# The category of Colombeau algebras

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## Abstract

Since the beginning of Colombeau’s the theory of algebras of generalized functions, the role of its characteristic polynomial growth versus a more general condition has been explored. Recently, we introduced the notion of asymptotic gauge (AG), and we used it to study Colombeau AG-algebras. This construction concurrently generalizes many different algebras used in Colombeau’s theory and, at the same time, allows for more general growth scales. In this paper, we study the categorical properties of Colombeau AG-algebras with respect to the choice of the AG. The main aim of the paper is to study suitable functors to relate differential equations framed in algebras having different growth scales.

### Keywords

Colombeau-type algebras Functorial properties Asymptotic gauges Embeddings of distributions Sets of indices### Mathematics Subject Classification

Principal 46F30 Secondary 18B99## 1 Introduction

- (a)
\(\Omega \in (\mathcal {T}\mathbb {R}^{n})^{\text {op}}\mapsto \mathcal {G}(\Omega )\) is a presheaf of commutative and associative differential algebras with respect to suitable derivations \(\partial _{\Omega }^{\alpha }:\mathcal {G}(\Omega )\longrightarrow \mathcal {G}(\Omega )\) for all \(\alpha \in \mathbb {N}^{n}\).

- (b)
The set of generalized scalars \({\widetilde{\mathbb {R}}}=\{f\in \mathcal {G}(\mathbb {R})\mid \partial _{\mathbb {R}}^{\alpha }f=0\ \forall \alpha \}\) is a non-Archimedean ring (with zero divisors).

- (c)
The embedding \(\iota _{\Omega }:\mathcal {D}'\longrightarrow \mathcal {G}\) is a natural transformation between sheaves of real vector spaces.

- (d)
The derivations are compatible, i.e. if \(D_{\Omega }^{\alpha }:\mathcal {D}'(\Omega )\longrightarrow \mathcal {D}'(\Omega )\) is the usual \(\alpha \)-derivation of distributions, then \(\partial _{\Omega }^{\alpha }(\iota _{\Omega }(T))=\iota _{\Omega }(D_{\Omega }^{\alpha }(T))\) for all \(T\in \mathcal {D}'(\Omega )\).

- (e)
Let \(S_{\Omega }:{\mathcal {C}}^{\infty }(\Omega )\longrightarrow \mathcal {D}'(\Omega )\), \(\langle S_{\Omega }(f),\varphi \rangle :=\int _{\Omega }f\varphi \), be the usual Schwartz embedding of smooth functions. Then \(\iota _{\Omega }[S_{\Omega }({\mathcal {C}}^{\infty }(\Omega )]\) is a differential subring of \(\mathcal {G}(\Omega )\) and is naturally isomorphic to \({\mathcal {C}}^{\infty }(\Omega )\). This natural identification \({\mathcal {C}}^{\infty }(\Omega )=\iota _{\Omega }[S_{\Omega }({\mathcal {C}}^{\infty }(\Omega )]\subseteq \mathcal {G}(\Omega )\) can also be understood as a suitable universal property.

The relevance of properties (a)–(e) can also be highlighted by mentioning that they are included in the axiomatic approach of [21, 22].

*special*one, which is the quotient algebra \(\mathcal {G}^s(\Omega ):={\mathcal E}_M(\Omega )/\mathcal {N}^s(\Omega )\), where

^{1}

In particular, if one considers the usual sheaf of smooth functions \({\mathcal {C}}^{\infty }(-)\), Colombeau AG-algebras are among the simplest and most general approaches (only [14] is actually more general). In fact, Colombeau AG-algebras include in the same abstract framework all known Colombeau-like algebras, like the special one \(\mathcal {G}^{\mathrm{s}}\), the full one \({\mathcal G}^{\mathrm{e}}\), [5], the NSA based algebra of asymptotic functions \(\hat{\mathcal {G}}\), [23], the diffeomorphism invariant algebras \({\mathcal G}^{\mathrm{d}}\), \(\mathcal {G}^{2}\) and \(\hat{\mathcal {G}}\), [5], the Egorov algebra, [10, 11, 25], and the algebra of non-standard smooth functions \(^{*}{\mathcal {C}}^{\infty }(\Omega )\), [4, 18]; see [12, 13] and below. The simplicity of the approach with AGs lies, for all the algebras mentioned above, in the use of the simple logical structure of quantifiers that characterizes the special algebra \(\mathcal {G}^{\mathrm{s}}\). In order to establish a conceptual knowledge of this multiplicity of differential algebras, the abstract framework of Colombeau AG-algebras is therefore a step towards the understanding of the core properties of Colombeau’s construction.

*not*one of the aim of the present paper. For the solution and uniqueness of all \({\widetilde{\mathbb {R}}}\)-linear ODE in a minimal Colombeau AG-algebra, see [12].

Is the construction of the Colombeau algebra \(\mathcal {G}(\Omega ,\mathcal {B})\) functorial with respect to the AG \(\mathcal {B}\)? Is this construction functorial with respect to the open set \(\Omega \)?

When can we consider two AGs as isomorphic? For instance, we will show that the AG of polynomial growth \(\mathcal {B}_{\text {pol}}\) is isomorphic to the AG of exponential growth \(\mathcal {B}_{\exp }\). This isomorphism holds in spite of the fact that using the latter we can solve ODE which are not solvable with the former, see Sect. 7.

How to relate the solutions of differential equations framed in a given Colombeau AG-algebra to those framed into another one?

Colombeau theory can be more clearly summarized by saying that it permits to define a differential algebra together with an embedding of Schwartz’s distributions. This embedding can be intrinsic, or diffeomorphism invariant, or it can be chosen in order to have properties like \(H(0)=\frac{1}{2}\), where

*H*is the Heaviside’s step function. Can we define a general*category of Colombeau algebras*having as objects triples \((G,\partial ,\iota )\) made of an algebra*G*, a family of derivations \(\partial \) and an embedding of distributions \(\iota \)? Can we see \(\mathcal {G}^{\mathrm{s}}\) as a suitable functor with values in this category? What is the domain of this functor?Similarly to the axiomatic approach of [21, 22], in defining this category of Colombeau algebras, we need to focus on peculiar properties of Colombeau-like algebras. This approach represents another way to establish a certain order of importance in the properties satisfied by all algebras of Colombeau type.

## 2 Sets of indices

### 2.1 Basic definitions

In [13], the general notion of sets of indices has been introduced. For reader’s convenience, in this section we recall the notations and notions from [13] that we will use in the present work. For all the proofs, we refer to [13].

### Definition 1

*set of indices*if the following conditions hold:

- (i)
\((I,\le )\) is a pre-ordered set, i.e.,

*I*is a non empty set with a reflexive and transitive relation \(\le \) ; - (ii)
\(\mathcal {I}\) is a set of subsets of

*I*such that \(\emptyset \notin \mathcal {I}\) and \(I\in \mathcal {I}\); - (iii)
\(\forall A,B\in \mathcal {I}\,\exists C\in \mathcal {I}:\ C\subseteq A\cap B\).

- (iv)
If \(e\le a\in A\in \mathcal {I}\) , the set \(A_{\le e}:=(\emptyset ,e]\cap A\) is downward directed by < , i.e., it is non empty and \(\forall b,c\in A_{\le e}\,\exists d\in A_{\le e}:\ d<b,\ d<c\).

The following are examples of sets of indices.

### Example 2

- (i)
Let \(I^{\mathrm{s}}:=(0,1]\subseteq \mathbb {R}\), let \(\le \) be the usual order relation on \(\mathbb {R}\), and let \(\mathcal {I}^{\mathrm{s}}:=\{ (0,\varepsilon _{0}]\mid \varepsilon _{0}\in I \} \). Following [13], we denote by \(\mathbb {I}^{\mathrm{s}}:=(I^{\mathrm{s}},\le ,\mathcal {I}^{\mathrm{s}})\) this set of indices.

