^{\otimes }$ , $\rho ^{\otimes }$ , $\alpha ^{\otimes }$ , and $\gamma ^{\otimes }$ are morphisms of distributive laws. The distributive laws involved with $\alpha ^{\otimes }$ are given by compositionality, and the distributive laws involved with $\lambda ^{\otimes }$ and $\rho ^{\otimes }$ are given by Proposition 30 below.

By Theorem 10, this is equivalent to the fact that $\lambda ^{\otimes }$ , $\rho ^{\otimes }$ , $\alpha ^{\otimes }$ , and $\gamma ^{\otimes }$ extend to natural transformations on $\mathscr {L}_{\underline {M}}$ . This is why the Kleisli category $\mathscr {L}_{\underline {M}}$ of a lax monoidal monad inherits from $\mathscr {L}$ the structure of a symmetric monoidal category.

Figure 3.

Extension and lifting of the monoidal structure.

3.3 Commutative monad

It is well known that symmetric monoidal monads are the same as commutative monads, see Kock (Reference Kock1970, Reference Kock1972). Let us recall what is a commutative monad. Let $\underline {M}$ be a monad on a symmetric monoidal category $\mathscr {L}$ .

Definition 32. A left strength for $\underline {M}$ is a natural transformation $\mathsf {t}_{X, Y} \in \mathscr {L}(X \otimes M Y, M (X \otimes Y))$ subject to the compatibility with the monoidal structure

and the compatibility with the monad structure

A right strength is a natural transformation $\mathsf {s}_{X, Y} \in \mathscr {L}(M X \otimes Y, M (X \otimes Y))$ subject to similar commutations.

When the category is symmetric monoidal, any left strength $\mathsf {t}$ induces a right strength

(10) \begin{equation} \mathsf {s}_{X,Y} = M \gamma ^{\otimes }_{Y,X} \circ \mathsf {t}_{Y,X} \circ \gamma ^{\otimes }_{X,Y} \end{equation}

and vice versa. This lead to the following naming convention in symmetric monoidal categories.

Definition 34. A commutative monad is a monad equipped with a strength $\mathsf {t}$ such that the following diagram commutes.

where $\mathsf {s}$ is the right-strength obtained from $\mathsf {t}$ by Eq. (10) .

It is well known that any commutative monad is a lax symmetric monoidal monad (see Kock (Reference Kock1970)), defining $m^2 \in \mathscr {L}(M X \otimes M Y, M (X \otimes Y))$ as the diagonal of the square above. Conversely, any symmetric monoidal monad is a commutative monad, see Kock (Reference Kock1972). The left and right strengths are defined from the lax monoidal structure by

Recall from Proposition 30 that $\mathsf {t}_{X, \_}$ is the distributive law associated to the extension $X \widehat {\otimes } \_$ of $X \otimes \_$ , and $\mathsf {s}_{\_, Y}$ is the distributive law associated to the extension $\_ \widehat {\otimes } Y$ of $\_ \otimes Y$ . So the equivalence between lax symmetric monoidal monads and commutative monads can also be understood as the fact that providing an extension $\_ \widehat {\otimes } \_$ is the same as providing two extensions $X \widehat {\otimes } \_$ and $\_ \widehat {\otimes } Y$ that follows the compatibility condition of the bifunctor theorem, see proposition 1 of MacLane (Reference MacLane1971).

4.1 Categories enriched over sigma-monoids

If $A$ is a countable set and $M$ is a set, an $A$ -indexed family of elements of $M$ is a function $\vec {x} : A \rightarrow M$ . We also write $\vec {x} = (x_a)_{a \in A}$ . We consider in this section non-empty sets $M$ together with a partial function $\Sigma$ from indexed family on $M$ to $M$ called the sum. An indexed family $(x_a)_{a \in A}$ is called summable when it is in the domain of $\Sigma$ , and we write its image as $\sum _{a \in A} x_a$ .

It follows from (PA) that any subfamily of a summable family is also summable. In particular, it follows from (PA) and (Unary) that the empty family is always summable (if $M$ is not empty). Let $0$ be its sum. For any family $\vec {x} = (x_a)_{a \in A}$ , we define

(12) \begin{equation} \mathsf {supp}(\vec {x}) = \{a \in A \mid x_a \neq 0 \}. \end{equation}

Then $0$ behaves as the neutral element, as shown in Proposition 38 below.

Definition 42. We generalize Definition 40 above. Given two indexing set $A$ and $B$ , an injection $\varphi :A \hookrightarrow B$ and a family $\vec {x}= (x_a)_{a \in A}$ we define a $B$ -indexed family $\varphi _{\ast }\overrightarrow {x}= (y_b)_{b \in B}$ by

\begin{equation*} y_b= \begin{cases} x_a & if \ b\in \varphi (A) \ and \ \varphi (a)=b\\ 0 & otherwise. \end{cases} \end{equation*}

We introduce the following terminology, as an analogue of the terminology of additive and left additive categories used in differential categories and cartesian differential categories, see Reference Blute, Cockett and SeelyBlute et al., (2006, 2009). The monoid structure on the hom-set is replaced by a $\Sigma$ -monoid structure.

