Categorical frameworks for generalized functions

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Fr\"olicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Fr\"olicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Fr\"olicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Fr\"olicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.

The problem of considering (generalized) derivatives of locally integrable functions arises frequently in Physics, e.g. in idealized models like in shock Mechanics, material points Mechanics, charged particles in Electrodynamics, gravitational waves in General Relativity, etc. (see e.g. [10,21,31]). Therefore, the need to perform calculations with discontinuous functions like one deals with smooth functions motivated the introduction of generalized functions (GF) as objects extending, in some sense, the notion of function. As such, generalized functions find deep applications in solutions of singular differential equations ( [22,30]) and are naturally framed in (several) theories of infinite dimensional spaces, from locally convex vector spaces ( [24]) and convenient vector spaces ( [26]) up to diffeological ( [23,25]) and Frölicher spaces ( [11]).
The foundation of a rigorous linear theory of generalized functions has been pioneered by L. Schwartz with a deep use of locally convex vector space theory ( [32,22]), but heuristic multiplications of distributions early appeared e.g. in quantum electrodynamics, elasticity, elastoplasticity, acoustics and other fields ( [10,31]). Despite the impossibility of a straightforward extension of Schwartz linear theory ( [33]) to an algebra extending pointwise product of continuous functions, the theory of Colombeau algebras (see e.g. [8,9,10,30,21,31]) permits to bypass this impossibility in a very simple way by considering an algebra of generalized functions which extends the pointwise product of smooth functions.
The main aim of the present work is to study different categories as frameworks for generalized functions. In particular, we introduce the category FDlg of functionally generated spaces. This category has very nice properties and strictly lies between Frölicher and diffeological spaces.
We start by defining the algebras from Colombeau theory that we will consider in this work. Henceforth, we will use the notations of [21,22] for the well-known Schwartz distribution theory.
1.1. The special and full Colombeau algebras.
The ring of constants in G s is denoted by R and is called the ring of Colombeau generalized numbers (CGN). It is an ordered ring with respect to the order defined by [x ε ] ≤ [y ε ] iff ∃[z ε ] ∈ R such that (z ε ) ∼ 0 and x ε ≤ y ε + z ε for ε sufficiently small. Even if this order is not total, we can still define the infimum [x ε ] ∧ [y ε ] := [min(x ε , y ε )], and analogously the supremum of two elements. More generally, the space of generalized points in Ω is Ω = Ω M / ∼, where Ω M = {(x ε ) ∈ Ω I | ∃N ∈ N : |x ε | = O(ε −N )} is called the set of moderate nets and (x ε ) ∼ (y ε ) if |x ε − y ε | = O(ε m ) for every m ∈ N. By N we will denote the set of all negligible nets of real numbers (x ε ) ∈ R I , i.e., such that (x ε ) ∼ 0.
The space of compactly supported generalized points Ω c is defined by Ω c / ∼, where Ω c := {(x ε ) ∈ Ω I | ∃K ⋐ Ω ∃ε 0 ∀ε < ε 0 : x ε ∈ K} and ∼ is the same equivalence relation as in the case of Ω. Any Colombeau generalized function (CGF) u ∈ G s (Ω) acts on generalized points from Ω c by u(x) := [u ε (x ε )] and is uniquely determined by its point values (in R) on compactly supported generalized points ( [21]), but not on standard points. A CGF [u ε ] is called compactly-bounded (c-bounded) from Ω into Ω ′ if for any K ⋐ Ω there exists K ′ ⋐ Ω ′ such that u ε (K) ⊆ K ′ for ε small. This type of CGF is closed with respect to composition. Moreover, if u ∈ G s (Ω) is c-bounded from Ω into Ω ′ and v ∈ G s (Ω ′ ), then [v ε • u ε ] ∈ G s (Ω). For x, y ∈ R n we will write x ≈ y if x − y is infinitesimal, i.e., if |x − y| ≤ r for all r ∈ R >0 .
Topological methods in Colombeau theory are usually based on the socalled sharp topology (see e.g. [2] and references therein), which is the topology generated by the balls B S ρ (x) = {y ∈ R n | |y − x| < ρ}, where | − | is the natural extension of the Euclidean norm on R n , i.e., |[x ε ]| := [|x ε |] ∈ R, and ρ ∈ R >0 is positive invertible. Henceforth, we will also use the notation R * := {x ∈ R | x is invertible}. Finally, Garetto in [12,13] extended the above construction to arbitrary locally convex spaces by functorially assigning a space of CGF G s E to any given locally convex space E. The seminorms of E can then be used to define pseudovaluations which in turn induce a generalized locally convex topology on the C-module G s E , again called sharp topology.
The full Colombeau algebra. Clearly, the embedding ι Ω defined in (1.1) depends on the net of maps (ψ ε ) ∈ D(R n ) I whose existence is given by Thm. 1. This shall not be considered only in a negative way: e.g. it is not difficult to choose (ψ ε ) so that the embedding satisfies the properties that is an infinite number of R (here H is the Heaviside function and δ is the Dirac delta function). These properties are informally used in several applications.
The main idea of the full Colombeau algebra is to consider a different set of indices, instead of I = (0, 1], so as to obtain an intrinsic embedding. (vii) We say that R ∈ N e (Ω) iff R ∈ E e (Ω) and On a Frölicher space X we consider only U = R, i.e., the smooth structure on the space is given by a set of smooth curves; moreover, these curves are determined by (and they determine) a given set of functionals, i.e., of smooth functions of the type l : X −→ R (see Def. 9). The category Fr of all Frölicher spaces is a full subcategory of the category Dlg of all diffeological spaces.
In the following subsections, we are going to focus on a family of diffeological spaces called functionally generated (diffeological) spaces, where the diffeological structure is determined by a given family of locally defined smooth functionals. As we will see in the present work, these spaces frequently appear in functional analysis, strictly lie between diffeological spaces and Frölicher spaces, and the category FDlg of all these spaces behaves nicely -it is complete, cocomplete and Cartesian closed.
To simplify the notation, we write OR ∞ for the category of open sets in Euclidean spaces and ordinary smooth functions.
is a set |X| together with a specified family of functions such that for any U, V ∈ OR ∞ , the following three axioms hold: For a diffeological space X = (|X|, D), every element in D is called a plot of X. We write d ∈ U X to denote that d ∈ D U , which will also be called a figure of type U of the space X.

Definition 4.