- (ii)
If \(\varphi \in \mathcal {D}(\mathbb {R}^{n}),\,r\in \mathbb {R}_{>0}\) and \(x\in \mathbb {R}^{n}\), we use the symbol \(r\odot \varphi \) to denote the function \(x\in \mathbb {R}^{n}\mapsto \frac{1}{r^{n}}\cdot \varphi (\frac{x}{r})\in \mathbb {R}\), see [13]. With the usual notations of [5], we define \(I^{\mathrm{e}}:=\mathcal {A}_{0},\mathcal {\,I}^{\mathrm{e}}:=\{ \mathcal {A}_{q}\mid q\in \mathbb {N}\} \), and for \(\varepsilon ,\,e\in I^{\mathrm{e}}\), we set \(\varepsilon \le e\) iff there exists \(r\in \mathbb {R}_{>0}\) such that \(r\le 1\) and \(\varepsilon =r\odot e\). Then \(\mathbb {I}^{\mathrm{e}}:=(I^{\mathrm{e}},\le ,\mathcal {I}^{\mathrm{e}})\) is a set of indices used in this framework to unify and simplify the full algebra \({\mathcal G}^{\mathrm{e}}\) (see [13, Sect. 3]).

- (iii)
Let \(I^{\text {E}}:=\mathbb {N}\) with the inverse \(\lesssim \) of the usual order notion on \(\mathbb {N}\) (namely, \(m\lesssim n\) iff \(m\ge n\)), and let \(\mathcal {I}^{\text {E}}\) be the Fréchet filter of cofinite sets. The set of indices \(\mathbb {I}^{\text {E}}:=(I^{\text {E}},\lesssim ,\mathcal {I}^{\text {E}})\) is used by [10, 11] to introduce the so-called Egorov algebra. The analogue in NSA is \(\mathbb {I}^{*}:=(\mathbb {N},\lesssim ,\mathcal {I}^{*})\), where \(\mathcal {I}^{*}\) is a free ultrafilter that contains \(\mathcal {I}^{\text {E}}\). The set of indices \(\mathbb {I}^{*}\) is used to define the algebra of non-standard smooth functions \(^{*}{\mathcal {C}}^{\infty }(\Omega )\), see [18].

- (iv)For every \(\varphi \in \mathcal {A}_{0}\), let us call
*order of*\(\varphi \) the natural numberand, for every \(\varphi \), \(\psi \in \mathcal {A}_{0}\), set$$\begin{aligned} o(\varphi ):=\min {\{n\in \mathbb {N}\mid \varphi \in \mathcal {A}_{n}{\setminus }\mathcal {A}_{n+1}\}} \end{aligned}$$We have that \(\widetilde{\mathbb {I}}^{\mathrm{e}}=({\mathcal {A}}_{0},\lesssim ,\{\mathcal {A}_{q}\mid q\in \mathbb {N}\})\) is a downward directed set of indices that can be used to try a simplification of the full algebra \({\mathcal G}^{\mathrm{e}}\). See Sect. 3.1 for the nicer properties that downward directed sets have with respect to the notions we are going to introduce.$$\begin{aligned} \varphi \lesssim \psi \quad \text{ iff } o(\varphi )<o(\psi )\quad \text{ or } \varphi \le \psi \,\text{ in }\,\mathbb {I}^{\mathrm{e}}. \end{aligned}$$

Henceforward, functions of the type \(f:I\longrightarrow \mathbb {R}\) are called *nets*, and for their evaluation we both use the notations \(f_{\varepsilon }\) or \(f(\varepsilon )\), the latter in case the subscript notation were too cumbersome. When the domain *I* is clear, we use also the notation \(f=(f_{\varepsilon })\) for the whole net. Analogous notations will be used for nets of smooth functions \(u=(u_{\varepsilon })\in \mathcal {C}^{\infty }(\Omega )^{I}\).

In each set of indices, we can define two notions of big-O for nets of real numbers. These two big-Os share the same (usual) properties of the classical one as preorders and concerning algebraic operations (see [13, Thm. 2.8, Thm. 2.14]). Since each set of the form \(A_{\le e}=(\emptyset ,e]\cap A\) is downward directed, the first big-O is the usual one:

### Definition 3

*I*. We write

The second notion of big-O is the following:

### Definition 4

For example, in the case of the set of indices \(\mathbb {I}^{\mathrm{e}}\) used for the full algebra, we have \(x_{\varepsilon }=O(y_{\varepsilon })\) as \(\varepsilon \in \mathbb {I}^{\mathrm{e}}\) if and only if \(\exists q\in \mathbb {N}\,\forall \varphi \in \mathcal {A}_{q}:\ x(\varepsilon \odot \varphi )=O[y(\varepsilon \odot \varphi )]\) as \(\varepsilon \rightarrow 0^{+}\), where the latter big-O is the classical one, see [13, Thm. 3.2]. We can hence recognise an important part of the usual definition of moderate and negligible nets for the full algebra \({\mathcal G}^{\mathrm{e}}\). The abstract approach we use in this paper can be easily understood by interpreting \(\mathbb {I}\) in the simplest case \(\mathbb {I}^{\mathrm{s}}\) of the special algebra and in the case \(\mathbb {I}^{\mathrm{e}}\) of the full algebra. In the former, any formula of the form \(\exists A\in \mathcal {I}\,\forall a\in A\) becomes \(\exists \varepsilon _{0}\in (0,1]\,\forall \varepsilon \in (0,\varepsilon _{0}]\). In the latter it becomes \(\exists q\in \mathbb {N}\,\forall \varphi \in \mathcal {A}_{q}\).

In every set of indices we can formalize the notion of *for*\(\varepsilon \)*sufficiently small* as follows.

### Definition 5

*for*\(\varepsilon \)

*sufficiently small in*\(A_{\le a}\)

*the property*\(\mathcal {P}(\varepsilon )\)

*holds*, if

*for*\(\varepsilon \)

*sufficiently small in*\(\mathbb {I}\)

*the property*\(\mathcal {P}(\varepsilon )\)

*holds, if*\(\exists A\in \mathcal {I}\,\forall a\in A\,\forall ^{\mathbb {I}}\varepsilon \in A_{\le a}:\ \mathcal {P}(\varepsilon )\).

Using this notion, we can define an order relation for nets.

### Definition 6

*i*, \(j:I\longrightarrow \mathbb {R}\) be nets. Then we say \(i>_{\mathbb {I}}j\) if

Finally, we recall the notion of limit of a net of real numbers:

### Definition 7

*l*

*is the limit of*

*f*

*in*\(\mathbb {I}\) if

Let us observe that if \(l=\lim _{\varepsilon \le a}f|_{A}(\varepsilon )\) and \(B\subseteq A\), \(B\in \mathcal {I}\), then \(l=\lim _{\varepsilon \le a}f|_{B}(\varepsilon )\); moreover, there exists at most one *l* verifying (2.5).

## 3 The category \(\mathbf Ind \)

We start by defining the notion of morphism between two sets of indices. This is also a natural step to define the concept of morphism of asymptotic gauges. A natural property to expect from a morphism \(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) between sets of indices \(\mathbb {I}_{1}\), \(\mathbb {I}_{2}\) is the preservation of the notion of “eventually” for properties \(\mathcal {P}\), i.e. that \(\forall ^{\mathbb {I}_{1}}\varepsilon _{1}\,\mathcal {P}(\varepsilon _{1})\) implies \(\forall ^{\mathbb {I}_{2}}\varepsilon _{2}\,\mathcal {P}(f(\varepsilon _{2}))\). Let us note that we start from a property \(\mathcal {P}(\varepsilon _{1})\), for \(\varepsilon _{1}\in I_{1}\), and we want to arrive at a property \(\mathcal {P}(f(\varepsilon _{2}))\), for \(\varepsilon _{2}\in I_{2}\).

### Definition 8

*infinitesimal*if

- (i)
\(f:I_{2}\longrightarrow I_{1}\);

- (ii)
\(\forall \alpha \in A_{\le a}\,\forall ^{\mathbb {I}_{2}}\varepsilon _{2}\in B_{\le b}:\ f(\varepsilon _{2})\in A_{\le \alpha }\).

*morphism*of sets of indices if

Therefore, a morphism \(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) is a map in the opposite direction \(f:I_{2}\longrightarrow I_{1}\) between the underlying sets. Only in this way we have that the map *f* preserves the asymptotic relations that hold in \(\mathbb {I}_{1}\), see Corollary 13 for a list of examples.

### Example 9

- (i)
For every set of indices \(\mathbb {I}=(I,\le ,\mathcal {I})\) if \(1_{\mathbb {I}}:I\longrightarrow I\) is the identity function then \(1_{\mathbb {I}}:\mathbb {I\longrightarrow \mathbb {I}}\) is a morphism.