Definition 45. A left $\Sigma$ -additive category is a category $\mathscr {C}$ such that for any objects $X, Y \in \mathbf {Obj}(\mathscr {C})$ , the hom-set $\mathscr {C}(X, Y)$ is a $\Sigma$ -monoid, and such that the sum is left distributive over composition: for all objects $X, Y, Z$ , for all indexed family $(g_b \in \mathscr {C}(Y, Z))_{b \in B}$ and $f \in \mathscr {C}(X, Y)$ ,

(13) \begin{equation} \left ( \sum _{b \in B} g_b \right ) \circ f \sqsubseteq \sum _{b \in B} g_b \circ f. \end{equation}

A morphism $h \in \mathscr {C}(Y, Z)$ is $\Sigma$ -additive if for all indexed family $(f_a \in \mathscr {C}(X, Y))_{a \in A}$

(14) \begin{equation} h \circ \left (\sum _{a \in A} f_a \right ) \sqsubseteq \sum _{a \in a} (h \circ f_a). \end{equation}

A $\Sigma$ -additive category is a left $\Sigma$ -additive category in which all morphisms are $\Sigma$ -additive. A $\Sigma$ -additive category can also be defined as a category enriched over the category of $\Sigma$ -monoids, see Haghverdi (Reference Haghverdi2000).

We want to stress however that (left) $\Sigma$ -additive categories are not necessarily (left) additive categories. The sum of a $\Sigma$ -monoid may be defined on some infinite indexed families, but undefined on some finite indexed family. None of the examples of $\Sigma$ -additive categories given in section 11 are additive categories.

Let $0^{X, Y}$ be the zero of the $\Sigma$ -monoid structure of $\mathscr {C}(X, Y)$ . It follows from the definition that for any $f \in \mathscr {C}(X, Y)$ , $0^{Y, Z} \circ f = 0^{X, Y}$ , and that for any $\Sigma$ -additive $h \in \mathscr {C}(Y, Z)$ , $h \circ 0^{X, Y} = 0^{X, Z}$ . We will often omit the superscript and simply write $0$ .

Distributivity of the left and on the right implies double distributivity, thanks to the axiom (PA).

Proposition 46. For any left $\Sigma$ -additive category $\mathscr {C}$ , any indexed family $(f_a \in \mathscr {C}(X, Y))_{a \in A}$ and any indexed family $(h_b \in \mathscr {C}(Y, Z))_{b \in B}$ such that $h_b$ is $\Sigma$ -additive for all $b \in B$ ,

\begin{equation*} \left (\sum _{b \in B} h_b \right ) \circ \left (\sum _{a \in A} f_a \right ) \sqsubseteq \sum _{(a, b) \in A \times B} (h_b \circ f_a)\,. \end{equation*}

Proof. By left distributivity and additivity of the $h_b$

\begin{equation*} \left (\sum _{b \in B} h_b \right ) \circ \left (\sum _{a \in A} f_a \right ) \sqsubseteq \sum _{b \in B} \left ( h_b \circ \left ( \sum _{a \in A} f_a \right ) \right ) \sqsubseteq \sum _{b \in B} \sum _{a \in A} (h_b \circ f_a).\end{equation*}

But the sets $A \times \{b\}$ are a partition of $A \times B$ , so (PA) ensures that

\begin{equation*} \sum _{(a, b) \in A \times B} (h_b \circ f_a) \cong \sum _{b \in B} \sum _{a \in A} (h_b \circ f_a). \end{equation*}

Proof. Let $(f_a \in \mathscr {C}(X, Y))_{a \in A}$ be a summable family. It follows from Proposition 46 that

\begin{equation*} \left (\sum _{b \in B} h_b \right ) \circ \left (\sum _{a \in A} f_a \right ) \sqsubseteq \sum _{(a, b) \in A \times B} (h_b \circ f_a) \end{equation*}

Then, the sets $\{a\} \times B$ are a partition of $A \times B$ , so it follows from (PA) that

\begin{equation*} \sum _{(a, b) \in A \times B} (h_b \circ f_a) \cong \sum _{a \in A} \sum _{b \in B} (h_b \circ f_a) \end{equation*}

Finally, we know by left distributivity that

\begin{equation*} \sum _{b \in B} (h_b \circ f_a) = \left ( \sum _{b \in B} h_b \right ) \circ f_a. \end{equation*}

It follows that

\begin{equation*} \left (\sum _{b \in B} h_b \right ) \circ \left (\sum _{a \in A} f_a \right ) \sqsubseteq \sum _{a \in A} \left ( \left ( \sum _{b \in B} h_b \right ) \circ f_a \right ). \end{equation*}

4.2 Sigma-summability structures

We assume in this section that $\mathscr {C}$ is a left $\Sigma$ -additive category.

Intuitively, SS-1 and SS-2 mean that $\mathsf {S} X$ can be seen as the object of ”summable sequences”. The morphism $\pi _i$ maps a sequence to its $i$ -th element. Then, SS-3 ensures that the sums between sequences is given by the sum on each component

\begin{equation*} \pi _i \circ \sum _{a \in A} \left \langle \!\left \langle f_{i,a} \right \rangle \!\right \rangle _{i \in \mathbb {N}} \sqsubseteq \sum _{a \in A} f_{i,a} \text { so } \sum _{a \in A} \left \langle \!\left \langle f_{i,a} \right \rangle \!\right \rangle _{i \in \mathbb {N}} = \left \langle \!\left \langle \sum _{a \in A} f_{i, a} \right \rangle \!\right \rangle _{i \in \mathbb {N}}. \end{equation*}

Finally, SS-4 crucially ensures that $(\left \langle \!\left \langle f_{i,a} \right \rangle \!\right \rangle _{i \in \mathbb {N}})_{a \in A}$ is summable if and only if $(f_{i,a})_{(i,a) \in \mathbb {N} \times A}$ is summable (the only if is a consequence of double distributivity, see Proposition 46). So the summability of sequences is completely determined by the global summability of all of their components.