A morphism (also called smooth map) f : X −→ Y between two diffeological spaces X = (|X|, D X ) and Y = (|Y |, D Y ) is a function |f | : |X| −→ |Y | such that f • d ∈ D Y U for any d ∈ D X U and U ∈ OR ∞ . If we write f (d) := f • d, by the covering condition of Def. 3 we have a generalization of the usual evaluation; moreover, f : X −→ Y is smooth if and only if for all U ∈ OR ∞ and d ∈ U X, we have f (d) ∈ U Y , i.e., f take figures of type U on the domain to figures of the same type in the codomain. Moreover, X = Y as diffeological spaces if and only if for all d and U , d ∈ U X if and only if d ∈ U Y . These and several other generalization of set-theoretical properties justify the use of the symbol ∈ U .
All diffeological spaces with smooth maps form a category, which will be denoted by Dlg. Given two diffeological spaces X and Y , we write C ∞ (X, Y ) for the set of all smooth maps X −→ Y .
Here is a list of basic properties of diffeological spaces. We refer readers to the standard textbook [23] for more details.

Remark 5. (i)
By a smooth manifold, we always assume it is Hausdorff and finite-dimensional. Every smooth manifold M is automatically a diffeological space M = (M, D) with d ∈ U M if and only if d : U −→ M is smooth in the usual sense. We call this D the standard diffeology on M , and without specification, we always assume a smooth manifold with this diffeology when viewed as a diffeological space. Moreover, given two smooth manifolds M and N , f : M −→ N is smooth if and only if f : M −→ N is smooth in the usual sense. In other words, the category Man of all smooth manifolds and smooth maps is fully embedded in Dlg. This justifies our notation C ∞ (X, Y ) for the homset Dlg(X, Y ). Limits of smooth manifolds that already exist in Man are preserved by this embedding (see Thm. 27). Generally speaking the same property does not hold for colimits of smooth manifolds that already exist in Man.
(ii) Given a set X, the set of all diffeologies on X forms a complete lattice.
The smallest diffeology is called the discrete diffeology, which consists of all locally constant functions, and the largest diffeology is called the indiscrete diffeology, which consists of all set functions. Let A = (X, D A ) and B = (X, D B ) be two diffeological spaces with the same underlying set. We simply write Therefore, given a family of functions {ι i : |X i | −→ Y } i∈I from the underlying sets of the diffeological spaces X i to a fixed set Y , there exists a smallest diffeology on Y making all these maps ι i smooth. We call this diffeology the final diffeology associated to I. In more detail, Dually, given a family of functions {p j : X −→ |Y j |} j∈J from a given set X to the underlying sets of the diffeological spaces Y j , there exists a largest diffeology on X making all these maps p j smooth. We call this diffeology the initial diffeology associated to J. In more detail, In particular, if Y is a quotient set of |X|, then the final diffeology on Y associated to the quotient map |X| −→ Y is called the quotient diffeology, and Y with the quotient diffeology is called a quotient diffeological space of X. Dually, if X is a subset of |Y |, then the initial diffeology on X associated to the inclusion map X −→ |Y | is called the sub-diffeology, and we write (X ≺ Y ) to denote this new diffeological space. We call (X ≺ Y ) the diffeological subspace of Y . Finally, the initial diffeology associated to the projection maps p i : i∈I |X i | −→ |X i | of an arbitrary product is called the product diffeology, and dually the final diffeology associated to the inclusion maps |X j | −→ j∈J |X j | of an arbitrary coproduct is called the coproduct diffeology.
(iii) The category Dlg is complete and cocomplete. In more detail, let G : I −→ Dlg be a functor from a small category I. Write | − | : Dlg −→ Set for the forgetful functor. Then both lim G and colim G exist in Dlg as lifting and co-lifting of limits and colimits in Set. In more detail, | lim G| = lim |G| and the diffeology of lim G is the initial diffeology associated to the universal cone {lim |G| −→ |G(i)|} i∈I in Set; dually | colim G| = colim |G| and the diffeology of colim G is the final diffeology associated to the universal co-cone {|G(i)| −→ colim |G|} i∈I in Set. (iv) The category Dlg is Cartesian closed. In more detail, given three diffeological spaces X, Y and Z, there is a canonical diffeology (called the functional diffeology) on C ∞ (X, Y ) defined by in the present work, we use the notations of [1]). Without specification, the set C ∞ (X, Y ) is always equipped with the functional diffeology when viewed as a diffeological space. Then Cartesian closedness means that where g ∧ (x)(y) := g(x, y)). Therefore, Cartesian closedness permits to equivalently translate an infinite dimensional problem like f ∈ C ∞ (X, C ∞ (Y, Z)) into a finite dimensional one f ∨ ∈ C ∞ (X × Y, Z) and vice versa. (v) Every diffeological space can be extended with infinitely near points X ∈ Dlg → • X ∈ • Dlg, X ⊆ • X, obtaining a nonArchimedean framework similar to Synthetic Differential Geometry (see e.g. [28] and references therein) but compatible with classical logic. The category • Dlg of Fermat spaces is defined by generalizing the category of diffeological spaces, but taking suitable smooth functions defined on the extension • U ⊆ • R n of open sets U ∈ OR ∞ . It is remarkable to note that the so called Fermat functor • (−) : Dlg −→ • Dlg has very good preservation properties strictly related to intuitionistic logic. See [14,15,16,17,20] for more details. (vi) Dlg is a quasi-topos and hence is locally Cartesian closed ( [3]).
Every diffeological space has an interesting canonical topology: Definition 6. Let X = (|X|, D) be a diffeological space. The final topology τ X induced by D is called the D-topology.
Without specification, every diffeological space X is equipped with the D-topology τ X . Elements in τ X are called D-open subsets.
The D-topology on any smooth manifold is the usual topology. (ii) The D-topology on any discrete (indiscrete) diffeological space is the discrete (indiscrete) topology.  1 We can recall the symbol TD by saying "topological space from diffeological space".
Analogously we can recall the plenty of symbols for the other functors related to our categories in this paper.
|X| and d ∈ U D T (X) iff d ∈ Top(U, X) (both functors act as identity on arrows).
As a consequence, the D-topology of a quotient diffeological space of X is same as the quotient topology of T D (X). However, the D-topology of a diffeological subspace of X may be different from the sub-topology of T D (X).
For more detailed discussion of the D-topology of diffeological spaces, see [23, Chapter 2] and [7]. Now, let's turn to Frölicher spaces. In several spaces of functional analysis (like all those listed in Section 2.5), smooth figures are "generated by smooth functionals". Therefore, smoothness can also be tested using smooth functionals, similarly as using projections in finite dimensional Euclidean spaces. In Frölicher spaces, we focus our attention also to smooth functions of the type X −→ R. Definition 9. A Frölicher space (C, X, F) is a set X together with two specified families of functions C ⊆ Set(R, X) and F ⊆ Set(X, R) with the following smooth compatibility conditions: Frölicher spaces is a function f : X −→ Y such that one of the following equivalent conditions hold: All Frölicher spaces and their morphisms form a category, which will be denoted by Fr. Here is a list of basic properties for Frölicher spaces. For details, we refer readers to [11].