- (ii)
Let \(f:(0,1]\longrightarrow (0,1]\) be a map, then \(f:\mathbb {I}^{\mathrm{s}}\longrightarrow \mathbb {I}^{\mathrm{s}}\) is a morphism if and only if \(\forall \varepsilon \in (0,1]\,\exists \delta \in (0,1]:\ f((0,\delta ])\subseteq (0,\varepsilon ]\), i.e. if and only if \(\lim _{\varepsilon \rightarrow 0^{+}}f(\varepsilon )=0\).

- (iii)
For the set of indices \(\mathbb {I}^{\mathrm{e}}\) of the full algebra, we recall that \((\mathcal {A}_{q})_{\le \varphi }=(\emptyset ,\varphi ]\) and \(\underline{\varphi }:=\min \{ \text {diam}(\text {supp}\varphi ),1\} \). If \(f:I^{\mathrm{e}}\longrightarrow I^{\mathrm{e}}\) is a map, then we have that \(f:(\emptyset ,\varphi ]\longrightarrow (\emptyset ,\psi ]\) is infinitesimal if and only if \(\forall \varepsilon \in (0,1]\,\exists \delta \in (0,1]:\ f(\{ r\odot \psi \mid r\in (0,\delta ]\})\subseteq \{ r\odot \varphi \mid r\in (0,\varepsilon ]\} \). Therefore, this implies that \(\lim _{\varepsilon \rightarrow 0^{+}}\underline{f(\varepsilon \odot \psi )}=0\). If we denote by \(\frac{f(\varepsilon \odot \psi )}{\varphi }\) the unique \(r\in (0,1]\) such that \(f(\varepsilon \odot \psi )=r\odot \varphi \) (in case it exists), then \(f:(\emptyset ,\varphi ]\longrightarrow (\emptyset ,\psi ]\) is infinitesimal if and only if \(\lim _{\varepsilon \rightarrow 0^{+}}\frac{f(\varepsilon \odot \psi )}{\varphi }=0\). Moreover, \(f:\mathbb {I}^{\mathrm{e}}\longrightarrow \mathbb {I}^{\mathrm{e}}\) is a morphism if and only if \(\forall m\in \mathbb {N}\,\forall \varphi \in \mathcal {A}_{m}\,\exists q\in \mathbb {N}\,\forall \psi \in \mathcal {A}_{q}:\ \lim _{\varepsilon \rightarrow 0^{+}}\frac{f(\varepsilon \odot \psi )}{\varphi }=0\). This and the previous example justify our use of the name

*infinitesimal*in Definition 8. - (iv)
Let \(\varphi \in \mathcal {A}_{0}\) be fixed, let \(\mathbb {I}_{\varphi }:=((\emptyset ,\varphi ],\le ,\{(\emptyset ,\varphi ]\})\), where the order relation on \(\mathbb {I_{\varphi }}\) is the restriction of the order relation on \(\mathbb {I}^{\mathrm{e}}\). If \(f:(0,1]\longrightarrow (\emptyset ,\varphi ]\) is the function \(f(r):=r\odot \varphi \) for every \(r\in (0,1]\) then we have that \(f:\mathbb {I}_{\varphi }\longrightarrow \mathbb {I^{\mathrm{s}}}\) is a morphism. Conversely, if \(g:(\emptyset ,\varphi ]\longrightarrow (0,1]\) maps every \(\psi \in (\emptyset ,\varphi ]\) to the unique \(r\in (0,1]\) such that \(\psi =r\odot \varphi \), i.e. \(g(\psi )=\frac{\psi }{\varphi }\), then \(g:\mathbb {I}^{\mathrm{s}}\longrightarrow \mathbb {I}_{\varphi }\) is a morphism. We have that \(f=g^{-1}\).

- (v)
Let \(f:\mathbb {N}\longrightarrow \mathbb {N}\) be a map, then \(f:\mathbb {I}^{\text {E}}\longrightarrow \mathbb {I}^{\text {E}}\) is a morphism if and only if \(\lim _{n\rightarrow +\infty }f(n)=+\infty \). Analogously, \(f:\mathbb {I}^{*}\longrightarrow \mathbb {I}^{*}\) is a morphism if and only if there exists an ultrafilter set \(B\in \mathcal {I}^{*}\) such that \(\lim _{\begin{array}{c} n\rightarrow +\infty \\ n\in B \end{array} }f(n)=+\infty \).

- (vi)
Let us denote by \(\overline{\mathbb {N}}\) the set of indices \((\mathbb {N}_{>0},\lesssim ,\mathcal {I}_{n})\) where \(\lesssim \) is the inverse of the usual order notion on \(\mathbb {N}\) (namely, \(m\lesssim n\) iff \(m\ge n\)) and, for every natural number

*n*, \(\mathcal {I}_{n}:=\{m\in \mathbb {N}\mid m\lesssim n\}.\) If \(f:\mathbb {N}_{>0}\rightarrow (0,1]\) is the function that maps \(n>0\) to \(\frac{1}{n}\), we have that \(f:\mathbb {I}^{\mathrm{s}}\longrightarrow \overline{\mathbb {N}}\) is a morphism. Conversely, if \(g:(0,1]\longrightarrow \mathbb {N}\) is the function that maps \(\varepsilon \) to the floor \(\lfloor \frac{1}{\varepsilon } \rfloor \) then \(g:\overline{\mathbb {N}}\longrightarrow \mathbb {I}^{\mathrm{s}}\) is a morphism. We have \(g\circ f=1_{\mathbb {N}_{>0}}\), but there does not exist any isomorphism between these two sets of indices \(\overline{\mathbb {N}}\) and \(\mathbb {I}^{\mathrm{s}}\) because they have different cardinalities. - (vii)
For every \(n\in \mathbb {N}\) let us fix \(\varphi _{n}\in \mathcal {A}_{n}\setminus \mathcal {A}_{n+1}\). Let \(f:\mathbb {N}\longrightarrow \mathcal {A}_{0}\) be the function that maps

*n*to \(\varphi _{n}\). Then we have that \(f:\widetilde{\mathbb {I}}^{\mathrm{e}}\longrightarrow \overline{\mathbb {N}}\) is a morphism. Conversely, if \(o:\mathcal {A}_{0}\longrightarrow \overline{\mathbb {N}}\) is the function that maps \(\varphi \) to \(o(\varphi )\) (see (iv) in Example 2) then \(o:\overline{\mathbb {N}}\longrightarrow \widetilde{\mathbb {I}}^{\mathrm{e}}\) is a morphism.

### Lemma 10

Let \(\mathbb {I}_{k}=(I_{k},\le _{k},\mathcal {I}_{k})\) be sets of indices for \(k=1\), 2, 3. Let \(a\in A\in \mathcal {I}_{1},\,b\in B\in \mathcal {I}_{2}\) and \(c\in C\in \mathcal {I}_{3}\). Then if \(f:A_{\le a}\longrightarrow B_{\le b}\) and \(g:B_{\le b}\longrightarrow C_{\le c}\) are infinitesimals, also the composition \(f\circ g:A_{\le a}\longrightarrow C_{\le c}\) is infinitesimal.

### Proof

The following results motivate our definition of morphism of sets of indices.

### Lemma 11

In the assumptions of Definition 8, let \(f:A_{\le a}\longrightarrow B_{\le b}\) be infinitesimal, and let \(\mathcal {P}(\varepsilon _{1})\) be a given property of \(\varepsilon _{1}\in I_{1}\). If \(\mathcal {P}(\varepsilon _{1})\) holds \(\forall ^{\mathbb {I}_{1}}\varepsilon _{1}\in A_{\le a}\) then \(\mathcal {P}(f(\varepsilon _{2}))\) holds \(\forall ^{\mathbb {I}_{2}}\varepsilon _{2}\in B_{\le b}\).

### Proof

Let \(e_{1}\in A_{\le a}\) be such that \(\mathcal {P}(\varepsilon _{1})\) holds for all \(\varepsilon _{1}\in A_{\le e_{1}}\). Since \(f:A_{\le a}\longrightarrow B_{\le b}\) is infinitesimal, there exists \(e_{2}\in B_{\le b}\) be such that \(f(\varepsilon _{2})\in A_{\le e_{1}}\) for all \(\varepsilon _{2}\in B_{\le e_{2}}\). Therefore \(\mathcal {P}(f(\varepsilon _{2}))\) holds for all \(\varepsilon _{2}\in B_{\le e_{2}}\). \(\square \)

### Theorem 12

Let \(\mathbb {I}_{k}=(I_{k},\le _{k},\mathcal {I}_{k})\) be sets of indices for \(k=1,2\). Let \(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) be a morphism of sets of indices and let \(\mathcal {P}(\varepsilon _{1})\) be a given property of \(\varepsilon _{1}\in I_{1}\). If \(\forall ^{\mathbb {I}_{1}}\varepsilon _{1}\,\mathcal {P}(\varepsilon _{1})\) then \(\forall ^{\mathbb {I}_{2}}\varepsilon _{2}\,\mathcal {P}(f(\varepsilon _{2}))\).