We now assume that $\mathscr {C}$ is equipped with a left $\Sigma$ -summability structure. Observe that by left distributivity,

(15) \begin{equation} \sigma \circ \left \langle \!\left \langle f_i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \sqsubseteq \sum _{i \in \mathbb {N}} (\pi _i \circ \left \langle \!\left \langle f_i \right \rangle \!\right \rangle _{i \in \mathbb {N}}) = \sum _{i \in \mathbb {N}} f_i. \end{equation}

Thus, the data of $(\mathsf {S}, (\pi _i)_{i \in \mathbb {N}}, \sigma )$ completely characterize the summability and the values of sums on $\mathbb {N}$ -indexed families. By Lemma 44, this is enough to characterize the $\Sigma$ -monoid structure on all indexed family, because any countable set has an injection into $\mathbb {N}$ . This explains why earlier versions of this paper define summability structures first, and the $\Sigma$ -monoid structure simply arise as a by-product. The definition of sum of those earlier version were similar to the characterization given by Proposition 54 below.

Proposition 54. A family $\vec {f} = (f_a \in \mathscr {C}(X, Y))_{a \in A}$ is summable if and only if there exists an injection $\varphi : A \hookrightarrow \mathbb {N}$ and a morphism $\langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi } \in \mathscr {C}(X, \mathsf {S} Y)$ such that $\langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi }$ is a witness for $\varphi _{\ast } f$ . That is,

\begin{equation*} \pi _i \circ \langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi } = \begin{cases} f_a\ if \ i = \varphi (a) \ for\ some\ a \in A \\ 0 \ otherwise. \end{cases} \end{equation*}

Furthermore, $\sum _{a \in A} f_a = \sigma \circ \langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi }$ .

The characterization of summability in terms of the existence of witnesses implies that properties involving summability can often be expressed in terms of the basic data of the summability structure. This is the case for instance of the additivity of morphisms.

Proof. The forward implication is a direct consequence of the definition of additivity applied on the empty family and on the summable family $(\pi _i)_{i \in \mathbb {N}}$ . For the backward implication, assume that $\vec {f} = (f_a)_{a \in A}$ is summable. By Proposition 54, there exists an injection $\varphi : A \hookrightarrow \mathbb {N}$ and a witness $\langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi }$ of $\varphi _{\ast } \vec {f}$ . Then we can check that $\left \langle \!\left \langle h \circ \pi _i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \circ \langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi }$ is a witness for $\varphi _{\ast } (h \circ f_a)_{a \in A}$ . Indeed,

\begin{equation*} h \circ \pi _i \circ \langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi } = \begin{cases} h \circ f_a \text { if } i \in \varphi (A) \text { and } \varphi (a) = i \\ h \circ 0 = 0 \text { otherwise.} \end{cases} \end{equation*}

Thus, by Proposition 54, $(h \circ f_a)_{a \in A}$ is summable with sum

\begin{equation*} \sum _{a \in A} h \circ f_a = \sigma \circ \left \langle \!\left \langle h \circ \pi _i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \circ \langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi } = h \circ \sigma \circ \langle \!\langle \vec {f} \rangle \!\rangle ^{\varphi } = h \circ \sum _{a \in A} f_a.\end{equation*}

It concludes the proof.

4.3 The category of sigma-additive morphisms

We assume in this section that $\mathscr {C}$ is a left $\Sigma$ -additive category. It is easy to show that the identity morphism $\mathsf {id}_X \in \mathscr {C}(X, X)$ is $\Sigma$ -additive, and that the composition of two $\Sigma$ -additive morphisms is also $\Sigma$ -additive.

There is an obvious forgetful functor from $\mathscr {C}^{\mathsf {add}}$ to $\mathscr {C}$ that we always keep implicit. It follows from Corollary 47 that $\mathscr {C}^{\mathsf {add}}$ is a $\Sigma$ -additive category. We now assume that $\mathscr {C}$ is equipped with a left $\Sigma$ -summability structure $(\mathsf {S}, \vec {\pi })$ .

Proof. Assume that $(f_a \in \mathscr {C}(X, Y))_{a \in A}$ is summable. We first want to show that $(\left \langle \!\left \langle h_i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \circ f_a)_{a \in A}$ is summable. By SS-4, it suffices to show that $(h_i \circ f_a)_{(i, a) \in \mathbb {N} \times A}$ is summable. This is the case, by double distributivity (Proposition 46). Then, we show that

\begin{equation*} \left \langle \!\left \langle h_i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \circ \left (\sum _{a \in A} f_a \right ) = \sum _{a \in A} (\left \langle \!\left \langle h_i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \circ f_a).\end{equation*}

This is done in a straightforward way using the additivity and the joint monicity of the $\pi _i$ .

