Exponential Functions in Cartesian Differential Categories

In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $e^x$ from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $e^0 = 1$, $e^{x+y} = e^x e^y$, and $\frac{\partial e^x}{\partial x} = e^x$ all hold. Every differential exponential map induces a commutative semiring, which we call a differential exponential semiring, and conversely, every differential exponential semiring induces a differential exponential map. In particular, differential exponential maps can be defined without the need for limits, converging power series, or unique solutions of certain differential equations -- which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, and the dual numbers exponential. And as another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.


Introduction
Cartesian differential categories [2], introduced by Blute, Cockett, and Seely, come equipped with a differential combinator D which provides a categorical axiomatization of the directional derivative from multivariable differential calculus. Important examples of Cartesian differential categories include the category of real smooth functions (Example 2.3), the coKleisli category of a differential category [4], the differential objects of a tangent category [10], and categorical models of Ehrhard and Regnier's differential λ-calculus [14] (which are in fact called Cartesian closed differential categories [19]). Other interesting (and surprising) examples include abelian functor calculus [1] and cofree Cartesian differential categories [12,17]. Since their introduction, Cartesian differential categories have a rich literature and have been successful in generalizing many concepts from classical differential calculus, and more recently have also started to find their way in applications.
In particular, Cockett and Cruttwell have introduced the notion of dynamical systems and their solutions in tangent categories [8], which generalize ordinary differential equations in this context, specifically initial value problems. As every Cartesian differential category is a tangent category, this implies that dynamical systems allow one to study differential equations in a Cartesian differential category. In classical differential calculus, one of the most important tools used for solving differential equations is the exponential function e x . Therefore, it is desirable to generalize the exponential function for Cartesian differential categories.
The exponential function e x admits numerous equivalent characterization. It can either be defined as the (partial) inverse of the natural logarithm function ln(x), or as the limit: or as the convergent power series: x n n! or even as the solution to f ′ (x) = f (x) with initial condition f (0) = 1. However, in arbitrary Cartesian differential categories, one does not necessarily have partial functions, a notion of convergence, infinite sums, or even (unique) solutions to initial value problems. Therefore one must look for a more algebraic characterization of the exponential function. In classical algebra, an exponential ring [22] is a ring equipped with an endomorphism e such that e(x + y) = e(x)e(y) and e(0) = 1, with the canonical example being the field of real numbers R with the exponential function e x . While this seems promising, arbitrary objects in a Cartesian differential category do not necessarily come equipped with a multiplication. Rather than requiring this extra ring structure on objects, it turns out that the differential combinator D will allow to bypass the need for a multiplication.
In the category of real smooth functions, which is the canonical example of a Cartesian differential category, the differential combinator D applied to the exponential function is precisely its directional derivative D[e x ](x, y) = e x y -so the multiplication of R appears in D[e x ]. Inspired by this observation, the generalization of the exponential function in a Cartesian differential category can be defined simply in terms of an endomorphism A e − − → A which is compatible with the differential combinator D in the sense that D[e](0, x) = x and e(x + y) = D[e](x, e(y)). We call such endomorphisms differential exponential maps, which is the main novel notion of study in this paper, and these generalize the exponential function for Cartesian differential categories. Indeed for e x , the differential exponential maps axioms correspond precisely to e 0 x = x and e x e y = e x+y .
While we mentioned that not every object has a multiplication, it turns out that every differential exponential map A e − − → A does induce a commutative semiring structure on A, and thus A does have a multiplication. The construction is once again inspired by the classical exponential function e x . Applying the differential combinator on e x twice we obtain D[e x ]((x, y), (z, w)) = e x yz + e x w. Setting x = 0 and w = 0, one re-obtains precisely the multiplication of R, D[e x ]((0, y), (z, 0)) = yz. This construction is easily generalized to an arbitrary Cartesian differential category, and one can then use the differential combinator axioms to show that every differential exponential map induces a commutative semiring. Conversly, one can alternatively axiomatize an object equipped with a differential exponential maps instead as a commutative semiring equipped with an endomorphism which satisfies the three fundamental properties of the exponential function: e 0 = 1, e x+y = e x e y , and ∂e x ∂x = e x . We call such semirings, differential exponential semirings, and there is a bijective correspondence between differential exponential maps and differential exponential semiringswhich is one of the main results of this paper.
As one of the main motivations for their development, differential exponentials maps does allow one to solve a certain class of linear dynamical systems in any Cartesian differential category. Furthermore, it turns out that a differential exponential map is indeed the unique solution to the dynamical system which generalizes the initial value problem f ′ (x) = f (x) with initial condition f (0) = 1. Hopefully in future work on solving differential equations in a Cartesian differential category, differential exponential maps will play a key role.
Outline and Main Results: Section 2 is a background section on Cartesian differential categories where we briefly review the basic definitions, as well as to introduce the notation and terminology used in this paper. In particular, we review the canonical commutative monoid structure ⊕ on every object in a Cartesian differential category (Lemma 2.10), which plays a key role throughout this paper. In Section 3 we introduce differential exponential maps (Definition 3.1) and in particular show that the category of differential exponential maps is a Cartesian tangent category (Proposition 3.6). In Example 3. 7 we provide examples of differential exponential maps in the category of smooth real functions, which include the classical exponential function, the complex exponential function, and the dual number exponential function. In Section 4 we introduce differential exponential semirings. We show that every differential exponential semiring induces a differential exponential map (Proposition 4.4) and conversely that every differential exponential map induces a differential exponential semiring (Proposition 4.5), and show that these constructions are inverses of each other (Theorem 4.8). As an immediate consequence, the category of differential exponential semirings is also a Cartesian tangent category (Proposition 4.9). In Section 5 we explain the relationship between differential exponential maps and solutions to differential equations in arbitrary Cartesian differential categories. In particular, we show that every differential exponential map is the unique solution to the expected differential equation (Proposition 5.5) and that a certain class of dynamical systems admit a solution (Proposition 5.12). We also show that in the presence of unique solutions to differential equations, one can build a differential exponential map (Proposition 5.13). In Section 6 we study differential exponential maps in the coKleisli category of a differential (storage) category and give equivalent characterizations in these cases (Proposition 6.3 and Proposition 6.5). We also introduce !-differential exponential algebras (Definition 6.6) for differential storage categories. We show that every !-differential exponential algebra induces a differential exponential map in the coKleisli category (Proposition 6.7), and conversely that every differential exponential map in the coKleisli category induces a !-differential exponential algebra (Proposition 6.7), and that these constructions are inverses of each other (Theorem 6.9). We conclude this paper with Section 7 which discusses some interesting potential future work to do with differential exponential maps.