### Proof

Let \(A\in \mathcal {I}_{1}\) be such that \(\forall a\in A\,\forall ^{\mathbb {I}_{1}}\varepsilon _{1}\in A_{\le a}\,\mathcal {P}(\varepsilon _{1})\) holds. Since \(\emptyset \notin \mathcal {I}_{1}\), there exists \(a\in A\). But \(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) is a morphism, so there exists \(B\in \mathcal {I}_{2}\) such that \(f:A_{\le a}\longrightarrow B_{\le b}\) is infinitesimal for all \(b\in B\). By Lemma 11, we deduce that \(\forall ^{\mathbb {I}_{2}}\varepsilon _{2}\in B_{\le b}\,\mathcal {P}(f(\varepsilon _{2}))\), which is our conclusion. \(\square \)

Three simple consequences of Theorem 12 are presented in the following corollary.

### Corollary 13

- (i)
If \(i>_{\mathbb {I}_{1}}j\) then \(i\circ f>_{\mathbb {I}_{2}}j\circ f\);

- (ii)
If \(x_{\varepsilon _{1}}=O(y_{\varepsilon _{1}})\) as \(\varepsilon _{1}\in \mathbb {I}_{1}\), then \(x_{f(\varepsilon _{2})}=O(y_{f(\varepsilon _{2})})\) as \(\varepsilon _{2}\in \mathbb {I}_{2}\);

- (iii)
For every net \(g:I_{1}\longrightarrow \mathbb {R}\) if \(l=\lim _{\mathbb {I}_{1}}g\) then \(l=\lim _{\mathbb {I}_{2}}g\circ f\).

### Proof

Property (i) follows directly from Theorem 12 because \(i>_{\mathbb {I}_{1}}j\) means \(\forall ^{\mathbb {I}}\varepsilon :\ i_{\varepsilon }>j_{\varepsilon }\). To prove (ii), let \(A\in \mathcal {I}_{1}\) be such that \(x_{\varepsilon _{1}}=O_{A,a}(y_{\varepsilon _{1}})\) for all \(a\in A\). Therefore, there exists \(H\in \mathbb {R}_{>0}\) such that \(\forall ^{\mathbb {I}_{1}}\varepsilon _{1}\in A_{\le a}\ |x_{\varepsilon _{1}}|\le H\cdot |y_{\varepsilon _{1}}|\). But \(A\ne \emptyset \), so we can pick \(a\in A\), and \(f:\,\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) yields the existence of \(B\in \mathcal {I}_{2}\) such that \(f:A_{\le a}\longrightarrow B_{\le b}\) is infinitesimal for all \(b\in B\). By Lemma 11 we get \(\forall ^{\mathbb {I}_{2}}\varepsilon _{2}\in B_{\le b}\ |x_{f(\varepsilon _{2})}|\le H\cdot |y_{f(\varepsilon _{2})}|\), from which the conclusion follows. Using the same ideas, we can prove (iii). \(\square \)

### Theorem 14

The class of all sets of indices together with their morphisms forms a category \(\mathbf {Ind}\).

### Proof

### 3.1 Downward directed and segmented sets of indices

In this section, we study suitable classes of sets of indices where the notion of morphism of the category \(\mathbf Ind \) simplifies.

### Definition 15

- (i)
\(\mathbb {I}\) is

*segmented*if \(\forall A\in \mathcal {I}\,\exists a:\ (\emptyset ,a]\subseteq A\); - (ii)
\(\mathbb {I}\) is

*downward directed*if \((I,\le )\) is downward directed, i.e. for every*a*, \(b\in I\) there exists \(c\in I\) such that \(c\le a,\)\(c\le b\).

*canonical*

*set of indices*generated by \(\mathbb {I}\), and we denote it by \(\overline{\mathbb {I}}\), the set of indices \(\overline{\mathbb {I}}=(I,\le ,\mathcal {S}_{I}),\) where

### Example 16

- (i)
The sets of indices \(\mathbb {I}^{\mathrm{s}}\) and \(\mathbb {I}^{\text {E}}\) are both segmented and downward directed.

- (ii)
If \(\mathbb {I}=\mathbb {I}^{\mathrm{s}}\) then \(\overline{\mathbb {I}}=\mathbb {I}\). If \(\mathbb {I}=\mathbb {I}^{\text {E}}\) then \(\bar{\mathbb {I}}\ne \mathbb {I}\), but we will see in Theorem 18 that they are isomorphic in the category \(\mathbf Ind \).

- (iii)
If \(\mathbb {I}=\widetilde{\mathbb {I}}^{\mathrm{e}}\) then \(\overline{\mathbb {I}}\ne \mathbb {I}.\)

As mentioned above, the notion of morphism is simplified when we work with this type of sets of indices.

### Theorem 17

- (i)
\(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) is a morphism of sets of indices;

- (ii)
\(\forall a\in I_{1}\,\exists b\in I_{2}:\ f((\emptyset ,b])\subseteq (\emptyset ,a]\);

- (iii)
\(\forall a\in I_{1}\,\forall b\in I_{2}:\ f:(I_{1})_{\le a}\longrightarrow (I_{2})_{\le b}\) is infinitesimal.

### Proof

To prove that (i) entails (ii), let \(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) be a morphism and let \(a\in I_{1}\). Setting \(A=I_{1}\) in the definition of morphism, we get the existence of \(B\in \mathcal {I}_{2}\) such that \(f:A_{\le a}\longrightarrow B_{\le \bar{b}}\) is infinitesimal for each \(\bar{b}\in B\). Take any \(\bar{b}\in B\ne \emptyset \). Setting \(\alpha =a\) in the definition of infinitesimal (Definition 8), we get the existence of \(b\in B_{\le \bar{b}}\subseteq I_{2}\) such that \(f(\varepsilon _{2})\in (\emptyset ,a]\) for all \(\varepsilon _{2}\in (\emptyset ,b]\), which is our conclusion.

To prove that (ii) entails (iii), let \(a\in I_{1},\,b\in I_{2}\) and let \(\overline{b}\in I_{2}\) be such that \(f(\emptyset ,\overline{b}]\subseteq (\emptyset ,a]\). Let \(\alpha \in (\emptyset ,a]\) and let \(\widetilde{b}\in B_{\le \overline{b}}\) be such that \(f(\emptyset ,\widetilde{b}]\subseteq (\emptyset ,\alpha ]\). Since \((I_{2},\le )\) is downward directed, we can find \(\beta \in I_{2}\) such that \(\beta \le b,\,\beta \le \widetilde{b}.\) By construction, \(f(\emptyset ,\beta ]\subseteq (\emptyset ,\alpha ]\) and \((\emptyset ,\beta ]\subseteq (\emptyset ,b]=(I_{2})_{\le b}.\) Therefore \(f:(I_{1})_{\le a}\longrightarrow (I_{2})_{\le b}\) is infinitesimal.

To prove that (iii) entails (i), assume that \(a\in A\in \mathcal {I}_{1}\). Set \(B:=I_{2}\) and take any \(b\in B\). By (iii) we obtain that \(f:(I_{1})_{\le a}\longrightarrow (I_{2})_{\le b}\) is infinitesimal. Therefore, for each \(\alpha \le a\) there exists \(\widetilde{\beta }\le b\) such that we have \(f(\varepsilon _{2})\le \alpha \) for every \(\varepsilon _{2}\le \widetilde{\beta }\). small, let’s say for \(\varepsilon _{2}\le \tilde{\beta }\le b\). But \(\mathbb {I}_{1}\) is segmented, so there exists \(a'\) such that \((\emptyset ,a']\subseteq A\). Once again from (iii) we also have that \(f:(I_{1})_{\le a'}\longrightarrow (I_{2})_{\le b}\) is infinitesimal. Hence for some \(\bar{\beta }\le b\) we have \(f(\varepsilon _{2})\le a'\) for each \(\varepsilon _{2}\le \bar{\beta }\). Since \((I_{2},\le )\) is downward directed, we can find \(\beta \in I_{2}=B\) such that \(\beta \le \tilde{\beta }\) and \(\beta \le \bar{\beta }\). Therefore, for each \(\varepsilon _{2}\le \beta \) we have both \(f(\varepsilon _{2})\le \alpha \) and \(f(\varepsilon _{2})\in (\emptyset ,a']\subseteq A\). This proves that \(f:A_{\le a}\longrightarrow B_{\le b}\) is infinitesimal, which completes the proof. \(\square \)

### Theorem 18

Every segmented downward directed set of indices \(\mathbb {I}\) is isomorphic to \(\bar{\mathbb {I}}\) in the category * Ind*.