Recall Proposition 55: a morphism $h$ is $\Sigma$ -additive if and only if $(h \circ \pi _i)_{i \in \mathbb {N}}$ is summable with sum $h \circ \sigma$ . We define

(16) \begin{equation} \mathsf {S} h = \left \langle \!\left \langle h \circ \pi _i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \end{equation}

so that $\pi _i \circ \mathsf {S} h = h \circ \pi _i$ . Furthermore, $\sigma \circ \mathsf {S} h = h \circ \sigma$ by Eq. (15). We can easily check by joint monicity of the $\pi _i$ that $\mathsf {S} h \circ \left \langle \!\left \langle x_i \right \rangle \!\right \rangle _{i \in \mathbb {N}} = \left \langle \!\left \langle h \circ x_i \right \rangle \!\right \rangle _{i \in \mathbb {N}}$ , so $\mathsf {S} h$ consists intuitively in applying $h$ in each argument.

Proof. Observe first that for all $h \in \mathscr {C}^{\mathsf {add}}(X, Y)$ , $\mathsf {S} h \in \mathscr {C}^{\mathsf {add}}(\mathsf {S} X, \mathsf {S} Y)$ by definition of $\mathsf {S} h$ and Lemma 58. Then

\begin{equation*} \pi _i \circ \mathsf {S} \mathsf {id}_X = \mathsf {id}_{X} \circ \pi _i = \pi _i = \pi _i \circ \mathsf {id}_{\mathsf {S} X} \end{equation*}

so $\mathsf {S} \mathsf {id}_X = \mathsf {id}_{\mathsf {S} X}$ by joint monicity of the $\pi _i$ . Then, assume that $h \in \mathscr {C}(X, Y)$ and $h' \in \mathscr {C}(Y, Z)$ are $\Sigma$ -additive.

\begin{equation*} \pi _i \circ \mathsf {S} h' \circ \mathsf {S} h = h' \circ \pi _i \circ \mathsf {S} h = h' \circ h \circ \pi _i = \pi _i \circ \mathsf {S} (h' \circ h) \end{equation*}

so $\mathsf {S} (h' \circ h) = \mathsf {S} h' \circ \mathsf {S} h$ by joint monicity of the $\pi _i$ . Thus, $\mathsf {S}$ is a functor, and we already know that $\pi _i \, \mathsf {S} h = h \, \pi _i$ , $\sigma \, \mathsf {S} h = h \, \sigma$ , so they are natural transformations.

Proposition 61. For all family $(h_a)_{a \in A}$ of $\Sigma$ -additive morphisms,

\begin{equation*} \mathsf {S} \left ( \sum _{a \in A} h_a \right ) \cong \sum _{a \in A} (\mathsf {S} h_a) \end{equation*}

Proof. Let $\varphi : A \hookrightarrow \mathbb {N}$ be an injection. Assume that $(h_a)_{a \in A}$ is summable. Then $(h_i')_{i \in \mathbb {N}} = \varphi _{\ast }{(h_a)_{a \in A}}$ is summable by Lemma 44. So by double distributivity,

\begin{equation*} (h_i' \, \pi _j)_{(i, j) \in \mathbb {N} \times \mathbb {N}} = (\pi _j \, \mathsf {S} h_i')_{(i, j) \in \mathbb {N} \times \mathbb {N}} \end{equation*}

is summable. It follows by SS-4 that $(\mathsf {S} h_i')_{i \in \mathbb {N}}$ is summable. But $\mathsf {S} 0 = 0$ (this directly follows from the joint monicity of the $\pi _i$ ). So $(\mathsf {S} h_i')_{i \in \mathbb {N}} = \varphi _{\ast }{(\mathsf {S} h_a)_{a \in A}}$ and $(\mathsf {S} h_a)_{a \in A}$ is summable by Lemma 44. Furthermore,

\begin{equation*} \pi _i \, \mathsf {S} \left ( \sum _{a \in A} h_a \right ) = \left ( \sum _{a \in A} h_a \right ) \, \pi _i \sqsubseteq \sum _{a \in A} (h_a \, \pi _i) \end{equation*}

\begin{equation*} \pi _i \, \left ( \sum _{a \in A} \mathsf {S} h_a \right ) \sqsubseteq \sum _{a \in A} (\pi _i \, \mathsf {S} h_a) = \sum _{a \in A} (h_a \, \pi _i) \end{equation*}

so by joint monicity of the $\pi _i$ , the two sums are equal.

To conclude the proof, assume that $(\mathsf {S} h_a)_{a \in A}$ is summable and let us show that $(h_a)_{a \in A}$ is summable. As observed above, $(\mathsf {S} h_i')_{i \in \mathbb {N}} = \varphi _{\ast }{(\mathsf {S} h_a)_{a \in A}}$ so this family is summable by Lemma 44. It follows by double distributivity that $(\pi _j \, \mathsf {S} h_i')_{(i, j) \in \mathbb {N} \times \mathbb {N}} = (h_i' \, \pi _j)_{(i, j) \in \mathbb {N} \times \mathbb {N}}$ is summable.

Then, let $\iota _i \in \mathscr {C}(X, \mathsf {S} X)$ be the morphism such that $\pi _i \circ \iota _i = \mathsf {id}_X$ and $\pi _j \circ \iota _i = 0$ if $j \neq i$ . This morphism exists thanks to (Unary) and SS-2. Then by left distributivity in $\mathscr {C}$ , $(h_i' \circ \pi _j \circ \iota _0)_{(i, j) \in \mathbb {N} \times \mathbb {N}}$ is summable. This indexed family has values $h_i'$ on indexes $(i, 0)$ , and values $0$ on any index $(i, j)$ such that $j \neq 0$ , by additivity of $h_i'$ . So by Proposition 38 and Lemma 41, $(h_i')_{i \in \mathbb {N}}$ is summable and thus $(h_a)_{a \in A}$ is summable by Lemma 44.