Remark 11. (i)
Every smooth manifold M is automatically a Frölicher space with C = C ∞ (R, M ) and F = C ∞ (M, R). Without specification, we always assume a smooth manifold with this Frölicher structure when viewed as a Frölicher space. Moreover, this gives a full embedding of Man in Fr.
is a Frölicher space and all these maps ι i are morphisms between Frölicher spaces. We call this Frölicher structure on Y the final Frölicher structure associated to Dually, let {p j : X −→ Y j } j∈J be a family of functions from a fixed set X to the underlying sets of the Frölicher spaces is a Frölicher space and all these maps p j are morphisms between Frölicher spaces. We call this Frölicher structure on X the initial Frölicher structure associated to The category Fr is complete and cocomplete. In more detail, let G : I −→ Fr be a functor from a small category I. Write |−| : Fr −→ Set for the forgetful functor. Then both lim G and colim G exist in Fr as lifting and co-lifting of limits and colimits in Set. In more detail, | lim G| = lim |G| and the Frölicher structure of lim G is the initial Frölicher structure associated to the universal cone {lim |G| −→ |G(i)|} i∈I in Set; dually | colim G| = colim |G| and the Frölicher structure of colim G is the final Frölicher structure associated to the universal co-cone {|G(i)| −→ colim |G|} i∈I . In the category of Hausdorff Frölicher spaces, limits and colimits of smooth manifolds that already exist in Man are preserved by the embedding Man −→ HFr (see Thm. 31). (iv) The category Fr is Cartesian closed. In more detail, given Frölicher spaces X and Y , set Then one can show that (C, Fr(X, Y ), F) is a Frölicher space. Without specification, Fr(X, Y ) is always equipped with this Frölicher structure when viewed as a Frölicher space.
This defines a full embedding D F : Fr −→ Dlg. So, there will be no confusion to call smooth maps also the morphisms between Frölicher spaces. Moreover, one can show that ) as diffeological spaces. This embedding functor has a left adjoint given as follows. For a diffeological space Both functors D F and F D are identities on the morphisms. For more discussion on the relationship between diffeological spaces and Frölicher spaces, see [34,4].

2.2.
Definition and examples of functionally generated diffeologies. Now, let's introduce a special class of diffeological spaces called functionally generated spaces, which are like Frölicher spaces but with locally defined smooth functionals. The idea is that, in this type of spaces, we can know whether a continuous function d ∈ Top(U, T D (X)) is a figure by testing for smoothness of the composition with a given family of locally defined smooth functionals l : (A ≺ X) −→ R.
We say that F generates D if for any open set U ∈ OR ∞ and any continuous map d ∈ Top(U, T D (X)), the condition is called a smooth functional of the space X. Finally, we say that the diffeological space X is functionally generated if its diffeology can be generated by some family F, and we denote with FDlg the full subcategory of Dlg of all functionally generated diffeological spaces.
If the codomain of a continuous map f : X −→ Y is functionally generated, then we can also test the smoothness of f by smooth functionals of Y : Assume that the diffeology of Y is generated by the family {F A } A∈τ Y . Then the following are equivalent Then for any A ∈ τ Y and any l ∈ F A , by Here is a list of basic properties and examples of functionally generated spaces: The notion of functionally generated space is of local nature, i.e., we can equivalently say that F generates D if for any U ∈ OR ∞ and any d ∈ Set(U, |X|), the condition (ii) We can also equivalently ask that F A ⊆ Set(A, R) and for all continuous d ∈ Top(U, T D (X)) and any open set U ∈ OR ∞ we have d ∈ U X if and only if (2.1) holds. Therefore, smooth functionals determine completely the figures (plots) of the underlying diffeological space, i.e., if D 1 , D 2 are diffeologies on |X| and F generates both D 1 and D 2 , then Then M X also generates D. Of course, M X is the maximum family of smooth functionals which can be used to test whether a continuous map d ∈ Top(U, T D (X)) is a figure or not, and the interesting problem is to find a smaller family F ⊆ M X generating the same set of plots of X.
(iv) The diffeology generated by a Frölicher space (C, X, F) is functionally generated by globally defined smooth functionals. That is, it suffices to considerF defined by l ∈F A if and only if A = X and l ∈ F. Therefore, the functor F D : Fr −→ Dlg has values in FDlg. In particular, every smooth manifold and every discrete diffeological space is functionally generated. However, there are functionally generated spaces which do not come from Frölicher spaces; see Ex. 25.
In a functionally generated space, besides the usual D-topology τ X , we can consider the initial topology τ F with respect to all smooth functionals A∈τ X F A (which is analogous to the weak topology, see e.g. [24]). In particular, the topology τ M X is called the functional topology on X. In general, τ F is coarser than the D-topology, see Ex. 25, but in every functionally generated space the functional topology and the D-topology coincide, as stated in the following theorem.
Proof. The topology τ F has the set as a subbase. For any l −1 (V ) in this subbase and any plot d ∈ U X the set Here are some examples of diffeological spaces which are not functionally generated.
Let (X, D) be the irrational torus R/(Z+θZ), for some θ ∈ R\Q, with the quotient diffeology D; see [23]. Then (X, D) is not functionally generated. Indeed, since Z + θZ is dense in R, the D-topology τ X is indiscrete, that is, τ X = {∅, X}. Hence, every smooth map X −→ R is constant. Therefore, for any function d : U −→ X, for any l ∈ M X , the composition l • d is constant, hence smooth. Therefore, there does not exist a family F that generates D. (ii) For any n ≥ 2, let R n w = (R n , D w ) be R n with the wire diffeology D w ; see [23]. Then the D-topology of R n w is the usual Euclidean topology, and by Boman's theorem [5], w is functionally generated, we would have 1 R n ∈ R n R n w , which is false for the wire diffeology. Therefore, R n w is not functionally generated.
2.3. Categorical properties of functionally generated spaces. In this subsection, we are going to prove some nice categorical properties for the category FDlg of all functionally generated spaces with smooth maps, that is, FDlg is complete, cocomplete and Cartesian closed.
Although the family F that generates a diffeology is a τ X -family of smooth functions, in practice, we usually only need a B-family with B ⊆ τ X . In other words, F A can be any subset (in particular, the empty set) of Assume that each D i is generated by F i , and let D be the initial diffeology on X associated to this family (i.e., d The proof follows directly from Def. 12.
for some i ∈ I and B ∈ τ X i , so that we are essentially considering only smooth functionals defined on D-open subsets . In this sense, F is also the smallest family of smooth functionals generating (X, D) and containing all the smooth functionals of the form λ • p i | p −1 i (B) . In particular, every subset of a functionally generated space with the subdiffeology is again functionally generated. Analogously, every product of functionally generated spaces with the product diffeology is functionally generated, so that Similarly, one can show that every coproduct of functionally generated spaces with the coproduct diffeology is functionally generated.