### Proof

It suffices to consider the identity \(1_{I}:i\in I\mapsto i\in I\), which is a morphism \(1_{I}\in \mathbf Ind (\mathbb {I},\bar{\mathbb {I}})\cap \mathbf Ind (\bar{\mathbb {I}},\mathbb {I})\) because of condition (ii) of Theorem 17. \(\square \)

Therefore, up to isomorphism, the only segmented downward directed set of indices having \((I,\le )\) as underlying pre-ordered set is \(\overline{\mathbb {I}}.\)

## 4 Asymptotic gauge Colombeau type algebras

### 4.1 Asymptotic gauges

In [12], we introduced the notion of asymptotic gauge. The idea was to use it as an asymptotic scale that generalizes the role of the polynomial family \((\varepsilon ^{n})_{\varepsilon \in (0,1],n\in \mathbb {N}}\) in classical constructions of Colombeau algebras. We recall the notations and notions from [12] that we will use in the present work. For all the proofs, we refer to [12].

### Definition 19

*asymptotic gauge on*\(\mathbb {I}\) (briefly: AG on \(\mathbb {I}\)) if

- (i)
\(\mathcal {B}\subseteq \mathbb {R}^{I}\);

- (ii)
\(\exists i\in \mathcal {B}:\ \lim _{\mathbb {I}}i=\infty \);

- (iii)
\(\forall i,j\in \mathcal {B}\,\exists p\in \mathcal {B}:\ i\cdot j=O(p)\);

- (iv)
\(\forall i\in \mathcal {B}\,\forall r\in \mathbb {R}\,\exists \sigma \in \mathcal {B}:\ r\cdot i=O(\sigma )\);

- (v)
\(\forall i,j\in \mathcal {B}\,\exists s\in \mathcal {B}:\ s>_{\mathbb {I}}0\ ,\ |i|+|j|=O(s)\).

*moderate nets generated by*\(\mathcal {B}\) is

Let us observe that \(\mathbb {R}_{M}(\mathcal {B})\) is an AG, and that \(\mathbb {R}_{M}(\mathbb {R}_{M}(\mathcal {B}))=\mathbb {R}_{M}(\mathcal {B}).\) Every asymptotic gauge formalizes a notion of “growth condition”. We can hence use an asymptotic gauge \(\mathcal {B}\) to define moderate nets. We can also use the reciprocals of nets taken from another asymptotic gauge \(\mathcal {Z}\) to define negligible nets. From this point of view, it is natural to introduce the following:

### Definition 20

*set of*\(\mathcal {B}\)

*-moderate nets*is

*set of*\(\mathcal {Z}\)

*-negligible nets*is

In [12], we proved that if \(\mathbb {R}_{M}(\mathcal {B})\subseteq \mathbb {R}_{M}(\mathcal {Z})\) then the quotient \(\mathcal {E}_{M}(\mathcal {B},\Omega )/\mathord {\mathcal {N}(\mathcal {Z},\Omega )}\) is an algebra. When this happens, we will use the following:

### Definition 21

*Colombeau AG algebra generated by*\(\mathcal {B}\)

*and*\(\mathcal {Z}\) is the quotient

- (i)
\((x_{\varepsilon })\sim _{\mathcal {Z}}(y_{\varepsilon })\) if and only if \(\forall z\in \mathcal {Z}_{>0}:\ x_{\varepsilon }-y_{\varepsilon }=O(z_{\varepsilon }^{-1})\), where \((x_{\varepsilon })\), \((y_{\varepsilon })\in \mathbb {R}_{M}(\mathcal {B})\).

- (ii)
\({\widetilde{\mathbb {R}}}(\mathcal {B},\mathcal {Z}):=\mathbb {R}_{M}(\mathcal {B})/\sim _{\mathcal {Z}}\). We simply use the notation \({\widetilde{\mathbb {R}}}(\mathcal {B})\) for \({\widetilde{\mathbb {R}}}(\mathcal {B},\mathcal {B})\).

### Example

Both Egorov algebra \(\mathcal {G}^{\text {E}}(\Omega )\) and the algebra of non-standard smooth functions \(^{*}{\mathcal {C}}^{\infty }(\Omega )\) are Colombeau AG algebras with \(\mathcal {B}=\mathcal {Z}=\mathbb {R}^{\mathbb {N}}\). In fact, in both cases \(\mathcal {E}_{M}(\mathcal {B},\Omega )={\mathcal {C}}^{\infty }(\Omega )^{\mathbb {N}}\) because the AG \(\mathcal {B}=\mathbb {R}^{\mathbb {N}}\) is trivial. It is also easy to see that in the former case \((u_{n})_{n\in \mathbb {N}}\in \mathcal {N}^{\text {E}}(\mathbb {R}^{\mathbb {N}},\Omega )\) if and only if for all \(K\Subset \Omega \), \(u_{n}|_{K}=0\) for all \(n\in \mathbb {N}\) sufficiently big. In the latter \((u_{n})_{n\in \mathbb {N}}\in \mathcal {N}^{*}(\mathbb {R}^{\mathbb {N}},\Omega )\) if and only if for all \(K\Subset \Omega \) there exists an ultrafilter set \(A\in \mathcal {I}^{*}\) such that \(u_{n}|_{K}=0\) for all \(n\in A\).

Morphisms between sets of indices can be used to construct asymptotic gauges, as the following theorem shows.

### Theorem 22

### Proof

All the defining properties of an asymptotic gauge for \(\mathcal {B}\circ f\) can be derived from Corollary 13. For example, let us prove that \(\forall i,j\in \mathcal {B}\circ f\,\exists s\in \mathcal {B}\circ f:\ s>_{\mathbb {I}}0\ ,\ |i|+|j|=O(s).\) Let *i*, \(j\in \mathcal {B}\circ f\) and let \(i=b_{1}\circ f,\,j=b_{2}\circ f\). Let \(b_{3}\in \mathcal {B}\) be such that \(|b_{1}|+|b_{2}|=O(b_{3}).\) Then by Corollary 13 we deduce that \(|b_{1}\circ f|+|b_{2}\circ f|=O(b_{3}\circ f).\) Setting \(s=b_{3}\circ f\) we therefore have that \(|i|+|j|=O(s).\)\(\square \)

## 5 The categories Ag\(_{2}\) and Ag\(_{1}\)

We want to prove that the Colombeau AG algebra construction of Definition 21 is functorial in the pair \((\mathcal {B},\mathcal {Z})\) of AGs. In proving this result, the following category arises naturally:

### Definition 23

- (i)
\((\mathcal {B},\mathcal {Z})\in \textsc {Ag}_{2}\) if \(\mathcal {B}\), \(\mathcal {Z}\) are AGs on some set of indices \(\mathbb {I}\) and \(\mathbb {R}_{M}(\mathcal {B})\subseteq \mathbb {R}_{M}(\mathcal {Z})\).

- (ii)
Let \((\mathcal {B}_{1},\mathcal {Z}_{1})\), \((\mathcal {B}_{2},\mathcal {Z}_{2})\in \textsc {Ag}_{2}\) be pairs of AGs on the sets of indices resp. \(\mathbb {I}_{1}\), \(\mathbb {I}_{2}\). We say that \(f\in \textsc {Ag}_{2}((\mathcal {B}_{1},\mathcal {Z}_{1}),(\mathcal {B}_{2},\mathcal {Z}_{2}))\) is a

*morphism of pairs of AGs*if \(f\in \mathbf Ind (\mathbb {I}_{1},\mathbb {I}_{2})\), \(\mathbb {R}_{M}(\mathcal {B}_{1}\circ f)\subseteq \mathbb {R}_{M}(\mathcal {B}_{2})\) and \(\mathbb {R}_{M}(\mathcal {Z}_{2})\subseteq \mathbb {R}_{M}(\mathcal {Z}_{1}\circ f)\).