Conventions: We use diagrammatic order for composition: this means that the composite map f g is the map which first does f then g. For a category with finite products, we denote the binary product of objects A and B by pairing operation −, − and thus f × g = π 0 f, π 1 g , and the chosen terminal object as ⊤.

Background: Cartesian Differential Categories
In this section, if only to introduce notation, we briefly review Cartesian differential categories, their underlying Cartesian left additive structure, and their induced Cartesian tangent category structure. That said, we assume that the reader is familiar with the theory of Cartesian differential categories. For a more in-depth discussion we refer the reader to [2,10].
and zero 0 ∈ X(A, B), such that composition on the left preserves the additive structure, that is: We note that the definition of a Cartesian left additive category presented here is not precisely that given in [2, Definition 1.2.1], but was shown to be equivalent in [17,Lemma 2.4]. Also, note that in a Cartesian left additive category, the unique map from an object A to the terminal object ⊤ is the zero map A 0 − − → ⊤. Also, an important map which will use throughout this paper is the which is a natural isomorphism such that cc = 1.
Definition 2.2 A Cartesian differential category [2, Definition 2.1.1] is a Cartesian left additive category X equipped with a differential combinator D, which is a family of operators , such that the following axioms hold: A discussion on the intuition for the differential combinator axioms can be found in [2, Remark 2.1.3]. We also note that the definition of a Cartesian differential category given here is not precisely that given in [2, Definition 2.1.1] but rather an equivalent version found in [11,Section 3.4]. Also note that unlike in [2,10,11], we use the convention used in the more recent work on Cartesian differential categories where the linear argument of D[f ] is its second argument rather than its first argument. The canonical example of a Cartesian differential category is the category of Euclidean spaces and smooth maps between them -which will be our main example throughout this paper.

Example 2.3
Let R be the set of real numbers and let SMOOTH be the category whose objects are the Euclidean vector spaces R n (including the singleton R 0 = { * }) and whose maps are smooth function R n F − − → R m , which of course are in fact tuples F = f 1 , . . . , f m for some smooth functions SMOOTH is a Cartesian differential category where the finite product structure and additive structure are defined in the obvious way, and whose differential combinator is given by the directional derivative of smooth functions. Explicitly, recall that for a smooth map R n f − − → R, its gradient R n ∇(f ) − −−− → R n is defined as: It is also possible to define D[F ] in terms of the Jacobian matrix of F . Many other interesting examples of Cartesian differential categories can be found throughout the literature such as categorical models of the differential λ-calculus [14], which are called Cartesian closed differential categories [19], the subcategory of differential objects of a Cartesian tangent category [10], and the coKleisli category of a differential category [2,4]. In particular, we will take a closer look at the coKleisli category of a differential category in Section 6.
An important class of maps in a Cartesian differential category is the class of linear maps. Later in Section 4, we will also discuss bilinear maps.
Here are now some useful properties about linear maps for this paper: Lemma 2.6 [2, Lemma 2.2.2] In a Cartesian differential category: For a Cartesian differential category X, define its subcategory of linear maps LIN[X] to be the category whose objects are the same as X and whose maps are linear in X. Lemma 2.6 tells us that LIN[X] is a well-defined category and also that it has finite biproducts, and thus is a Cartesian left additive category where every map is additive. LIN[X] also inherits the differential combinator from X and so LIN[X] is a Cartesian differential category where every map is linear. Therefore the obvious forgetful functor LIN[X] U − − → X preserves the Cartesian differential structure strictly. The differential combinator of a Cartesian differential category induces an endofunctor and this endofunctor makes a Cartesian differential category a Cartesian tangent category. We will not review the full definition of a tangent category here but we will highlight certain properties that will be important for this paper. We invite the reader to read the full definition of a tangent category in [10,11].  Here are now some useful properties involving the tangent functor (which we leave to the reader to check for themselves): Lemma 2.9 In a Cartesian differential category: We conclude this section with the observation that in a Cartesian differential category, the additive structure induces a commutative monoid structure for every object. Cartesian left additive categories can be axiomatized in terms of equipping each object with a commutative monoid structure such that the projection maps are monoid morphisms.
Lemma 2.10 [2, Proposition 1.2.2] In a Cartesian differential category, for an object A define the map A × A ⊕ − − → A as ⊕ = π 0 + π 1 . Then the triple (A, ⊕, 0) is a commutative monoid, that is, the following diagrams commute Here are now some useful properties involving this commutative monoid structure (which we leave to the reader to check for themselves): Lemma 2.11 In a Cartesian differential category: (iv) A map f is additive if and only if ⊕f = (f × f )⊕ and 0f = f .

Differential Exponential Maps
In this section, we introduce differential exponential maps, which generalizes the notion of the classical exponential function e x for arbitrary Cartesian differential categories.