Let f, g : X −→ Y be smooth maps between functionally generated spaces. In general, the coequalizer in Dlg may not be functionally generated. For example, let X = R be equipped with the discrete diffeology, and let Y = R be equipped with the standard diffeology. Let θ be some irrational number. Fix a representative in R for each element in R/(Z+θZ), i.e., a function ρ : Let f : X −→ Y be the identity function and let g : X −→ Y be the function defined by g(r) := ρ(c) for all r ∈ c ∈ R/(Z + θZ), i.e., sending every point in the subset ρ(c) + Z + θZ to the fixed representative ρ(c) ∈ R. It is clear that both f and g are smooth because we equip X with locally constant figures, and the coequalizer in Dlg is the irrational torus because the equivalence relation of R/(Z + θZ) is the smallest one where f (r) = r is equivalent to g(r) = ρ(c) for r ∈ c. We already know from (i) of Ex. 16 that the irrational torus is not functionally generated. However, we will show below that the category FDlg is cocomplete, and the coequalizer of the above diagram in FDlg is the underlying set of the irrational torus with the indiscrete diffeology. Now, we want to see how to define a new functionally generated diffeology starting from a diffeological space and a τ X -family of smooth functionals.
We set DF := ∪ U ∈OR ∞ DF U and callX F := (|X|, DF) the diffeological space generated by X and F. If Y ∈ Dlg is a diffeological space, we will always apply the above construction with respect to the D-topology, i.e., with X = T D (Y ) and considering only smooth functions: One can show directly from the definitions that Remark 20. In the hypotheses of the previous Def. 19, the following properties hold: We can trivially extend the B-family F to the whole σ-family by set- We will always assume to have extended F in this way; (vii) the D-topology onX F coincides with the D-topology τ X on X; And if X is functionally generated, then X =X M X ; (ix) DM X is the smallest functionally generated diffeology on |X| containing D.
In particular, if we take F to be the empty τ X -family, that is, Theorem 21. The inclusion functor D FG : FDlg / / Dlg is a right adjoint of the functor FG D : X ∈ Dlg →X M X ∈ FDlg (both functors act as identity on arrows). Therefore, for all X ∈ Dlg and Y ∈ FDlg, we have Proof. It follows by applying Def. 12, Def. 19 and Rem. 20.
Corollary 22. Let G : I −→ FDlg be a functor from a small category I. Then Therefore, the category FDlg is cocomplete.
Proof. Since FG D is a left adjoint, it preserves colimits But G i ∈ FDlg is functionally generated, so FG D (G i ) = G i from (viii) of Rem. 20. Here is another interesting example that colimit in FDlg is different from the corresponding colimit in Dlg: Example 24. Let X be the pushout of in Dlg. Then we have a commutative diagram in Dlg with i(x) = (x, 0) and j(y) = (0, y). This induces a smooth injective map X −→ R 2 . Write Y ∈ Dlg for the image of this map with the subdiffeology of R 2 . One can show that (i) the induced smooth map X −→ Y is not a diffeomorphism; (ii) the D-topology on both X and Y coincide with the sub-topology of which implies that X is not functionally generated; (iv) Y is Frölicher because Fr is closed with respect to subobjects, so Y ∈ FDlg.
Hence, by Cor. 22, the pushout of (2.3) in FDlg is FG D (X) ≃ Y ≃ X. Now we show that the embedding FG F : Fr −→ FDlg is not essentially surjective, that is, there are functionally generated spaces which are not from Frölicher spaces: Example 25. Let Y = (−∞, 0) ∪ (0, ∞), and let X be the pushout of R). Since no element in C ∞ (X, R) can detect the double points at origin, there is no Frölicher space such that its image under the embedding D F : Fr −→ Dlg is X. But since the two colimit maps R −→ X are injective and open, X is functionally generated. In other words, for any U ∈ OR ∞ , C ∞ (X, R) can not detect whether an arbitrary function U −→ X is smooth, but it can detect whether a continuous function U −→ X is smooth. Moreover, the initial topology on X with respect to C ∞ (X, R) is strictly coarser than the D-topology.
Theorem 26. The category FDlg is Cartesian closed.
Proof. Since Dlg is Cartesian closed and the product in FDlg is the same as the product in Dlg, it suffices to show that if X is a diffeological space and Y is a functionally generated space, then the functional diffeology of the space C ∞ (X, Y ) is functionally generated.
We split the proof of the claim into three steps.
Step 1: We prove that if C ∞ (R n , Y ) is functionally generated for all n ∈ N, then C ∞ (X, Y ) is functionally generated.
To prove that C ∞ (X, Y ) is functionally generated, by (vi) of Rem. 20, for any d ∈ U FG D (C ∞ (X, Y )) we need to show that d ∈ U C ∞ (X, Y ), i.e., that d ∨ : U × X −→ Y is smooth. This is equivalent to show that for any plot p : R n −→ X, the composition This is again equivalent to show that the composition is smooth. By assumption C ∞ (R n , Y ) is functionally generated, and the map p * is smooth, so the adjunction FG D ⊣ D FG (Thm. 21) implies that is smooth, which prove our first claim.
Step 2: We prove below that if d : which implies that the map d ∨ is continuous.
Step 3: We prove below that C ∞ (R n , Y ) is functionally generated. Let d ∈ U FG D (C ∞ (R n , Y )).We need to show that the induced function d ∨ : U × R n −→ Y is smooth. From Step 2, we know that d ∨ is continuous. Since Y is functionally generated, it is enough to show that for any D-open subset A of Y and any l ∈ C ∞ (A ≺ Y, R), the composition , use the notationsÃ, V and W defined in Step 2. Since smoothness is a local condition, it is enough to show that the composition Equivalently, we need to show that the composition is smooth, where Res is the restriction map. It is easy to see that the map Res : Y ))). But both V and R are Frölicher spaces, so C ∞ (V, R) is functionally generated, and the adjunction FG D ⊣ D FG (Thm. 21) implies that the map l * • Res : Y ))), so the conclusion follows.

Preservation of limits and (suitable) colimits of manifolds.
In this subsection, we are going to discuss the question that if a limit (or colimit) exists in Man, the category of smooth manifolds and smooth maps, then is it the same as the corresponding limit (or colimit) in FDlg? The statements of the main results and the idea of proofs mainly come from [29]. Proof. By the universal property of limit in FDlg, there is a canonical smooth map η : First we prove that |η| is surjective. Note that any x ∈ | lim(FG M • F )| corresponds to a smooth map x : R 0 −→ lim(FG M • F ). So we have a cone x −→ F . Since R 0 is a smooth manifold and lim F exists in Man, by the universal property of limit in Man and FDlg, there exists y : R 0 −→ lim F such that x = η • FG M (y), which implies that |η| is surjective.