### Theorem 24

\(\textsc {Ag}_{2}\) with set-theoretical composition and identity is a category.

### Proof

The generalization with two AGs is a relatively new step in considering Colombeau like algebras, and its main aim is to highlight what peculiar properties are used to derive the fundamental properties (a)–(e), in particular the specific embedding property (e). It is therefore natural to consider also the following

### Definition 25

We say that \(\mathcal {B}\in \textsc {Ag}_{1}\) if \((\mathcal {B},\mathcal {B})\in \textsc {Ag}_{2}\). We set \(f\in \textsc {Ag}_{1}(\mathcal {B}_{1},\mathcal {B}_{2})\) if \(f\in \textsc {Ag}_{2}((\mathcal {B}_{1},\mathcal {B}_{1}),(\mathcal {B}_{2},\mathcal {B}_{2}))\). We call such an *f* a *morphism of AG*s. In this case, Definition 23 (ii) becomes \(\mathbb {R}_{M}(\mathcal {B}_{1}\circ f)=\mathbb {R}_{M}(\mathcal {B}_{2})\).

Of course \(\textsc {Ag}_{1}\) is embedded into \(\textsc {Ag}_{2}\) by means of \(\mathcal {B}\mapsto (\mathcal {B},\mathcal {B})\) and of the identity on arrows. By an innocuous abuse of language, we can hence say that \(\textsc {Ag}_{1}\) is a subcategory of \(\textsc {Ag}_{2}\).

### Example 26

- (i)
Let \(\mathbb {I}=\mathbb {I}^{\mathrm{s}}\), let \(\mathcal {B}_{1}=\{(\varepsilon ^{-n})\mid n\in \mathbb {N}\}\), \(\mathcal {B}_{2}=\{(\varepsilon ^{-2n})\mid n\in \mathbb {N}\}\). Then

*f*, \(g:\mathbb {I\longrightarrow \mathbb {I}}\) such that \(f(\varepsilon )=\varepsilon ^{2}\) and \(g(\varepsilon )=\sqrt{\varepsilon }\) induce morphisms \(\mathcal {B}_{1}\overset{f}{\longrightarrow }\mathcal {B}_{2}\) and \(\mathcal {B}_{2}\overset{g}{\longrightarrow }\mathcal {B}_{1}\). Clearly \(f\circ g=1_{\mathcal {B}_{2}}\) and \(g\circ f=1_{\mathcal {B}_{1}}\), therefore \(\mathcal {B}_{1}\) and \(\mathcal {B}_{2}\) are isomorphic. - (ii)
Let \(\mathbb {I}_{1}=\mathbb {I}^{\mathrm{s}},\)\(\mathbb {I}_{2}=\overline{\mathbb {N}}\) (see Example 9 (vi)), let \(\mathcal {B}_{1}=\{(\varepsilon ^{-n})\mid n\in \mathbb {N}\}\), \(\mathcal {B}_{2}=\{(n^{m})_{n}\mid m\in \mathbb {N}\}\). Then \(f:\mathbb {I}_{1}\longrightarrow \mathbb {I}_{2}\) such that \(f(n)=\frac{1}{n+1}\) for every \(n\in \mathbb {N}\) induces a morphism \(\mathcal {B}_{1}\overset{f}{\longrightarrow }\mathcal {B}_{2}\) and \(g:\mathbb {I}_{2}\longrightarrow \mathbb {I}_{1}\) such that \(g(\varepsilon )=\lfloor \frac{1}{\varepsilon }\rfloor \) for every \(\varepsilon \in (0,1]\) induces a morphism \(\mathcal {B}_{2}\overset{g}{\longrightarrow }\mathcal {B}_{1}\).

- (iii)
Let \(o:\overline{\mathbb {N}}\longrightarrow \widetilde{\mathbb {I}}^{\mathrm{e}}\) be the morphism given by the maps

*o*that maps every \(\varphi \in \mathcal {A}_{0}\) to the order \(o(\varphi )\) of \(\varphi \). Let \(\mathcal {B}_{1}=\{(b_{n}^{m})_{n}\mid m\in \mathbb {N}\}\) where \(b_{n}=\frac{1}{n+1}\) for every \(n\in \mathbb {N}\) and let \(\mathcal {B}_{2}=\{(b_{\varphi }^{n})_{\varphi }\mid n\in \mathbb {N}\}\), where \(b_{\varphi }=\frac{1}{o(\varphi )+1}\) for every \(\varphi \in \mathcal {A}_{0}\). Then \(\mathcal {B}_{1}\overset{o}{\longrightarrow }\mathcal {B}_{2}\) is a morphism. - (iv)
Set \(f(\varepsilon )=\varepsilon +\varepsilon ^{2}\cdot \sin (\frac{1}{\varepsilon })\) for \(\varepsilon \in (0,1]\) and \(\mathcal {B}^{\mathrm{s}}:=\{ \varepsilon ^{-a}\mid a\in \mathbb {R}_{>0}\} \). Then \(\varepsilon -\varepsilon ^{2}\le f(\varepsilon )\le \varepsilon +\varepsilon ^{2}\), and this implies \(f\in \textsc {Ag}_{1}(\mathcal {B}^{\mathrm{s}},\mathcal {B}^{\mathrm{s}})\). Let us note that

*f*is not invertible in any neighbourhood of 0 so that it is not an isomorphism of AGs.

In [12], we defined two asymptotic gauges \(\mathcal {B}_{1}\), \(\mathcal {B}_{2}\) to be equivalent if and only if \(\mathbb {R}_{M}(\mathcal {B}_{1})=\mathbb {R}_{M}(\mathcal {B}_{2}).\) Within the present categorical framework, this definition is motivated by the following result.

### Theorem 27

Let \(\mathcal {B}\) be an asymptotic gauge on \(\mathbb {I}\). Then \(\mathcal {B}\) is isomorphic to \(\mathbb {R}_{M}(\mathcal {B})\).

### Proof

It is sufficient to observe that, by definition, \(\mathcal {B}\overset{1_{I}}{\longrightarrow }\mathbb {R}_{M}(\mathcal {B})\) is a morphism and, since \(\mathbb {R}_{M}(\mathbb {R}_{M}(\mathcal {B}))=\mathbb {\mathbb {R}}_{M}(\mathcal {B})\), also \(\mathbb {R}_{M}(\mathcal {B})\overset{1_{I}}{\longrightarrow }\mathcal {B}\) is a morphism. \(\square \)

In particular, it follows that for every two asymptotic gauges \(\mathcal {B}_{1}\), \(\mathcal {B}_{2}\) defined on the same set of indices \(\mathbb {I}\), we have that if \(\mathcal {B}_{1}\) is equivalent to \(\mathcal {B}_{2}\) then they are isomorphic. Conversely, if \(f\in \textsc {Ag}_{1}(\mathcal {B}_{1},\mathcal {B}_{2})\) is an isomorphism, then \(\mathbb {R}_{M}(\mathcal {B}_{1})=\mathbb {R}_{M}(\mathcal {B}_{1}\circ f\circ f^{-1})=\mathbb {R}_{M}(\mathcal {B}_{2}\circ f^{-1})=\mathbb {R}_{M}(\mathcal {B}_{1})\), and hence \(\mathbb {R}_{M}(\mathcal {B}_{1})=\mathbb {R}_{M}(\mathcal {B}_{2}\circ f^{-1})\). Analogously, \(\mathbb {R}_{M}(\mathcal {B}_{2})=\mathbb {R}_{M}(\mathcal {B}_{1}\circ f)\). In particular, the identity \(1_{I}\in \textsc {Ag}_{1}(\mathcal {B}_{1},\mathcal {B}_{2})\cap \textsc {Ag}_{1}(\mathcal {B}_{2},\mathcal {B}_{1})\) if and only if these AGs are equivalent. For example \(\{ (\varepsilon ^{-a})\mid a\in \mathbb {R}_{>0}\} \) and \(\{(\varepsilon ^{-n})\mid n\in \mathbb {N}\} \) are equivalent. Nevertheless, it is not difficult to prove that not all isomorphic AGs on the same set of indices are equivalent. To prove this result, we need to recall (see [12, Def. 36]) that an AG \(\mathcal {B}\) is called *principal* if there exists a *generator*\(b\in \mathcal {B}\) such that \(\mathbb {R}_{M}(\text {AG}(b))=\mathbb {R}_{M}(\mathcal {B})\), where \(\text {AG}(b):=\{ b^{m}\mid m\in \mathbb {N}\} \).