Definition 3.1 A differential exponential map in a Cartesian differential category is a map A e − − → A, such that the following diagrams commute: where ⊕ is defined as in Lemma 2.10.
The intuition for a differential exponential map is best explained in Example 3.7.i, which shows that the classical exponential function e x (which is, of course, the main motivating example) is a differential exponential map. Briefly since e x is its own derivative, its directional derivative D[e x ] is simply e x y. The left diagram of (5) generalizes that e 0 y = y (since e 0 = 1), while the right diagram generalizes that e x e y = e x+y . The differential combinator is the key piece that allows one to bypass the need for a multiplication operation and a multiplicative unit in the definition of a differential exponential map. That said, in Section 4 we will see that every differential exponential map does induce a multiplication and that analogues of the three essential properties of the classical exponential function are satisfied (Proposition 4.5). And conversely, we will also see how one can also axiomatize differential exponential maps in terms of a multiplication and analogues of the three essential properties of the classical exponential function (Proposition 4.4). And, as mentioned in the introduction, we also highlight that the definition of a differential exponential map does not require any added structure or property on the Cartesian differential category such as a notion of converging limits or infinite sums. Before giving examples of differential exponential maps, which can be found in Example 3.7, let us first consider the category of differential exponential maps and constructions of differential exponential maps. For a Cartesian differential category X, define its category of differential exponential maps as the category DEM[X] whose objects are pairs (A, e) consisting of an object A ∈ X and a differential exponential map A  , which we denote lim X∈D U F(X) with projections π X : lim X∈D U F(X) − → U F(X) . Note that U F(X) is the underlying object of F(X), and so it comes equipped with a differential exponential , the following diagram commutes: which is the unique map which makes the following diagram commute: We now wish to show that lim X∈D e X is a differential exponential map. To do so, first note that for each X ∈ D, π X is linear and so by Lemma 2.6 it follows that: Now since π X is linear, it is also additive (Lemma 2.6) and therefore we obtain the following: = (π X × π X ) ⊕ e X (π X is additive + Lemma 2.11) Therefore for each X ∈ D, we have that: Then by the universal property of the limit, it follows that: and so we conclude that lim X∈D e X is a differential exponential map. From here, it is straightforward By Lemma 2.6, the projection maps of the product are linear in any Cartesian differential category. Therefore, an immediate consequence of Lemma 3.2 is that the product of differential exponential maps is again a differential exponential map.

Corollary 3.3 In a Cartesian differential category:
(i) For the terminal object ⊤, ⊤ 1 − − → ⊤ is a differential exponential map; It is important to note that a differential exponential map of type A × B − → A × B is not necessarily the product of differential exponential maps, that is, of the form e × e ′ . That said, another consequence of Lemma 3.2 is that the category of differential exponentials maps has finite products.

Corollary 3.4 For a Cartesian differential category X, DEM[X]
has finite products where the terminal object is (⊤, 1), and where the product of (A, e) and (B, Our next observation about the category of differential exponential maps is that it is a Cartesian tangent category. We first show that the tangent of a differential exponential map is again a differential exponential map. = 1 (T is a functor) Next we show that (1 × T(e))D T(e) = ⊕T(e): So we conclude that T(e) is a differential exponential map. (which is the same as for X and can be found in [10,Proposition 4.7]).
Proof: The tangent functor T is well defined by Lemma 3.5. Since all the maps of the tangent structure of X are linear, it follows that by their respective naturality with the tangent functor of X, they are also maps in DEM[X] which are natural for its tangent functor. The existence of the necessary limits for tangent structure in DEM[X] will follow from Corollary 2.8 and Lemma 3.2. And lastly, the required equalities for tangent structure will hold since they hold in LIN[X]. So we conclude that DEM[X] is a tangent category. Furthermore, since X is a Cartesian tangent category, it follows that DEM[X] is also a Cartesian tangent category, that is, It may be tempted to think that DEM[X] is also a Cartesian differential category, but this is since it is not necessarily the case that (f + g)e equals e ′ (f + g). This implies that DEM[X] is not a Cartesian left additive category, and so in particular not a Cartesian differential category.

Example 3.7
Here are now examples of differential exponential maps in the Cartesian differential category SMOOTH (as defined in Example 2.3).
Since ∂e x ∂x (a) = e a , we have that: The exponential function satisfies the left diagram of (5) since e 0 = 1: while the right diagram of (5) is also satisfied since e a e b = e a+b : Then the exponential function R e x −−→ R is a differential exponential map.
(ii) Consider the complex exponential function e x+iy = e x cos(y) + ie x sin(y), which can be seen as the smooth real function R 2 ǫ −→ R 2 , (x, y) → (e x cos(y), e x sin(y)). By the Leibniz rule and the derivative identities for both e x and the trig functions, we have that: Or using complex number notation: D[ǫ] (a + ib, c + id) = e a+ib (c + id). Then by similar arguments as the real exponential function, R 2 ǫ −→ R 2 is also a differential exponential map.
(iii) Applying Corollary 3.3.ii to the exponential function R . , e xn ), are differential exponential maps in SMOOTH. One could also take the product of the real exponential function and the complex exponential function to obtain another example of a differential exponential map.
(iv) Applying Lemma 3.5 to the exponential function R e x −−→ R, the tangent exponential function , is a differential exponential map. To better understand this differential exponential map, consider that for dual numbers: e x+yε = e x + e x yε (which can be worked out directly using the power series expression of e x and that ε 2 = 0). Writing dual numbers x + yε instead as (x, y), it becomes clear that T(e x ) is the exponential function for the ring of dual numbers R[ε]. This relation was to be expected since tangent structure is closely related to dual numbers and Weil algebras [18]. Corollary 3.3.i tells us that the identity map of the terminal object is a differential exponential map. So we conclude this section with the observation that a differential exponential map is linear if and only if it is the identity map of a terminal object.
if and only if A is a terminal object (and therefore e = 1 A = 0). Therefore, a differential exponential map A e − − → A is additive or linear if and only if A is a terminal object.