Next we prove that |η| is injective. If a, a ′ ∈ |FG M (lim F )| such that |η|(a) = |η|(a ′ ), then the two cones a −→ F and a ′ −→ F have the same image in the target. By the universal property of limit in Man, a = a ′ .
Finally, we prove that η −1 is smooth. Let d ∈ U lim(FG M • F ). Since the functor FG M is fully faithful, we get a cone U −→ F . Note that FG M (U ) = U . By the universal property of limit in Man and FDlg, we get a smooth map f : Remark 28. Note that the category OR ∞ with the usual open coverings is a site. [3] showed that OR ∞ is a concrete site, and the category Dlg is equivalent to the category CSh(OR ∞ ) of concrete sheaves over OR ∞ . We write CPre(OR ∞ ), Pre(OR ∞ ) and Sh(OR ∞ ) for the category of concrete presheaves over OR ∞ , the category of presheaves over OR ∞ and the category of sheaves over OR ∞ , respectively. There are embedding functors and Moreover, if a limit exists in Man, then the corresponding limits in all the other categories listed above are isomorphic to that limit in the corresponding categories.
In general, if a colimit in Man exists, when viewed as a functionally generated space it may be different from the corresponding colimit in FDlg.
Example 29. Recall that in Ex. 25, we showed that the underlying set of the pushout X of in FDlg has double points at origin. Moreover, the D-topology τ X is not Hausdorff. One can also show by continuity that the pushout of this diagram in Man exists, and it is R. Therefore, X ≃ R in FDlg. Assume that |η| is not surjective. Say y ∈ |FG M (colim G)| is not in the image. Then A := colim G\{y} is a smooth manifold, and G(j) −→ colim G factors through A ֒→ colim G for each j ∈ J since | colim(FG M • G)| = colim |FG M • G|. Hence, by the universal property of colimit in Man, the identity map colim G −→ colim G must factor through A ֒→ colim G, which is impossible. Therefore, |η| is surjective. Proof. Let X be a diffeological space. It is direct to show that (1) If C ∞ (X, R) separates points, that is, then the D-topology τ X is Hausdorff. (2) If the D-topology τ X is Hausdorff, then any plot R −→ X with finite image must be constant. Since colim(FG M • G) is Frölicher and its D-topology is Hausdorff, from the above we know that C ∞ (colim(FG M • G), R) separates points. Then the equality FG M (f ) • η = l implies that |η| is injective. For Here is an immediate application: Example 32. Recall from Ex. 24 that the pushout of in FDlg is the union of the two axes in R 2 with the sub-diffeology, which is a Frölicher space with Hausdorff D-topology, but clearly not a smooth manifold. By Thm. 31, the pushout of the above diagram does not exist in Man.
The following list of remarks permits to restrict the range of choices: Remark 33.
(i) Schwartz distribution theory is classically framed using locally convex topological vector spaces (LCTVS), so it is natural to search for a category which contains the category LCS of LCTVS and continuous linear maps as a subcategory. (ii) The space A 0 (Ω) is an affine space and is usually identified with its underlying vector space (see e.g. [21]). However, it seems that the necessity of this identification is only due to the choice of a category like LCS, which is not closed with respect to arbitrary subspaces. It would be better to choose a complete category. (iii) U (Ω) and A 0 (Ω) can be viewed as manifolds modelled in convenient vector spaces (CVS, [11,26]). However, the category of this type of manifolds is not Cartesian closed ( [26]), whereas Cartesian closedness is a basic choice preferred by several mathematicians working with infinite dimensional spaces (see e.g. [16] and references therein). (iv) The candidate category shall contain the category Con ∞ of convenient vector spaces and generic smooth maps ( [11,26]) as a (full) subcategory because the differential calculus of these spaces is used in the study of Colombeau algebras ( [21]). Note that we have embeddings Con ⊆ LCS ⊆ Con ∞ , where Con is the category of CVS and continuous linear maps. (v) The candidate category must be closed with respect to arbitrary quotient spaces, so as to contain the quotient algebras G s (Ω) and G e (Ω).
Of course, a better choice would be to consider a cocomplete category. Since, generally speaking, CVS are not closed with respect to quotient spaces (see [26, page 22]), the candidate category cannot be Con ∞ . (vi) The candidate category must also contain nonlinear maps like the product of GF, e.g.
or, more generally, any nonlinear smooth operation f ∈ C ∞ (R n , R) which can be extended to an operation of our algebras of GF, e.g.
Another feature of the candidate category we are looking for is to contain as arrows the maps between infinite dimensional spaces like convolutions, derivatives and integrals of GF.
In the literature, there are only two categories satisfying all these requirements: the category Dlg of diffeological spaces, and the category Fr of Frölicher spaces. In the present work, we will also introduce the category FDlg of functionally generated spaces as another framework for GF, trying to take the best ideas and properties of both Dlg and Fr. We will see that all the spaces D K (Ω), D(Ω), D ′ (Ω), A q (Ω), U (Ω), G s (Ω) and G e (Ω) are objects of these categories, and in this paper, we in particular study them as functionally generated spaces.

Topologies for spaces of generalized functions
[26, page 2] declared that "locally convex topology is not appropriate for non-linear questions in infinite dimensions". Indirectly, this is also confirmed by the fact that topology plays a less important role in categories like Dlg or Fr. The main aim of this section is to highlight some relationship between Cartesian closedness and locally convex topology.

Locally convex vector spaces and Cartesian closed categories.
The problems that arise in relating locally convex topology and Cartesian closedness can be expressed as follows: Theorem 34. Let F ∈ LCS, and let (T , U ) be a Cartesian closed concrete category over Top, with exponential objects given by the hom-functor T (−, −) and the forgetful functor U : T −→ Top acting as identity on arrows. Assume that R,F ∈ T , U (R) = R and U (F ) = F . Set F ′ := LCS(F, R) for the continuous dual of F and assume that and the topology of the space U (F × T (F , R)) is coarser than the product topology of U (F ) × U (T (F , R)). Finally assume that for all g ∈ F ′ the map (λ ∈ R → λ · g ∈ F ′ ) is continuous with respect to the topology induced on F ′ by (3.1). Then the locally convex topology on the space F is normable.