Finally, we show that naturality in $\mathscr {C}^{\mathsf {add}}$ is preserved by witness and sums.

Proof. Assume first that $A = \mathbb {N}$ . We want to show that $\left \langle \!\left \langle \alpha _X^i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \in \mathscr {C}^{\mathsf {add}}(F X, \mathsf {S} G X)$ is a natural transformation $F \Rightarrow \mathsf {S} G$ . That is, for all $f \in \mathscr {D}(X, Y)$ ,

\begin{equation*} \left \langle \!\left \langle \alpha _Y^i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \, F f = \mathsf {S} G f \, \left \langle \!\left \langle \alpha _X^i \right \rangle \!\right \rangle _{i \in \mathbb {N}}.\end{equation*}

For all $i$ ,

\begin{equation*} \pi _i \, \left \langle \!\left \langle \alpha _Y^i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \, F f = \alpha _Y^i \, F f = G f \, \alpha _X^i = G f \, \pi _i \, \left \langle \!\left \langle \alpha _X^i \right \rangle \!\right \rangle _{i \in \mathbb {N}} = \pi _i \, \mathsf {S} G f \, \left \langle \!\left \langle \alpha _X^i \right \rangle \!\right \rangle _{i \in \mathbb {N}}. \end{equation*}

By joint monicity of the $\pi _i$ , we conclude that $(\left \langle \!\left \langle \alpha _X^i \right \rangle \!\right \rangle _{i \in \mathbb {N}})_X$ is natural. Furthermore, $\sum _{i \in \mathbb {N}} \alpha ^i_X = \sigma _{G X} \, \left \langle \!\left \langle \alpha _X^i \right \rangle \!\right \rangle _{i \in \mathbb {N}}$ is natural since $\sigma$ is natural. The case where $A$ is an arbitrary countable set immediately follows by Lemma 44 and naturality of $0$ .

4.4 The bimonad of sigma-summabibility structures

In this section, we endow $\mathsf {S}$ with a structure of bimonad in the sense of Definition 17. The corresponding natural transformations are essential in our approach to Taylor expansion and are similar to natural transformations introduced in tangent categories Rosicky (Reference Rosicky1984) and in coherent differentiation Ehrhard and Walch (Reference Ehrhard and Walch2023).

We will see in section 5 that Taylor expansionFootnote ⁶ consists in a functor $\mathsf {T}$ that extends $\mathsf {S}$ to $\mathscr {C}$ (in particular, $\mathsf {T} X = \mathsf {S} X$ ), and that the usual properties of differentiation correspond to the naturality with respect to $\mathsf {T}$ of the families of morphisms associated with the bimonad structure of $\mathsf {S}$ . We provide an intuition on those natural transformations as operations on formal power series.

It is quite enlightening to write a sequence $\left \langle \!\left \langle f_i \right \rangle \!\right \rangle _{i \in \mathbb {N}} \in \mathscr {C}(X, \mathsf {S} Y)$ as a formal power series $\sum _{i=0}^{\infty } f_i t^i$ (this series is called the generating function). The reason is that as we will see in section 6, the element at position $i$ in $\left \langle \!\left \langle f_i \right \rangle \!\right \rangle _{i \in \mathbb {N}}$ can be seen as an order $i$ degree of differentiation. This is in fact the reason why we write the projections $\pi _i$ starting from index $0$ and not from index $1$ . Then the projection $\pi _i$ maps $\sum _{i=0}^{\infty } f_i t^i$ to $f_i$ , and the sum maps $\sum _{i=0}^{\infty } f_i t^i$ to $\sum _{i \in \mathbb {N}} f_i$ (it evaluates the formal power series in $t = 1$ ). We introduce and recall the following families of morphisms on $\mathscr {C}$ .

1. (1) A morphism $\iota _{i,X} \in \mathscr {C}(X, \mathsf {S} X)$ that intuitively maps an element $x \in X$ to the monomial $x t^i$ , this morphism was used in the proof of Proposition 61;

2. (2) Remember that we have a morphism $\sigma _X \in \mathscr {C}(\mathsf {S} X, X)$ that intuitively maps a power series $\sum _{i \in \mathbb {N}} x_i t^i$ to $\sum _{i \in \mathbb {N}} x_i$ (it evaluates the series in $t = 1$ );

3. (3) A morphism $\theta _X \in \mathscr {C}(\mathsf {S}^2 X, \mathsf {S} X)$ that intuitively maps a multivariate formal power series $\sum x_{i,j} t^i u^j$ to the power series $\sum _{k \in \mathbb {N}} (\sum _{i+j = k} x_{i,j}) v^k$ ;

4. (4) A lift morphism $\mathsf {l}_X \in \mathscr {C}(\mathsf {S} X, \mathsf {S}^2 X)$ that intuitively maps the power series $\sum _{i \in \mathbb {N}} x_i t^i$ to the power series with two indeterminate $\sum _{i \in \mathbb {N}} x_i u^i v^i$ ;

5. (5) A flip operator $\mathsf {c}_X \in \mathscr {C}(\mathsf {S}^2 X, \mathsf {S}^2 X)$ that intuitively maps the power series $\sum _{i, j \in \mathbb {N}} x_{i,j} t^i u^j$ to $\sum _{i, j \in \mathbb {N}} x_{i,j} t^j u^i$ .