Proof: Suppose that A e − − → A is a differential exponential map which satisfies 0e = 0. Then we have that: So 1 A = 0, and so A is a terminal object. Conversely, if A is a terminal object, then we must have that e = 1 A = 0, and so clearly e is reduced. For the second statement, note that by definition every additive map is reduced, and since linear maps are additive (Lemma 2.6), they are also reduced.
So if e is additive or linear, it is reduced and therefore A is a terminal object. Conversely, if A is a terminal object, then e must be the identity, and identity maps are always linear and additive (Lemma 2.6). ✷ Example 3.9 Every category with finite biproducts is a Cartesian differential category with differential combinator defined as D[f ] = π 1 f , which means that every map is linear. Therefore, in this case, the only differential exponential maps are the identity maps on terminal objects.
Note that for a Cartesian differential category X, Lemma 3.8 also implies that the only differential object [10] in the Cartesian tangent category DEM[X] is the terminal object.

Differential Exponential Semirings
In this section, we introduce differential exponential semirings, which provide an equivalent alternative characterization of differential exponential maps. We will show that every differential exponential map induces a differential exponential semiring (Proposition 4.5) and that conversely, every differential exponential semiring induces a differential exponential map (Proposition 4.4). We will also show that for a Cartesian differential category, its category of differential exponential maps is isomorphic to its category of differential exponential semirings (Theorem 4.8). We begin by reviewing differential semirings, which are semirings in a Cartesian differential category whose multiplication is bilinear in the Cartesian differential category sense.
Definition 4.1 A differential semiring in a Cartesian differential category is a triple (A, ⊙, u) consisting of an object A and two maps is a commutative monoid, that is, the following diagrams commute: and A × A ⊙ − − → A is bilinear, that is, the following equality holds: We should justify the use of the term semiring in differential semiring. Indeed, the term (commutative) semiring should imply that there are two (commutative) monoid structures that satisfy the expected distributivity axioms. But we know that in a Cartesian differential category, as discussed in Lemma 2.10, every object A already comes equipped with a commutative monoid structure (A, ⊕, 0). So every differential semiring (A, ⊙, u) does come with two commutative monoid structures. The required distributivity axioms are captured by the fact that ⊕ is bilinear, and therefore additive in each argument -which is an equivalent way of saying that ⊕ and ⊙ distribute over one another in the expected semiring sense.
is a differential semiring, then the following diagrams commute: and so (A, ⊙, u, ⊕, 0) is a commutative semiring (where ⊕ is defined as in Lemma 2.10).
A differential exponential semiring is a differential semiring equipped with an endomorphism which satisfies analogues of the three essential properties of the classical exponential function. This endomorphism will, of course, turn out to be a differential exponential map. Definition 4.3 A differential exponential semiring in a Cartesian differential category is a quadruple (A, ⊙, u, e) consisting of a differential semiring (A, ⊙, u) and a map A e − − → A, such that the following diagrams commute: where ⊕ is defined as in Lemma 2.10.
If one keeps in mind the classical exponential function e x , then the axioms of a differential exponential semiring are straightforward to understand. The leftmost diagram of (11) generalizes that the directional derivative of e x is D[e x ](x, y) = e x y, the middle diagram generalizes that e 0 = 1, and lastly the rightmost diagram generalizes that e x+y = e x e y . Examples of differential exponential semirings can be found below in Example 5.6 after we have proven Proposition 4.5 that every differential exponential map induces a differential exponential semiring. We first show that the endomorphism of a differential exponential semiring is indeed a differential exponential map.
So we conclude that e is a differential exponential map. ✷ In order to show the converse of Proposition 4.4, consider the classical exponential function e x and note that its second order directional derivative (that is, the directional derivative of its directional derivative) is D[e x ]((x, y), (z, w)) = e x yz + e x w. Setting x = 0 and w = 0, one obtains yz, the multiplication of real numbers. The unit for this multiplication is obtain from e 0 = 1. Generalizing this construction allows one to show how a differential exponential map induces a differential exponential semiring. Proof: We will first prove that e satisfies the three identities of (11), as these will help simplify the proof that (A, ⊙ e , u e ) is a differential semiring. We first prove that D The remaining identity, 0e = u e is automatic by construction. So e satisfies three identities of (11). Next we show that (A, ⊙ e , u e ) is a differential semiring. We first explain why ⊙ e is bilinear. In [12,Section 3] it was shown that for every map − −−− → B was bilinear in context A, so bilinear in its last two arguments A or equivalently bilinear with respect to the differential combinator of the simple slice category over A [2, Section 4.5]. By [5,Proposition 4.1.3], bilinear maps in context are preserved by pre-composition with maps which leave the bilinear arguments unaffected, that is, by pre-composition by maps of the form (g × 1) × 1. Therefore is bilinear in context ⊤. However maps which are bilinear in context ⊤ correspond precisely to bilinear maps without context. In this case, we obtain that A × A Now we show that (A, ⊙ e , u e ) is a commutative monoid. First that ⊙ e is commutative, that is, π 1 , π 0 ⊙ e = ⊙ e , follows from [CD.7]: Finally we prove associativity, which in the author's opinion is the most complex proof in this paper.