Proof. The idea for the proof is only a reformulation of the corresponding result in [26, page 2]. Since T is Cartesian closed, every evaluation is an arrow of T (this is a general result in every Cartesian closed category, see e.g. [1]). Thus, U (ev XY ) = ev XY is a continuous function. In particular, evF R : U (F × T (F , R)) −→ U (R) = R is continuous. By assumption, also evF R : F × U (T (F , R)) −→ R is continuous. Therefore, also its restriction to the subspace F ′ = LCS(F, R) ⊆ U (T (F , R)) is (jointly) continuous: Hence, we can find neighborhoods U ⊆ F and V ⊆ F ′ of zero such that But then, because the map (λ ∈ R → λ · g ∈ F ′ ) is continuous, taking a generic functional g ∈ F ′ , we can always find λ ∈ R =0 such that λg ∈ V , and hence |g(u)| ≤ 1/λ for every u ∈ U . Any continuous functional is thus bounded on U , so the neighborhood U itself is bounded (see e.g. [24]). Since the topology of any locally convex vector space with a bounded neighborhood of zero is normable (see e.g. [24]), we get the conclusion.
If, in this theorem, we take F = C ∞ (R, R) or F = D(Ω) or any other non-normable LCTVS, there are two possibilities to make the space F an object in a Cartesian closed category: (a) F belongs to a Cartesian closed category T , but T is not a concrete category over Top. This is the solution used in CVS theory which are embedded in the Cartesian closed category Con ∞ . Note that LCS is not a full subcategory of Con ∞ since not every arrow of Con ∞ is continuous. A typical example of a Con ∞ -smooth but not continuous map (with respect to the given locally convex topology instead of the D-topology) is the evaluation (b) For the solution adapted by using diffeological spaces, we can take T = Dlg. But then several assumptions of Thm. 34 fail: e.g. in general τ X×Y ⊇ τ X × τ Y , but not the opposite as required; moreover, the Dtopology on D(Ω) is not normable since it is finer than the usual locally convex topology, which is also not normable. On the contrary, there is no problem regarding the continuity of the product by scalar, as stated in the following: Theorem 35. Let F be any one of the spaces C ∞ (Ω, R) or D(Ω), and let τ F be the D-topology on F . Let R)) be the smooth dual of F , and let τ F ′ s be the D-topology on F ′ s . Then, with respect to pointwise operations, both spaces (F, τ F ) and (F ′ s , τ F ′ s ) are topological vector spaces.
Proof. We proceed for the case F = C ∞ (Ω, R) since the other one is very similar. For simplicity set Y X := C ∞ (X, Y ), and It is easy to prove that the pointwise sum and pointwise product by scalars are given by (−) + (−) = γ 1 (−, s R ) • −, − and (−) · (−) = γ 2 • (p ∧ R • q 1 , q 2 ), where q 1 : R × F −→ R and q 2 : R × F −→ F are the projections. Therefore, both sum and product in F are composition or pairing of smooth functions, and hence they are smooth and continuous in the D-topology. Analogously, we can proceed with the smooth dual F ′ s by considering the properties of the operator (− ≺ −).

Spaces of compactly supported functions as functionally generated spaces
It is very easy to see that the spaces D(Ω) = {f ∈ C ∞ (Ω, R) | supp(f ) ⋐ Ω} and D K (Ω) = {f ∈ D(Ω) | supp(f ) ⋐ K} with K ⋐ Ω are functionally generated spaces. Recall that D K (Ω) is a LCTVS whose topology is induced by the family of norms (Fréchet structure) Therefore (see Def. 19 and Rem. 20), (V, D(V )) ∈ FDlg and, since the functionals are globally defined, this is also a Frölicher space, i.e., We will continue to denote our spaces by D(Ω) and D K (Ω) even when we think of them as diffeological spaces with the canonical diffeology. When we want to underscore that we are considering them only as LCTVS with the topology given by (4.2) and (4.1), we will use the notations D LC (Ω) and D LC K (Ω).

4.1.
Plots of D K (Ω), D(Ω) and Cartesian closedness. It is also interesting to reformulate the property of being a plot d ∈ U D(Ω), or d ∈ U D K (Ω), using Cartesian closedness. This permits to compare better the canonical diffeology on these spaces as LCTVS with the diffeology induced on them as subspaces of C ∞ (Ω, R). We will denote with D s (Ω) this diffeological space, so that d ∈ U D s (Ω) if and only if d : holds if and only if (i • d) ∨ ∈ C ∞ (U × Ω, R). Analogously, we define D s K (Ω). In performing this comparison, we will use Lem. 2.1, Lem. 2.2 and Thm. 2.3 of [25] which are cited here for reader's convenience. In this subsection, without confusion we use the same notation for morphisms in different categories when the functions for the underlying sets are the same.
To state the other cited results of [25], we need the following: Definition 38. Let U ∈ OR ∞ and let f : U × Ω −→ R be a map. We say that f is of uniformly bounded support (with respect to U ) if ∃K ⋐ Ω ∀u ∈ U : supp(f (u, −)) ⊆ K.
We say that f is locally of uniformly bounded support if ∀u ∈ U ∃V open neigh. of u in U : f | V ×Ω is of uniformly bounded support.
Finally we say that Using this definition, we can state Lemma 39 (2.2 of [25]). Let U ∈ OR ∞ and assume that f ∈ C ∞ (U × Ω, R) is pointwise of bounded support. Then the following are equivalent (i) f is locally of uniformly bounded support; Theorem 40 (2.3 of [25]). Let U ∈ OR ∞ . Then the following are equivalent: In other words, Thm 40 says that d ∈ U D(Ω) if and only if d ∈ U D s (Ω) and d ∨ is locally of uniformly bounded support, and hence we have D(Ω) ⊆ D s (Ω).
From these results we can also solve the same problem for the spaces D K (Ω) and D s K (Ω). The following lemma is analogous to Lem. 37 for D K (Ω). Lemma 41. If f ∈ C ∞ (U, D K (Ω)) with U ∈ OR ∞ and K ⋐ Ω, then f : U −→ D LC K (Ω) is continuous. Proof. Since the inclusion map i K : D LC K (Ω) ֒→ D LC (Ω) is continuous linear, by post-composition it also takes continuous linear functionals l : D LC (Ω) −→ R into continuous linear functionals l • i K : D LC K (Ω) −→ R. From Thm. 13 it follows that i K ∈ C ∞ (D K (Ω), D(Ω)) and hence i K • f ∈ C ∞ (U, D(Ω)). Therefore, Lem. 37 implies that i K • f is continuous and hence the conclusion since the topology on D LC K (Ω) coincides with the initial topology induced by i K .
The following lemma is analogous to Lem. 39 for D K (Ω).