We show that those morphisms formally exists thanks to the axioms of (left) $\Sigma$ -summability structure, that they are $\Sigma$ -additive, that they are natural transformations when considered as morphisms in $\mathscr {C}^{\mathsf {add}}$ , and that they provide a bimonad structure to $\mathsf {S}$ on $\mathscr {C}^{\mathsf {add}}$ (we will recall the commutations that are involved on the fly).

Proof. The existence of $\iota _i$ is a direct consequence of (Unary) and SS-2. The family $(\pi _i \, \pi _j)_{(i, j) \in \mathbb {N}^2}$ is summable by double distributivity (Proposition 46). Thus, $\theta$ exists by (PA) and SS-2, and $\mathsf {c}$ exists by SS-4. Finally, the family $(\iota _i \circ \pi _i)_{i \in \mathbb {N}}$ is summable if and only if $(\pi _j \circ \iota _i \circ \pi _i)_{(i, j) \in \mathbb {N}^2}$ is summable by SS-4. But

\begin{equation*}\pi _j \circ \iota _i \circ \pi _i = \begin{cases} \pi _i \text { if } i = j \\ 0 \text { otherwise.} \end{cases} \end{equation*}

Thus, $(\pi _j \circ \iota _i \circ \pi _i)_{(i, j) \in \mathbb {N}^2} = \varphi _{\ast } (\pi _i)_{i \in \mathbb {N}}$ where $\varphi : \mathbb {N} \hookrightarrow \mathbb {N}^2$ maps $i$ to $(i,i)$ . So this family is summable by Lemma 44, and $\mathsf {l}$ exists by SS-2.

Proposition 68. The tuple $\underline {\mathsf {S}} = (\mathsf {S}, \iota _0, \theta )$ is a monad on $\mathscr {C}^{\mathsf {add}}$ , that is the following diagrams of natural transformations commute.

Proof. As usual, we use the joint monicity of the $\pi _i$ .

\begin{align*} \pi _i\,\theta _X\,\iota _{0, \mathsf {S} X} = \sum _{l+r=i}\pi _l\,\pi _r\,\iota _{0, \mathsf {S} X} = \sum _{l+r=i}\pi _l\, (\delta _{r, }0 \mathsf {id}) = \pi _i\,\mathsf {id} = \pi _i \end{align*}

\begin{align*} \pi _i\,\theta \,\mathsf {S}\iota _{0, X} = \sum _{l+r=i}\pi _l\,\pi _r\,\mathsf {S}\iota _{0, X} = \sum _{l+r=i}\pi _l\,\iota _{0, X}\,\pi _r = \sum _{l+r=i}(\delta _{l, }0 \mathsf {id}) \,\pi _r = \pi _i \end{align*}

using the naturality of $\pi _r$ . Hence, $\theta \,\iota _{0, \mathsf {S} X}=\theta \,\mathsf {S}\iota _{0, X}=\mathsf {id}$ by joint monicity of the $\pi _i$ ’s.

Next we have

\begin{align*} \pi _i\,\theta _X\,\theta _{\mathsf {S} X} &= \sum _{l+r=i}\pi _l\,\pi _r\,\theta _{\mathsf {S} X}\\ &= \sum _{l+r=i}\pi _l\sum _{s+t=r}\pi _s\,\pi _t\\ &= \sum _{l+r+s=i}\pi _l\,\pi _r\,\pi _s \end{align*}

and

\begin{align*} \pi _i\,\theta _X\,\mathsf {S}\theta _{X} &= \sum _{l+r=i}\pi _l\,\pi _r\,\mathsf {S}\theta _{X}\\ &= \sum _{l+r=i}\pi _l\,\theta _{X}\,\pi _r\\ &= \sum _{l+r=i}(\sum _{j+k=l}\pi _j\,\pi _k)\,\pi _r\\ &= \sum _{j+k+r=i}\pi _j\,\pi _k\,\pi _k \end{align*}

and hence $\theta _X\,\theta _{\mathsf {S} X}=\theta _X\,\mathsf {S}\theta _{X}$ .

We use $\mathscr {C}^{\mathsf {add}}_{\mathsf {S}}$ for the Kleisli category of the monad $\underline {\mathsf {S}}$ on $\mathscr {C}^{\mathsf {add}}$ . Observe that the composition of $f \in \mathscr {C}^{\mathsf {add}}_{\mathsf {S}}(X, Y)$ with $g \in \mathscr {C}^{\mathsf {add}}_{\mathsf {S}}(Y, Z)$ , defined as $\theta \, \mathsf {S} g \, f$ , is characterized by the equations for each $k\in \mathbb {N}$

(17) \begin{equation} \pi _k \, \theta \, \mathsf {S} g \, f = \sum _{i+j = k} \pi _j \, \pi _i \, \mathsf {S} g \, f = \sum _{i+j = k} (\pi _j \, g) \, (\pi _i \, f) \end{equation}

Coming back to formal power series, a morphism $f \in \mathscr {C}^{\mathsf {add}}_{\mathsf {S}}(X, Y)$ can be equivalently written as a formal series $\sum _{i \in \mathbb {N}} f_i t^i$ where $f_i = \pi _i \, f_i$ . Then Eq. (17) above means that the composition of two series corresponds to their Cauchy product (which is the usual product of power series), in which the multiplication of coefficients is composition of linear morphisms.

\begin{equation*} \left (\sum _{j \in \mathbb {N}} g_j t^j\right ) \, \left (\sum _{i \in \mathbb {N}} f_i t^i\right ) = \sum _{k \in \mathbb {N}} \left (\sum _{i+j = k} g_j \, f_i\right ) \, t^k. \end{equation*}

The functor $K_{\mathsf {S}} : \mathscr {C}^{\mathsf {add}} \rightarrow \mathscr {C}^{\mathsf {add}}_{\mathsf {S}}$ maps a morphism in $f \in \mathscr {C}^{\mathsf {add}}$ to the formal power series $f t^0$ .