be the canonical associativity isomorphism α = π 0 , π 1 × 1 . By Lemma 2.6, α is linear and so is its inverse α −1 . First note that by [CD.5] and [CD.6], one can show that (which we leave as an exercise to the reader): Using the above identity, we compute that: (Lemma 2.10) (Lemma 2.10) So we have that: Using this identity, we can simplify α(1 × ⊙ e )⊙ e : (Lemma 2.6) So we have that: Now using the above identity and that we've already shown that ⊙ e is commutative, we can show that ⊙ e is associative, that is, (⊙ e × 1)⊙ e = α(1 × ⊙ e )⊙ e : (⊙ e × 1)⊙ e = (⊙ e × 1) π 1 , π 0 ⊙ e (8) So (A, ⊙ e , u e ) is a differential semiring, and therefore we conclude that (A, ⊙ e , u e ) is a differential exponential semiring. ✷ In the proof of Theorem 4.8, we will show that constructions of Proposition 4.4 and Proposition 4.5 are in fact inverse of each other. The construction from Proposition 4.5 is also compatible with some of the constructions of differential exponential maps in the following sense: Lemma 4.6 In a Cartesian differential category: (i) For the terminal object ⊤, the following equality holds: − − → B are differential exponential maps, then the following equality holds where recall that c = π 0 × π 0 , π 1 × π 1 .
(iii) If A e − − → A is a differential exponential map, then the following equality holds for the differ- Proof: (i) This is automatic by uniqueness of maps into the terminal object.
(ii) For the unit, this is mostly straightforward: For the multiplication, we have that: (i) For the exponential function R e x −−→ R, the induced multiplication is precisely given by the standard multiplication of real numbers, that is: ⊙ e x (x, y) = xy and u e x ( * ) = 1.
For a Cartesian differential category X, define its category of differential exponential semirings as the category DES[X] whose objects are differential exponential semirings (A, ⊙, u, e) and where a map (A, ⊙, u, e) f − − → (B, ⊙ ′ , u ′ , e ′ ) is a linear map A f − − → B in X such that the following diagrams commutes: and where composition and identity maps are as in X. Note that the two right most diagrams above imply that f is a monoid morphism.
Theorem 4.8 For a Cartesian differential category X, its category of differential exponential maps DEM[X] is isomorphic to its category of differential exponential semirings DES[X] via the functors , we must show that f also satisfies the three identities (15). By definition, one already has that ef = f e ′ , and so it remains to show that f is also a monoid morphism. Recall that since f is linear, f is also additive (Lemma 2.6). Now we first show that u e f = u e ′ : = 0e ′ (f additive) , so E is well-defined on maps. And clearly E preserves composition and identity, and therefore E is also a well-defined functor. Next we show that E and E −1 are inverses of each other. Clearly E −1 E(A, e) = (A, e) and E −1 E(f ) = f . For the other direction, clearly EE −1 (f ) = f and so it remains to show that EE −1 (A, ⊙, u, e) = (A, ⊙, u, e), that is, we must show that ⊙ = ⊙ e and u e = u. Starting with the unit: Next for the multiplication, we first observe that: = c T(e) × 1 c(π 0 × π 1 ) ⊙ +c T(e) × 1 c(π 1 × π 0 )⊙ = c T(e) × 1 (π 0 × π 1 ) ⊙ +c T(e) × 1 (π 1 × π 0 ) π 1 , π 0 ⊙ = c T(e) × 1 (π 0 × π 1 ) ⊙ +c T(e) × 1 (π 1 × π 0 )⊙ = c(π 0 × π 1 )(e × 1) ⊙ +c(D[e] × π 0 )⊙ (Definition of T) Using the above identity, we can easily show that ⊙ e = ⊙: We conclude this section with the observation that as an immediate consequence of both Theorem 4.8 and Lemma 4.6: the category of differential exponential semirings is a Cartesian tangent category and that it is isomorphic as a Cartesian tangent category to the category of differential exponential maps. Proposition 4.9 For a Cartesian differential category X, DES[X] has finite products where the terminal object is (⊤, 0, 1 ⊤ , 1 ⊤ ), and where the product of (A, ⊙, u, e) and (B, ⊙ ′ , u ′ , e ′ ) is: In this section, we explain how differential exponential maps provide solutions to certain differential equations and conversely how one can obtain a differential exponential map if one assumes that solutions are unique. These concept generalize the fact that in the classical case, the exponential function e x can be defined as the unique solution to the initial value problem f ′ (x) = f (x) with f (0) = 1. As introduced in [8, Section 5], ordinary differential equations in a Cartesian differential category are described as dynamical systems, while solutions for these differential equations are described as morphisms between these dynamical systems. We note that in [8], dynamical systems were defined for tangent categories and thus involves the tangent functor. Here we present the resulting definition specific to Cartesian differential categories, where dynamical systems can be described in terms of the differential combinator. (ii) A morphism of dynamical systems (A, a 0 , a 1 ) is a morphism of dynamical systems, we say that f is an (A, a 0 , a 1 )-solution of (A ′ , a ′ 0 , a ′ 1 ).
It is important to note that solutions need not be unique (though this assumption will come soon in Proposition 5.13). See [8,Section 5] or Example 5.4 below for how morphisms of dynamical systems do indeed correspond to solutions of ordinary differential equations.
For a differential semiring (A, ⊙, u), there is a canonical dynamical system (A, 0, u) where the differential state A u − − → A is defined as follows: and we can ask that (A, 0, u)-solutions be compatible with the semiring multiplication. Proof: The top triangle of (17) is precisely the left diagram of (19). So it remains to show that f also satisfies the bottom square of (17): So we conclude that f is an (A, 0, u)-solution of (A, a 0 , a 1 ). ✷ Example 5.4 A dynamical system in SMOOTH can be seen as a triple (R n , a, F ) where R n F − − → R n is a smooth function, which is a tuple F = f 1 , . . . , f n where R n f j − − → R, and a = (a 1 , . . . , a n ) ∈ R n . Consider the differential semiring induced from the exponential function (R, ⊙ e x , u e x ) (as defined in Example 5.6.i). Its canonical dynamical system is (R, 0, u e x ) where u e x (x) = 1. A (R, 0, u e x )solution of a dynamical system (R n , a, F ) is a smooth function R G − − → R n such that G(0) = a (which is the top triangle) and D[G](x, 1) = F (G(x)) (which is the bottom square). Since G is a tuple G = g 1 , . . . , g n where R g j − − → R, this amounts to saying that G solves the following system of differential equations: g 1 (x), . . . , g n (x)) g n (0) = a n Furthermore, note that when n = 1, G is always an (R, ⊙ e x , u e x )-solution.