### Theorem 28

For every principal AGs \(\mathcal {B}_{1}\), \(\mathcal {B}_{2}\) on \(\mathbb {I}^{\mathrm{s}}\), if \(\mathbb {R}_{M}(\mathcal {B}_{1})\subsetneq \mathbb {R}_{M}(\mathcal {B}_{2})\) then there exists a principal AG \(\mathcal {B}_{3}\) such that \(\mathbb {R}_{M}(\mathcal {B}_{1})\subsetneq \mathbb {R}_{M}(\mathcal {B}_{3})\subsetneq \mathbb {R}_{M}(\mathcal {B}_{2}).\)

### Proof

### Corollary 29

### Proof

This is an immediate consequence of Theorem 28. \(\square \)

In particular, if we let \(\mathcal {B}_{\text {pol}}:=\{ (\varepsilon ^{-n})\mid n\in \mathbb {N}\} \) and \(\mathcal {B}_{\text {exp}}:= \{(e^{n/\varepsilon })\mid n\in \mathbb {N}\} \), by Corollary 29 we have that there are infinitely many principal non equivalent AGs between \(\mathcal {B}_{\text {pol}}\) and \(\mathcal {B}_{\text {exp}}\). However, as we will show in Sect. 7, \(\mathcal {B}_{\text {pol}}:=\{ (\varepsilon ^{-n})\mid n\in \mathbb {N}\} \) and \(\mathcal {B}_{\text {exp}}:=\{ (e^{n/\varepsilon })\mid n\in \mathbb {N}\} \) are isomorphic, and this shows that not all isomorphic AGs are equivalent.

In [12], we proved that \(\mathbb {R}_{M}(\mathcal {B})\) is the minimal (with respect to inclusion) asymptotically closed solid ring containing the AG \(\mathcal {B}\). Therefore, we deduce that, modulo isomorphism, all the objects in a skeleton subcategory of \(\mathbf {\textsc {Ag}}_{1}\) are asymptotically closed solid rings.

In [12], we introduced the notion of “exponential of an AG”, which was crucial to study linear ODE with generalized constant coefficients. We recall its definition.

### Definition 30

*exponential of*\(\mathcal {B}\).

The following results will be needed to prove Theorem 35.

### Lemma 31

In the hypotheses of Definition 30, we have that \(\mu (\mathcal {B})\) is an AG.

### Proof

### Lemma 32

Let \(\mathbb {I}=(I,\le ,\mathcal {I})\) be a set of indices, and let *x*, *y*, \(z\in \mathbb {R}^{I}\). Let \(\mu :\mathbb {R}\longrightarrow \mathbb {R}_{\ge 0}\) be a non decreasing function. Then \(x_{\varepsilon }=O [\mu (y_{\varepsilon })]\) and \(y<_{\mathbb {I}}z\) imply \(x_{\varepsilon }=O [\mu (z_{\varepsilon })]\).

### Proof

### Corollary 33

Let \(\mathcal {B}_{1}\), \(\mathcal {B}_{2}\) be AGs on the same set of indices \(\mathbb {I}\), and let \(\mu :\mathbb {R}\longrightarrow \mathbb {R}_{\ge 0}\)verify the assumptions of Definition 30. Then \(\mathbb {R}_{M}(\mathcal {B}_{1})\subseteq \mathbb {R}_{M}(\mathcal {B}_{2})\) implies \(\mathbb {R}_{M}(\mu (\mathcal {B}_{1}))\subseteq \mathbb {R}_{M}(\mu (\mathcal {B}_{2}))\).

### Proof

Let \((y_{\varepsilon }')=(\mu (H\cdot b'_{\varepsilon }))\in \mu (\mathcal {B}_{1})\), with \(b'\in \mathcal {B}_{1}\), and let \(b''\in \mathcal {B}_{2}\) be such that \(b'<_{\mathbb {I}}b''\). Then we have that \((y'_{\varepsilon })<_{\mathbb {I}}(\mu (H\cdot b_{\varepsilon }''))\in \mu (\mathcal {B}_{2})\) since \(\mu \) is monotone. \(\square \)

### Definition 34

- (i)
\(E(\mathcal {B}):=\mu (\mathcal {B})\) for each \(\mathcal {B}\in \textsc {Ag}_{\le }\);

- (ii)
\(E(f):=f\) for each \(f\in \textsc {Ag}_{\le }(\mathcal {B}_{1},\mathcal {B}_{2})\).

### Theorem 35

\(\textsc {Ag}_{\le }\) is a subcategory of \(\textsc {Ag}_{1}\). If \(\mu \) verifies the assumptions of Definition 30, then \(E_{\mu }:\textsc {Ag}_{\le }\longrightarrow \textsc {Ag}_{\le }\) is a functor.

### Proof

To prove the first part, assume that \(\mathbb {R}_{M} (\mathcal {B}_{1}\circ f )=\mathbb {R}_{M} (\mathcal {B}_{2})\) and \(\mathbb {R}_{M} (\mathcal {B}_{2}\circ g )=\mathbb {R}_{M} (\mathcal {B}_{3})\). Then if \(b_{1}\circ f<_{\mathbb {I}_{2}}b_{2}\) and \(b_{2}\circ g<_{\mathbb {I}_{3}}b_{3}\), for \(b_{i}\in \mathcal {B}_{i}\), then \(b_{1}\circ f\circ g<_{\mathbb {I}_{3}}b_{2}\circ g<_{\mathbb {I}_{3}}b_{3}\) by Corollary 13. If \(b_{3}<_{\mathbb {I}_{3}}b_{2}\circ g\) and \(b_{2}<_{\mathbb {I}_{2}}b_{1}\circ f\) then \(b_{3}<_{\mathbb {I}_{3}}b_{2}\circ g<_{\mathbb {I}_{3}}b_{1}\circ f\circ g\) once again by Corollary 13. This implies that \(\mathbb {R}_{M} (\mathcal {B}_{1}\circ (f\circ g))=\mathbb {R}_{M} (\mathcal {B}_{3})\), hence \(f\circ g\in \textsc {Ag}_{\le }(\mathcal {B}_{1},\mathcal {B}_{3})\). This and Corollary 33 show that \(\textsc {Ag}_{\le }\) is a category. By Corollary 33, we also have that \(\textsc {Ag}_{\le }\) is a subcategory of \(\textsc {Ag}_{1}\). To show the second part, since \(E_{\mu }\) is the identity on arrows, it suffices to observe that \(\mathcal {B}\) and \(\mu (\mathcal {B})\) have the same set of indices for every AG \(\mathcal {B}\), and that \(\mu (\mathcal {B}_{1})\circ f=\mu (\mathcal {B}_{1}\circ f)\). Thus it follows by Corollary 33 that \(E_{\mu }(\mathcal {B}_{1})=\mu (\mathcal {B}_{1})\overset{f}{\longrightarrow }\mu (\mathcal {B}_{2})=E_{\mu }(\mathcal {B}_{2})\) is an arrow in \(\textsc {Ag}_{\le }\) for every arrow \(\mathcal {B}_{1}\overset{f}{\longrightarrow }\mathcal {B}_{2}\) in \(\textsc {Ag}_{\le }\). \(\square \)

## 6 Functoriality of Colombeau AG-algebras

In this section, we want to prove that the map \((\mathcal {B},\mathcal {Z},\Omega )\mapsto \mathcal {G}(\mathcal {B},\mathcal {Z},\Omega )\) is a functor. Clearly, \((\mathcal {B},\mathcal {Z})\in \textsc {Ag}_{2}\), so we need to introduce a category having open sets like \(\Omega \) as objects:

### Definition 36

We denote by \(\mathcal {O}\mathbb {R}^{\infty }\) the category having as objects \(\{\Omega \subseteq \mathbb {R}^{n}\mid n\in \mathbb {N},\,\Omega \,\text {open}\}\) and as morphisms \({\mathcal {O}}\mathbb {R}^\infty (U,V):={\mathcal {C}}^{\infty }(U,V)\).

Therefore, we can now prove the following:

### Theorem 37

\(\mathcal {G}:\textsc {AG}_{2}\times ({\mathcal {O}}\mathbb {R}^\infty )^{\text {{op}}}\longrightarrow \textsc {Alg}_{\mathbb {R}}\) is a functor, where \(\textsc {Alg}_{\mathbb {R}}\) is the category of commutative algebras over \(\mathbb {R}\).