Theorem 43. Let U ∈ OR ∞ and let K ⋐ Ω. Then the following are equivalent: Proof. (i) ⇒ (ii). We already proved in Lem. 41 that the inclusion map i K ∈ C ∞ (D K (Ω), D(Ω)), so i K • f ∈ C ∞ (U, D(Ω)). By Thm. 40 we have . The second part of the conclusion follows from the codomain D K (Ω) of f in (i). (ii) ⇒ (i). Assumption (ii) implies that f is locally of uniformly bounded support. From Thm. 40 we thus obtain that f ∈ C ∞ (U, D(Ω)). But our assumption implies that f (U ) ⊆ |D K (Ω)|. So the conclusion follows from the following Lem. 44.
Proof. We have to prove that figures of both spaces are equal. (|D K (Ω)| ≺ D(Ω)) ⊇ D K (Ω): This follows directly from the fact that the inclusion map i K ∈ C ∞ (D K (Ω), D(Ω)).
. We need to prove that l • d ∈ C ∞ (U, R) for all continuous linear maps l : D LC K (Ω) −→ R. So the problem is to extend any such given l to some λ ∈ D ′ (Ω). To this end, we can repeat the usual proof of the local form of distributions as derivatives of continuous functions to obtain the following: Theorem 45. For any continuous linear map l : D LC K (Ω) −→ R there exist g ∈ C 0 (Ω, R) and α ∈ N n such that l(ϕ) = ∂ α g, ϕ ∀ϕ ∈ D K (Ω).
The conclusion follows by applying this theorem.
The locally convex topology and the D-topology on D K (Ω) and D(Ω). In this section we present some results about functionals on the spaces D K (Ω) and D(Ω) which are continuous with respect to the locally convex topology and the D-topology. The first result follows at once from Lem. 37 and Lem. 41: Corollary 47. On the spaces D K (Ω) and D(Ω), the D-topology is finer than the locally convex topology.
It remains an open problem whether the D-topology is strictly finer than the locally convex topology or not. We first study the behaviour of maps of the form λ : D(Ω) −→ D(Ω ′ ), where henceforth we always assume that The same results hold for D K (Ω).
The following lemma is a trivial consequence of Thm. 13, but we prefer to state it for completeness. In the following results, we show that if a linear map |D(Ω)| −→ R is continuous with respect to the D-topology, then it is a distribution: The schema to prove this theorem is the following: we need to prove that l•d ∈ C ∞ (U, R) whenever d ∈ U D(Ω), i.e., by Thm. 40, if d ∨ ∈ C ∞ (U ×Ω, R) and d ∨ is locally of uniformly bounded support. We are going to prove that: Proof. Note that for any v ∈ V , the evaluation maps We now prove Thm. 50: Proof. To prove the existence of the limit in (i), we first fix d ∈ U D(Ω), u ∈ U , e i = (0, i−1 . . . . . . , 0, 1, 0, . . . , 0) ∈ R n ⊇ U and r ∈ R >0 such that B r (u) ⊆ U . Then there exist an open neighbourhood V of u in U and a ∈ R >0 such that v + he i ∈ B r (u) for all v ∈ V and h ∈ (−a, a). Set  H := (−a, a), and for any h ∈ H, define Clearly Also note that for any non-zero h ∈ H and for any (v, x) ∈ V × Ω, by the fundamental theorem of calculus, we have We prove below that lim h→0 δ(h) = ∂d ∨ ∂e i | V ×Ω in the space R V ×(Ω) which has the underlying set and figures defined by p ∈ W R V ×(Ω) iff p ∨ : W × V × Ω −→ R is smooth and locally of uniformly bounded support with respect to W × V .
Since d ∨ is locally of uniformly bounded support (Thm. 40), we may assume that V and H are sufficiently small so that δ ∨ : H × V × Ω −→ R is of uniformly bounded support with respect to H × V . Thus (4.6) To prove the above mentioned limit equality, let A be a D-open subset of . Now, we apply this limit to the adjoint map where the domain is the diffeological space R V ×(Ω) , and the codomain is the space D(Ω) ↑ V with D(Ω) V = |C ∞ (V, D(Ω))| the underlying set and figures defined by q ∈W D(Ω) ↑ V iff (q ∨ ) ∨ :W × V × Ω −→ R is smooth and locally of uniformly bounded support. We claim that the adjoint map (4.7) is also smooth with respect to these diffeological structures on its domain and codomain. In fact, if p ∈ W R V ×(Ω) , then ((−) ∧ • p) ∨ ∨ = p ∨ which is locally of uniformly bounded support by the definition of the diffeology on R V ×(Ω) . Therefore (−) ∧ : R V ×(Ω) −→ D(Ω) ↑ V is smooth and hence it is also D-continuous: Now consider the evaluation at v ∈ V ⊆ U : We claim that ev v : D(Ω) ↑ V −→ D(Ω) is smooth. In fact, for any q ∈W D(Ω) ↑ V , i.e., q ∨ ∨ :W ×V ×Ω −→ R is smooth and locally of uniformly bounded support. (4.8) We need to prove that (ev v • q) ∨ :W × Ω −→ R is also smooth and locally of uniformly bounded support. Take w ∈W , and from (4.8) we have open neighbourhoods C of w and D of v so that (q ∨ ) ∨ | C×D×Ω is of uniformly bounded support. We may assume that supp ( By Cartesian closedness, ev v • q is smooth and hence it is a figure of D(Ω). This proves that ev v : D(Ω) ↑ V −→ D(Ω) is smooth and hence it is also D-continuous. So we have: Therefore, this limit exists in D(Ω). By assumption, l : |D(Ω)| −→ R is D-continuous and linear, so This proves that the first partial derivatives of l • d exist and are continuous because both l and ∂d ∨ ∂e i are D-continuous. We can now apply the same procedure to the figure obtaining that also the second partial derivatives of l • d exist and are continuous. By applying inductively this process, we get the conclusion l • d ∈ C ∞ (U, R). Finally, from Thm. 48 we have l ∈ D ′ (Ω).
We also have the following By applying this result to a curve d ∈ R D(Ω), and knowing that the D-topology is finer than the usual locally convex topology, we get an independent proof that D(Ω) is a CVS.
We close this section with the following result, which underscores the difference between D(Ω) and its counterpart D s (Ω); in its statement, if F ∈ Dlg is also a vector space, then we set F ′ s := ({l ∈ C ∞ (F, R) | l is linear} ≺ C ∞ (F, R)) for its smooth dual space (this notation has been used for the special cases in Thm. 35).

Spaces for Colombeau generalized functions as diffeological spaces
It is natural to view all the spaces used to define CGF as diffeological spaces. We will start with C ∞ (Ω) I , E s M (Ω), A q (Ω), U (Ω), E e (Ω) and E e M (Ω), with the aim to prove that also the quotient spaces G s (Ω), G e (Ω) are smooth differential algebras.