For a differential exponential semiring, its differential exponential map is the solution to the dynamical system which generalizes the initial value problem f ′ (x) = f (x) and f (0) = 1, and that it is unique among certain solutions. Proposition 5.5 Let (A, ⊙, u, e) be a differential exponential semiring. Then e is the unique (A, ⊙, u)-solution of the dynamical system (A, u, 1) such that ⊕e = (e × e)⊙.
Proof: The left diagram of (19) is precisely the middle diagram of (11) that 0e = u. While the right diagram of (19) is precisely the left diagram of (11) that D[e] = (e × 1)⊙. Now suppose that there was another map A f − − → A which was an (A, ⊙, u)-solution of (A, u, 1) such that ⊕f = (f × f )⊙. This would imply that (A, ⊙, u, f ) was also a differential exponential semiring. However by Theorem 4.8, (A, ⊙, u, e) = (A, ⊙, u, f ) and so e = f . ✷ Example 5.6 Here we apply Proposition 5.5 to the examples of differential exponential semirings in SMOOTH from the previous section.
(i) The exponential function R e x −−→ R is the unique solution to the following differential equation: (ii) The complex exponential function R 2 ǫ −→ R 2 is the unique solution to the following system of differential equations: such that, using complex number notation: is the unique solution to the following system of differential equations: such that: (f 1 (x 1 + y 1 ), . . . , f n (x n + y n )) = (f 1 (x 1 )f 1 (y 1 ), . . . , f n (x n )f n (y n )) (iv) The dual numbers exponential function R 2 T(e x ) −−−−→ R 2 is the unique solution to the following system of differential equations: such that, using dual number notation: Differential exponential maps also provide solutions to certain dynamical systems in context.
Definition 5.7 In a Cartesian differential category, (ii) If (A, a 0 , a 1 ) is a dynamical system and (B, b 0 , b 1 ) a parametrized dynamical system, then such that the following diagram commutes: (iii) For a dynamical system (A, a 0 , a 1 ), an endomorphism B Dynamical systems can be described as parametrized dynamical systems over the terminal object ⊤ and in this case (17) is the same as (20), modulo the isomorphism A ∼ = A × ⊤.
Definition 5.8 Let (A, ⊙, u) be a differential semiring and (A, a 0 , a 1 ) a parametrized dynamical system over X.
(i) A parametrized (A, ⊙, u)-solution of (A, a 0 , a 1 ) is a map A × X f − − → A such that the following equalities hold: there is a parametrized (A, ⊙, u)-solution of the parametrized dynamical system (A, a 0 , a 1 ). Lemma 5.9 Let (A, ⊙, u) be a differential semiring.
(i) Let (A, a 0 , a 1 ) be a parametrized dynamical system over X. (A, a 0 , a 1 ), then f is also a parametrized (A, 0, u)-solution of (A, a 0 , a 1 ). (A, a 0 , a 1 ). The top triangle of (20) is the left equality of (21). So we need only show the bottom square of (20): So we conclude that f is a parametrized (A, 0, u)-solution of (A, a 0 , a 1 ). Now suppose that an endomorphism A (A, a 0 , a 1 ), which is therefore also a parametrized (A, 0, u)solution of (A, a 0 , a 1 ). So we conclude that a 0 is also (A, 0, u)-complete. ✷ We wish to show that for a differential exponential semiring, a certain class of linear endomorphisms are complete, that is, always have a parametrized solution.
(iii) For every point ⊤ a − − → A, ⊙ a is linear and the following diagrams commute: (iv) For an endomorphism A f − − → A, ⊙ uf = f if and only if the following diagram, Proof: We leave these as an exercise for the reader to check for themselves. ✷ Proposition 5.12 Let (A, ⊙, u, e) be a differential exponential semiring. Then for every point ⊤ Proof: Let ⊤ a − − → A be point and let X a 0 −−→ A be an arbitrary map. Then (A, a 0 , ⊙ a ) is a parametrized dynamical system over X. Now consider the following composite: We need to show both equalities of (21). Starting with the left identity of (21), using that since by Lemma 5.11 ⊙ a is linear it is also additive, we have that: And for the right identity of (21): (Lemma 2.9) = π 0 × 1, π 0 π 1 (⊙ a × ⊙ a ) × a 0 ((e × 1) × 1)(⊙ × 1)⊙ (Lemma 5.11 + Lemma 2.9 + (11)) = ((⊙ a × a 0 ) × 1)((e × 1) × 1)(⊙ × 1)⊙ (Lemma 5.11 + (8)) So we conclude that (⊙ a × a 0 )D[e] is an (A, ⊙, u)-solution of (A, a 0 , ⊙ a ) and therefore that ⊙ a is (A, ⊙, u)-complete. ✷ We would now like to prove the "converse" of Proposition 5.5, that is, we would like differential exponential maps to arise as solutions to certain dynamical systems. To do so, we will require the extra assumption that solutions are unique.
Proposition 5.13 Let (A, ⊙, u) be a differential semiring and suppose that: Then (A, ⊙, u, e) is a differential exponential semiring.
Proof: We must show that e satisfies the three identities of (11). By definition of e being an (A, ⊙, u)-solution of (A, u, 1), 0e = u and D[e] = (e×1)⊙. So it remains to show that ⊕e = (e×e)⊙.