Remark 69. The (left) summability structures of Ehrhard (Reference Ehrhard2023b) and Ehrhard and Walch (Reference Ehrhard and Walch2023) follow a similar principle, except that $\mathsf {S} X$ is intuitively the set of summable pairs $\left \langle \!\left \langle x_0, x_1 \right \rangle \!\right \rangle$ with $x_0, x_1 \in X$ . An element $\left \langle \!\left \langle x_0, x_1 \right \rangle \!\right \rangle \in \mathsf {S} X$ can be seen as a polynomial $x_0 + x_1 t$ with the convention that the formal variable $t$ follows the equation $t^2 = 0$ . Then the natural transformations $\iota _i$ (with $i \in \{0,1\}$ ) $\sigma, \mathsf {l}$ and $\mathsf {c}$ follow the same intuition. On the other hand, $\theta$ maps a polynomial $x_{0,0} + x_{1,0} t + x_{0,1} u + x_{1,1} t u$ to the polynomial $x_{0,0} + (x_{0,1} + x_{1,0}) v$ , since $v^2 = 0$ .

In that case, a morphism $\mathscr {C}^{\mathsf {add}}_{\mathsf {S}}(X, Y)$ can be seen as a polynomial $f_0 + f_1 t$ with $f_i \in \mathscr {C}^{\mathsf {add}}(X,Y)$ , and we have the following composition

\begin{equation*} (g_0 + g_1 t) \, (f_0 + f_1 t) = g_0 \, f_0 + (g_0 \, f_1 + g_1 \, f_0) t \end{equation*}

Which is similar to the multiplication on Clifford’s dual numbers. Similarly, an $n$ -ary counterpart of (left) summability structures for any $n \in \mathbb {N}^*$ would induce a monad and a category of polynomials quotiented by the equation $t^{n+1} = 0$ .

Proposition 70. The tuple $\overline {\mathsf {S}} = (\mathsf {S}, \sigma, \mathsf {l})$ is a comonad on $\mathscr {C}^{\mathsf {add}}$ , meaning that the following diagrams of natural transformations commute.

Proof. As usual, we use the joint monicity of the $\pi _i$ . Remember that $\pi _i \, \mathsf {l}_X = \iota _i \, \pi _i$ .

\begin{equation*} \pi _i \, \mathsf {S} \sigma _X \, \mathsf {l}_X = \sigma _X \, \pi _i \, \mathsf {l}_X = \sigma _X \, \iota _i \, \pi _i = \pi _i \end{equation*}

\begin{equation*} \pi _i \, \sigma _{\mathsf {S} X} \, \mathsf {l}_X = \pi _i \, \left ( \sum _{j \in \mathbb {N}} \pi _j \right ) \, \mathsf {l}_X = \sum _{j \in \mathbb {N}} \pi _i \, \pi _j \, \mathsf {l}_X = \sum _{j \in \mathbb {N}} \delta _{i, }j \pi _i = \pi _i \end{equation*}

so by joint monicity of the $\pi _i$ , $\mathsf {S} \sigma _X \, \mathsf {l}_X = \sigma _{\mathsf {S} X} \, \mathsf {l}_X = \mathsf {id}_X$ . Next, we have

\begin{equation*} \pi _i \, \pi _j \, \mathsf {S} \mathsf {l}_X \mathsf {l}_X = \pi _i \, \mathsf {l}_X \, \pi _j \, \mathsf {l}_X = \iota _i \, \pi _i \, \iota _j \, \pi _j = \begin{cases} \iota _i \, \pi _i \text { if } i=j \\ 0 \text { otherwise} \end{cases} \end{equation*}

\begin{equation*} \pi _i \, \pi _j \, \mathsf {l}_{\mathsf {S} X} \, \mathsf {l}_X = \pi _i \, \iota _j \, \pi _j \, \mathsf {l} = \pi _i \, \iota _j \, \iota _j \, \pi _j = \begin{cases} \iota _i \, \pi _i \text { if } i=j \\ 0 \text { otherwise} \end{cases} \end{equation*}

So by joint monicity of the $\pi _i \, \pi _j$ we have $\mathsf {S} \mathsf {l}_X \mathsf {l}_X = \mathsf {l}_{\mathsf {S} X} \, \mathsf {l}_X$ .

The monad structure and the comonad structure are compatible, in the sense that they form a $\mathsf {c}$ -bimonad, see Definition 17. Here are some useful observations on $\mathsf {c}$ .