The space C ∞ (Ω) I . Elements (u ε ) of C ∞ (Ω) I are arbitrary nets, indexed in ε ∈ I, of smooth functions on Ω. There are studies of Colombeau-like algebras with smooth or continuous ε-dependence (see [6,19] and references therein). In [18] it has been proved that a very large class of equations have no solution if we request continuous dependence with respect to ε ∈ I. For this reason, it is natural to think I as a space with the discrete diffeology (see (ii) of Rem. 5), i.e., where only locally constant maps d : U −→ I are figures d ∈ U I. With this structure, the space I is functionally generated by Set(I, R). If we think of C ∞ (Ω) as the space C ∞ (Ω, R) ∈ Dlg, then by Cartesian closedness (Thm. 26) we have u ∈ C ∞ (Ω) I , i.e., u ∈ C ∞ (I, C ∞ (Ω, R)), iff u ∈ Set(I, C ∞ (Ω)). The space C ∞ (Ω) I with this diffeological structure will be denoted by C ∞ (Ω, R) I . The space A q (Ω). The set A 0 (Ω) = ϕ ∈ |D(Ω)| |´ϕ = 1 has a natural diffeology, the sub-diffeology of D(Ω). So Analogously, the set A q (Ω) = ϕ ∈ A 0 (Ω) | ∀α ∈ N n : 1 ≤ |α| ≤ q ⇒ˆx α ϕ(x) dx = 0 has a natural diffeology, the sub-diffeology of A 0 (Ω) So where we used the property (S ≺ (T ≺ X)) = (S ≺ X) if S ⊆ T ⊆ |X| and X ∈ Dlg. Therefore, figures d ∈ U A q (Ω) are maps d : U −→ A q (Ω) such that d ∨ ∈ C ∞ (U × Ω, R) and d ∨ is locally of uniformly bounded support (Thm. 40).
The space U (Ω). In Def. 2 of the full Colombeau algebra, the set U (Ω) serves as domain of the representatives R : U (Ω) −→ R of CGF in G e (Ω). These representatives are requested to be smooth in the Ω slot (note that U (Ω) ⊆ A 0 × Ω) but with no particular regularity in the A 0 slot (which serves as an index set for the full Colombeau algebra, analogous to the interval I as an index set for the special one). This means that we shall consider the discrete diffeology on A 0 and the standard diffeology on Ω. If we identify the set A 0 with the corresponding diffeological space with the discrete diffeology, then The space E e (Ω). The space E e (Ω) (see Def. 2) inherits its diffeological structure from C ∞ (U (Ω), R) ∈ Dlg: E e (Ω) := (E e (Ω) ≺ C ∞ (U (Ω), R)) .
. We give an equivalent characterization of E e (Ω) as follows: For ϕ ∈ A 0 , set As a convention, when Ω ϕ = ∅, we think of C ∞ (Ω ϕ , R) as a set with a single element. Since R ∈ E e (Ω) iff R ∧ : A 0 −→ ϕ∈A 0 C ∞ (Ω ϕ , R) and R(ϕ, −) ∈ C ∞ (Ω ϕ , R) for all ϕ ∈ A 0 , R ∧ ∈ ϕ∈A 0 C ∞ (Ω ϕ , R). By Cartesian closedness of Dlg: Therefore, up to smooth isomorphism, figures of E e (Ω) can be described as The space E e M (Ω). The natural diffeology on the space of moderate functions E e M (Ω) is the sub-diffeology of E e (Ω). Hence, The special and full Colombeau algebras. Since the category Dlg of diffeological spaces is cocomplete, both quotient algebras G s (Ω) and G e (Ω) can be viewed as objects of Dlg: is the projection on the quotient. Analogously, we can describe figures of the full Colombeau algebra.
We can now state the following natural result: Proof. We prove that the maps are smooth for the case G(Ω) = G s (Ω), since the proof is similar for the case G(Ω) = G e (Ω).
Concerning the smoothness of the sum map, let d ∈ U G s (Ω) × G s (Ω), i.e., p i • d ∈ U G s (Ω), where p i : G s (Ω) × G s (Ω) −→ G s (Ω), i = 1, 2, are the projections. Hence, by the definition of the quotient diffeology on G s (Ω), for any u ∈ U we can write (p i • d) . Thus, we can write the composition Since (δ 1 + δ 2 ) | V 1 ∩V 2 ∈ V 1 ∩V 2 E s M (Ω), the conclusion follows from the definition of the quotient diffeology.
Proof. We only proceed for S, since the other two cases are similar. If ε ∈ I and p ∈ U D(R n ), then p ∨ ∈ C ∞ (U × R n , R) and p ∨ is locally of uniformly bounded support with respect to U (Thm. 40). But [S(ε, −) • p] ∨ (u, x) = 1 ε n p ∨ u, x ε for all (u, x) ∈ U × Ω, and this shows that [S(ε, −) • p] ∨ ∈ C ∞ (U × R n , R) and it is locally of uniformly bounded support with respect to U .
5.1. Colombeau ring of generalized numbers and evaluation of generalized functions. In this subsection we consider only the case of the special Colombeau algebra G s (Ω) since it is mostly studied in the literature. The case of the full algebra can be treated analogously.

Conclusions and open problems
We explore why the categories Fr of Frölicher spaces, Dlg of diffeological spaces and FDlg of functionally generated (diffeological) spaces work as good frameworks both for the classical spaces of functional analysis and for the Colombeau algebras. On the one hand, there seem to be few differences between FDlg and Fr: we can say that the former seems better than the latter because in FDlg we don't have the problem of extending to the whole space locally defined functionals; but in the latter, it is easier to work directly with globally defined functionals when the D-topology of the space is unknown. On the other hand, the usual counter-examples about locally Cartesian closedness of Fr do not seem to work in FDlg. Moreover, if compared to Dlg, functionally generated spaces seem to be closer to spaces used in functional analysis, where testing smoothness using functionals is customary. On the other hand, Thm. 55 and Thm. 58, show that Dlg can be considered a promising categorical framework for Colombeau algebras. Some open problems underscored by the present work are the following: • A clear and useful example of functionally generated space which is not Frölicher and where locally defined functionals cannot be extended to the whole space is missing. • The problem to show that Dlg gives also a sufficiently simple infinite dimensional calculus for the diffeomorphism invariant Colombeau algebra (see [21]) remains open. In particular, we note that the differentiable uniform boundedness principle ([21, Thm. 2.2.7]) is used in [21] only to prove the analogy of Lem. 56, whereas the other results of [21, Section 2.2.1] seem repeatable in Dlg without the need to know the calculus on convenient vector spaces. • The relationship between the locally convex topology and the Dtopology on D(Ω) is only partially solved (see Cor. 47). • The relationship between the space of Schwartz distributions D ′ (Ω) and the smooth dual D(Ω) ′ s is only partially solved (see Cor. 54). • The preservation of colimits from the category of smooth manifolds to FDlg is only partially solved.