Differential Exponential Maps for Differential Categories
An interesting and important source of Cartesian differential categories are coKleisli categories of differential categories [2,4]. In this section, we study differential exponential maps the coKleisli category of a differential (storage) category and also introduce !-differential exponential algebras and show that these are bijective correspondence with these sorts of differential exponential maps. If only to introduce notation, we first briefly review the full definition of a differential category. Here we present the definition of differential category found in [3], which is mostly the same as the one found in the original paper [4] but with the addition of the interchange rule, which has now become part of the definition. Also, for simplicity and following the convention of other differential category papers, in this section we allow ourselves to work in strict monoidal categories, that is, the associator and the unitor of the tensor product ⊗ are strict equalities. For a symmetric monoidal category, we let K be the monoidal unit and σ : A ⊗ B − → B ⊗ A be the natural symmetry isomorphism. , (f, g) → f + g, and zero 0 ∈ X(A, B), such that composition and the tensor product preserves the additive structure, that is, the following equalities hold: (ii) Equipped with a coalgebra modality [3, Definition 1], which is a quintuple (!, δ, ε, ∆, ι) consisting of an endofunctor ! and natural transformations !A is a comonad on X and (!A, ∆, ι) is a cocommutative comonoid, that is, the following equalities hold: and δ is a comonoid morphism, that is, the following equalities hold: For a full detailed explanation of the deriving transformation axioms and a string diagram representation, see [4,3]. Examples of differential categories can be found at the end of this section, while numerous other interesting examples can be found in [3,Section 9].
For a differential category with finite products, its coKleisli category is a Cartesian differential category. As we will be working with coKleisli category, we will use the notation found in [5] and use interpretation brackets − to help distinguish between composition in the base category and coKleisli composition. So for a comonad (!, δ, ε) on a category X, let X ! denote its coKleisli category, which is the category whose objects are the same as X and where X ! (A, B) = X(!A, B) with composition and identity defined as: If X if has finite products then so does X ! where on objects the product is defined as in X and where the remaining data is defined as follows: If X is a Cartesian left additive category then so is X ! where: where ⊕ is defined as in Lemma 2.10. Since every category with finite biproducts is a Cartesian left additive category (where every map is additive), it follows that every differential category with finite products (which by the additive enrichment are in fact biproducts) is a Cartesian left additive category. Lastly, one uses the deriving transformation to define the differential combinator of the coKleisli category.
= !(⊕)e By Proposition 6.3, e is a differential exponential map in the coKleisli category. ✷ In a differential storage category, differential exponential maps in the coKleisli category can also be characterized by commutative monoids in the base category Definition 6.6 A !-differential exponential algebra in a differential storage category is a quadruple (A, , v, e) consisting of an object A and maps is a commutative monoid, that is, the following diagrams commute: and also that the following diagrams commute: = (e ⊗ e) = ∇e (36) So we conclude that e is a differential exponential map in the coKleisli category. Next we show that u e = χ ⊤ v: Using the above identity, we can show that: (Naturality of ε) And so we have that ⊙ e = χ(ε ⊗ ε) . ✷ Proposition 6.8 Let !A e − − → A be a differential exponential map in the coKleisli category of a differential storage category. Define the maps A ⊗ A e − −− → A and K ve −−→ A respectively as follows: Then (A, e , v e , e) is a !-differential exponential algebra.
Proof: We first show that (A, e , v e ) is a commutative monoid. Starting with showing that e is commutative: Now that we've shown commutativity, we need only show one of the unit identities: Lastly, we show that e is also associative: So we conclude that (A, e , v e ) is a commutative monoid. Next we show that e satisfies the three identities of (36). The left most diagram of (36) is precisely the left diagram of (34) and v e = νe by construction. So it remains only to show that ∇e = (e ⊗ e) e : (e ⊗ e) e = (e ⊗ e)(η ⊗ η)∇e = (e ⊗ 1)(η ⊗ 1)∇e (34) So we conclude that (A, e , v e , e) is a !-differential exponential algebra. ✷ Finally we will now show that the constructions of Proposition 6.7 and Proposition 6.8 are inverses of each other by showing that the category of !-differential exponential algebras is isomorphic to the category of differential exponential maps in the coKleisli category. For a differential storage category X, define its category of !-differential exponential algebras !DEA[X] as the category whose objects are !-differential exponential algebras (A, , v, e) and where a map (38) and where composition and identity maps are as in X. Note that the two right most diagrams above imply that f is a monoid morphism. Theorem 6.9 For a differential storage category X, its category of !-differential exponential algebras !DEA[X] is isomorphic to the category of differential exponential maps of the coKleisli category = !(f )e ′ (38) and therefore F is well-defined. On the other hand, by Proposition 6.7, F −1 is well-defined on objects and so it again remains to check that it is also well-defined on maps. Note that since every map in DEM[X ! ] is of the form g = εg ′ , it follows from [cd.3] that F −1 ( g ) = ηεg ′ = g ′ . Since g is a map in DEM[X ! ], by Theorem 4.8, we also have that the following equalities hold: Now since g = εg ′ , the above identities can easily be simplified out to be: Using these identities, we now show that F −1 ( g ) satisfies (38): = (g ′ ⊗ g ′ )(η ⊗ η)χ −1 ⊙ e ′ (Naturality of χ and η) = v ′ (Proposition 6.7) So F −1 ( g ) is a map in !DEA[X] and therefore F −1 is well-defined. We leave it to the reader to check for themselves that F and F −1 preserves both identities and composition, and are thus indeed functors. Lastly, we need to show that F and F −1 are inverses of each other. Clearly FF −1 (A, e) = (A, e) and FF −1 ( g ) = g . In the other direction, we clearly have that F −1 F(f ) = f and so it remains to show that (A, , v, e) = FF −1 (A, , v, e) = (A, e , v e , e), that is, we need to show that = e and v = v e -both of which follow immediately from (36): We conclude this section by briefly studying !