Proof. By Remark 19 it is only necessary to show that $\mathsf {c}$ is a distributive law $\underline {\mathsf {S}} \overline {\mathsf {S}} \Rightarrow \overline {\mathsf {S}} \underline {\mathsf {S}}$ since $\mathsf {c}$ is involutive. The first condition is that $\mathsf {c}$ is a distributive law $\underline {\mathsf {S}} \mathsf {S} \Rightarrow \mathsf {S} \underline {\mathsf {S}}$ which corresponds to the two diagrams below. For the sake of readability, we write $\underline {\mathsf {S}}$ instead of just $\mathsf {S}$ to make it clear which part is playing the role of the monad.

We use the joint monicity of the $\pi _i \, \pi _j$ . Observe that

\begin{equation*} \pi _i \, \pi _j \, \mathsf {c} \, \iota _0 = \pi _j \, \pi _i \, \iota _0 = \begin{cases} \pi _j \text { if } i=0 \\ 0 \text { otherwise} \end{cases} \end{equation*}

\begin{equation*} \pi _i \, \pi _j \, \mathsf {S} \iota _0 = \pi _i \, \iota _0 \, \pi _j = \begin{cases} \pi _j \text { if } i=0 \\ 0 \text { otherwise} \end{cases} \end{equation*}

so by joint monicity of the $\pi _i \, \pi _j$ , $\mathsf {c} \, \iota _0 = \mathsf {S} \iota _0$ and the left diagram holds. Next,

\begin{equation*} \pi _i \, \pi _j \, \mathsf {c} \, \theta _{\mathsf {S} X} = \pi _j \, \pi _i \, \theta _{\mathsf {S} X} = \pi _j \, \left (\sum _{k = 0}^i \pi _k \, \pi _{i-k} \right ) \end{equation*}

\begin{align*} \pi _i \, \pi _j \, \mathsf {S} \theta _X \, \mathsf {c}_{\mathsf {S} X} \, \mathsf {S} \mathsf {c}_X &= \pi _i \, \theta _X \, \pi _j \, \mathsf {c}_{\mathsf {S} X} \, \mathsf {S} \mathsf {c}_X\qquad \qquad \text { naturality of } \pi _j \\ &= \pi _i \, \theta _X \, \mathsf {S} \pi _j \, \mathsf {S} \mathsf {c}_X\qquad \qquad \text { Lemma 72} \\ &= \pi _i \, \theta _X \, \mathsf {S}^2 \pi _j \qquad \qquad \text {functoriality of } \mathsf {S} \text { and Lemma 72} \\ &= \left(\sum _{k=0}^i \pi _k \, \pi _{i-k}\right) \, \mathsf {S}^2 \pi _j \\ &= \pi _j \, \left(\sum _{k=0}^i \pi _k \, \pi _{i-k}\right)\qquad \qquad \text { naturality, thanks to Proposition 64} \end{align*}

So by joint monicity of the $\pi _i \, \pi _j$ , $\mathsf {c} \, \theta _{\mathsf {S} X} = \mathsf {S} \theta _X \, \mathsf {c}_{\mathsf {S} X} \, \mathsf {S} \mathsf {c}_X$ and the right diagram holds. Next, we show that $\mathsf {c}$ is a distributive law $\mathsf {S} \overline {\mathsf {S}} \Rightarrow \overline {\mathsf {S}} \mathsf {S}$ , that is, the two diagrams below commute. For the sake of readability, we write $\overline {\mathsf {S}}$ instead of just $\mathsf {S}$ to make clear which part is playing the role of the comonad.

By naturality of $\sigma$ , of $\pi _i$ , and using the fact that $\mathsf {S} \pi _i \, \mathsf {c} = \pi _i$ (see Lemma 72), we have

\begin{equation*} \pi _i \, \sigma _{\mathsf {S} X} \, \mathsf {c} = \sigma _{\mathsf {S} X} \, \mathsf {S} \pi _i \, \mathsf {c} = \sigma _{\mathsf {S} X} \, \pi _i = \pi _i \, \mathsf {S} \sigma _{X} \end{equation*}

so by joint monicity of the $\pi _i$ , $\sigma _{\mathsf {S} X} \, \mathsf {c}_X = \mathsf {S} \sigma _X$ and the left diagram holds. Next, we have

\begin{align*} \pi _i \, \pi _j \, \mathsf {S} \mathsf {c}_X \, \mathsf {c}_{\mathsf {S} X} \, \mathsf {S} \mathsf {l}_X &= \pi _i \, \mathsf {c}_X \, \pi _j \, \mathsf {c}_{\mathsf {S} X} \, \mathsf {S} \mathsf {l}_X \\ &= \mathsf {S} \pi _i \, \mathsf {S} \pi _j \, \mathsf {S} \mathsf {l}_X \\ &= \mathsf {S} (\pi _i \, \pi _j \, \mathsf {l}) \\ &= \begin{cases} \mathsf {S} 0 = 0 \text { if } i \neq j \\ \mathsf {S} \pi _i \text { otherwise} \end{cases} \end{align*}

\begin{equation*} \pi _i \, \pi _j \, \mathsf {l}_{\mathsf {S} X} \, \mathsf {c}_X = \begin{cases} 0 \text { if } i \neq j \\ \pi _i \, \mathsf {c} = \mathsf {S} \pi _i \text { otherwise } \end{cases} \end{equation*}

so by joint monicity of the $\pi _i \, \pi _j$ we have $\mathsf {S} \mathsf {c}_X \, \mathsf {c}_{\mathsf {S} X} \, \mathsf {S} \mathsf {l}_X = \mathsf {l}_{\mathsf {S} X} \, \mathsf {c}_X$ and the right diagram holds.