-differential exponential algebras in two examples of differential storage categories. Example 6.10 Let REL be the category of sets and relations, that is, the category whose objects are sets X and where a map X , and where the codereliction η ⊆ X × !X is defined as follows: For more details on this example, see [4,Proposition 2.7]. It turns out that in fact ! is the free exponential modality [20] on REL, that is, !X is the cofree cocommutative comonoid over X in REL and therefore !-coalgebras are precisely cocommutative comonoids in REL. By self-duality of REL, ! is also a monad such that !X is the free commutative monoid over X in REL and !-algebras are precisely commutative monoids in REL. In fact, the codereliction η is the unit of the monad structure of !. It turns out that the !-differential exponential algebras are precisely the commutative monoids in REL (or equivalently the !-algebras). Indeed, every !-differential exponential algebra (A, , v, e) is by definition a commutative monoid (A, , v) and its associated !-algebra structure is precisely e ⊂ !A × A. Conversely, given a commutative monoid (A, , v), its associated !-algebra structure e ⊆ !A×A satisfies by definition that ηe = 1 and is also a monoid morphism, and therefore (A, , v, e) is a !-differential exponential algebra. In particular, since for every X, (!X, ∇, ν) is a commutative monoid, there is a natural transformation µ ⊆ !!X × !X such that (!X, ∇, ν, µ) is a !-differential exponential algebra. Explicitly, µ is defined as follows: Therefore, µ ⊆ !!X × !X is a differential exponential map in the coKleisli category REL ! . Also, it turns out that every X comes equipped with a monoid structure in REL given by the dual of the copying relation, and so the induced !-algebra structure e ⊆ !X × X is given by: (f, x) ∈ e ⇔ f (y) = n if y = x 0 if x = y and so for every X, e ⊆ !X × X is a differential exponential map in the coKleisli category REL ! .
For more examples, monoids in REL are studied in [15].
Example 6.11 Let k be a field and let VEC k be the category of k-vector spaces and k-linear maps between them. For a k-vector space V , let Sym(V ) be the symmetric algebra over V [16, Section 8, Chapter XVI], that is, the free commutative k-algebra over V . In particular, if X is a basis set for V , then Sym(V ) ∼ = k[X] where k[X] is the polynomial ring over the set X. This induces a monad Sym on VEC k such that the Sym-algebras are precisely the commutative k-algebras. Now suppose that k has characteristic 0, then VEC k is a differential storage category where: and where the codereliction V η − − → !V is defined as the injection of V into the 0 ∈ V component of !V . For full details on this example, see [7]. Similarly to the previous example, ! is the free exponential modality on [20] on VEC k , that is, !V is the cofree cocommutative k-coalgebra over V [21] and therefore !-coalgebras are precisely cocommutative k-coalgebras. It turns out that once again !-differential exponential algebras correspond precisely to commutative monoids in VEC k which are precisely the commutative k-algebras or equivalently the Sym-algebras. By definition, every every !-differential exponential algebra (A, , v, e) is a commutative k-algebra and it turns out that its Sym-algebra structure is given by pre-composing !A then it follows that (A, , v, e ω ) is a !-differential exponential algebra. In particular, since for every V , (!V, ∇, ν) is a commutative k-algebra, there is a natural transformation !!V µ − − → !V such that (!X, ∇, ν, µ) is a !-differential exponential algebra and therefore µ is a differential exponential map in the coKleisli category.
Other examples of !-differential exponential algebras should include the exponential function in the differential category of convenient vector spaces [6], and the power series associated with the exponential function as defined in [13,Lemma 19] for the differential category of finiteness spaces.

Conclusion and Future Work
As the exponential function e x (and its generalizations) is so prominent and important throughout various fields and has numerous applications, this opens the door to numerous possibilities and applications for differential exponential maps. In particular, as the theory of differential equations in Cartesian differential categories develops, differential exponential maps should be a key component for this theory in the same way that the exponential function is a fundamental tool in solving classical differential equations.
Another important path to take is to find and study more examples of differential exponential maps. Indeed, while in this paper we provided interesting examples of such in the category of real smooth functions and certain coKleisli categories of differential categories, one should also consider exploring differential exponential maps in more exotic examples of Cartesian differential categories such as in cofree Cartesian differential categories [12,17] or abelian functor calculus [1]. Another possible source of examples is to construct differential exponential maps in the presence of infinite sums, which many categorical models of the differential λ-calculus [14,19] have.
There are also certain interesting potential generalizations of differential exponential maps to consider. For example, the exponential function e x can also be defined as the inverse of the natural logarithm function ln(x). Since ln(x) is only partially defined, one must work in a differential restriction category [9] to generalize the natural logarithm function in such a way that differential exponential maps arise as their (partial) inverse. On the other hand, one could also generalize differential exponential maps to tangent category [10] and differential bundle [11], and this notion should be a generalization of exponential maps for manifolds and Lie groups.
Regarding !-differential exponential algebras, it is interesting to point out that in both examples of differential storage categories studied in this paper, there was a natural transformation !!A µ − − → !A which induced !A with a !-differential exponential algebra structure. A natural question to ask is when does the codereliction A η − − → !A and µ provide a monad structure on ! (with one of the monad identities already being a requirement for a !-differential exponential algebra), and conversely when does a monad structure on ! induce a natural !-differential exponential algebra structure. Lastly, another possible direction would be to generalize the trigonometric functions in the same way for arbitrary Cartesian differential categories.
In conclusion, there are many potential interesting paths to take for future work with differential exponential maps.