### Proof

*h*is an arbitrary smooth function. For this reason, when we want to deal with differential algebras, we will always consider \(\mathcal {T}\mathbb {R}^{n}\) instead of the category \({\mathcal {O}}\mathbb {R}^\infty \).

As a consequence of Theorem 37, we also have that essentially all the constructions of Colombeau-like algebras are functorial. For example, we can consider the set of indices \(\mathbb {I}^{\mathrm{s}}\) of the special algebra, the AG \(\mathcal {B}^{\mathrm{s}}:= \{ (\varepsilon ^{-a})\mid a\in \mathbb {R}_{>0}\} \), and the full subcategory \(\textsc {Ag}_{\mathbb {I}^{\mathrm{s}}}\) of \(\textsc {Ag}_{1}\) of all the AGs on \(\mathbb {I}^{\mathrm{s}}\). Clearly, \(\mathcal {G}^{\mathrm{s}}(\mathcal {B},\Omega ):=\mathcal {G}(\mathcal {B},\mathcal {B},\Omega )\) is a functor \(\mathcal {G}^{\mathrm{s}}:\textsc {Ag}_{\mathbb {I}^{\mathrm{s}}}\times ({\mathcal {O}}\mathbb {R}^\infty )^{\text {op}}\longrightarrow \textsc {Alg}_{\mathbb {R}}\) which corresponds to the usual sheaf via the restriction \(\mathcal {G}^{\mathrm{s}}(\mathcal {B}^{\mathrm{s}},\Omega )\) only for \(\Omega \in \mathcal {T}\mathbb {R}^{n}\). Analogously, we can consider \(\hat{\mathcal {G}}\), \(\mathcal {G}^{\mathrm{e}}\), \(\mathcal {G}^{\mathrm{d}}\), \(\mathcal {G}^{2}\), \(\mathcal {G}^{\text {E}}\) and \(^{*}{\mathcal {C}}^{\infty }(\Omega )\).

We also finally note that if we consider an inclusion \(h\in \mathcal {T}\mathbb {R}^{n}(\Omega _{2},\Omega _{1})\) and a morphism of pairs of AGs \(i\in \textsc {Ag}_{2}((\mathcal {B}_{1},\mathcal {Z}_{1}),(\mathcal {B}_{2},\mathcal {Z}_{2}))\), then \(\mathcal {G}(i,h):\mathcal {G}(\mathcal {B}_{1},\mathcal {Z}_{1},\Omega _{1})\longrightarrow \mathcal {G}(\mathcal {B}_{2},\mathcal {Z}_{2},\Omega _{2})\) preserves all polynomial and differential operations. Of course, it also takes generalized functions in the domain \(\mathcal {G}(\mathcal {B}_{1},\mathcal {Z}_{1},\Omega _{1})\) into generalized functions in the codomain \(\mathcal {G}(\mathcal {B}_{2},\mathcal {Z}_{2},\Omega _{2})\). We can therefore state that \(\mathcal {G}(i,h)\) permits to relate differential problems framed in \(\mathcal {G}(\mathcal {B}_{1},\mathcal {Z}_{1},\Omega _{1})\) to those framed in \(\mathcal {G}(\mathcal {B}_{2},\mathcal {Z}_{2},\Omega _{2})\); see also the next Theorem 38.

## 7 An unexpected isomorphism

*unexpected*.

This example is generalized in the following theorem, where we talk, essentially for the sake of simplicity, of ODE.

### Theorem 38

*b*, \(c\in \mathbb {R}^{I}\) be infinite nets, i.e. such that \(\lim _{\varepsilon \in \mathbb {I}}b_{\varepsilon }=\lim _{\varepsilon \in \mathbb {I}}c_{\varepsilon }=+\infty \). Set

*b*(and analogously for

*c*). Assume that \(\eta \), \(\lambda \in \textsc {Ind}(\mathbb {I},\mathbb {I})\) are morphisms of \(\mathbb {I}\) such that \(\eta =\lambda ^{-1}\):

- (i)
\(\text {{AG}}(b)\simeq \text {{AG}}(c)\) as AGs;

- (ii)
\(\mathcal {G}(\text {{AG}}(b),\Omega )\simeq \mathcal {G}(\text {{AG}}(c),\Omega )\) and \({\widetilde{\mathbb {R}}}(\text {{AG}}(b))\simeq {\widetilde{\mathbb {R}}}(\text {{AG}}(c))\) in the category \(\textsc {Alg}_{\mathbb {R}}\);

- (iii)let \(F=[F_{\varepsilon }]\in \mathcal {G}(\text {{AG}}(b),\mathbb {R}^{n}\times \mathbb {R})\), \(\bar{x}=[\bar{x}_{\varepsilon }]\in {\widetilde{\mathbb {R}}}^{n}\), \(\bar{t}=[\bar{t}_{\varepsilon }]\in {\widetilde{\mathbb {R}}}\). Then the Cauchy problemhas a solution \(x\in \mathcal {G}(\text {{AG}}(b),(t_{1},t_{2}))\) if and only if the Cauchy problem$$\begin{aligned} {\left\{ \begin{array}{ll} x'(t)=F(x(t),t);\\ x(\bar{t})=\bar{x}, \end{array}\right. } \end{aligned}$$(7.5)has a solution \(y\in \mathcal {G}(\text {{AG}}(c),(t_{1},t_{2}))\).$$\begin{aligned} {\left\{ \begin{array}{ll} y'(t)=\left[ F_{\lambda (\varepsilon )}\right] (y(t),t);\\ y\left( \left[ \bar{t}_{\lambda (\varepsilon )}\right] \right) =\left[ \bar{x}_{\lambda (\varepsilon )}\right] , \end{array}\right. } \end{aligned}$$(7.6)

### Proof

Assumption (7.4) yields \((b^{n}\circ \eta )(\varepsilon )=b_{\eta (\varepsilon )}^{n}=O_{\mathbb {\mathcal {I}}}(c_{\varepsilon }^{n})\) so \(\mathbb {R}_{M}(\text {AG}(b)\circ \eta )=\mathbb {R}_{M}(\text {AG}(c))\). Analogously, we have \(\mathbb {R}_{M}(\text {AG}(c)\circ \lambda )=\mathbb {R}_{M}(\text {AG}(b))\). This shows that \(\eta \) and \(\lambda \) are morphisms of AGs, and hence it proves (i). Property (ii) follows from the functorial property of \(\mathcal {G}(-,\Omega ):\text {AG}_{1}\longrightarrow \textsc {Alg}_{\mathbb {R}}\). To show (iii), let \(x=[x_{\varepsilon }]\in \mathcal {G}(\text {{AG}}(b),(t_{1},t_{2}))\) be a solution of (7.5) and set \(y:=\mathcal {G}(\lambda ,1_{(t_{1},t_{2})})(x)=[x_{\lambda (\varepsilon )}]\). Therefore, Theorem 37 and (6.2) yield the conclusion. \(\square \)

For instance, if *b*, \(c:(0,1]\longrightarrow (0,1]\) are homeomorphisms such that \(\lim _{\varepsilon \rightarrow 0^{+}}b_{\varepsilon }=0=\lim _{\varepsilon \rightarrow 0^{+}}c_{\varepsilon }\), then \(\eta :=c\circ b^{-1}\) and \(\lambda :=\eta ^{-1}\) verify the assumptions of this theorem.

In this categorical framework, it is unnatural to expect that isomorphisms as in (ii) hold for every pair of generators *b*, *c*. It is simple to see that this is the case if *b*, *c* are generators defined, respectively, on two sets of indices \(\mathbb {I}_{1}=(I_{1},\le _{1},\mathcal {I}_{1}),\,\mathbb {I}_{2}=(I_{2},\le _{2},\mathcal {I}_{2})\) such that \(I_{1}\) and \(I_{2}\) have different cardinalities, as in this case there can not be a bijection between \(I_{1}\) and \(I_{2}\). A more interesting question is if there exist principal AGs on the same set of indices that are not isomorphic. This is actually the case, as we will show after Definition 39, where we make precise the definition of the category of principal AGs.

We finally note that an isomorphism similar to (ii) has been proved by [21, 22, 24] in the context of NSA. However, note that in [21, 22, 24] the non-constructive isomorphism strongly depends on the condition that the cardinality of the field of generalized numbers equals the successor of \(\text {card}(\mathbb {R})\). On the contrary, here we have a constructive, but less general, isomorphism strictly depending on the notion of morphism of AGs and the functorial properties of Colombeau AG-algebras.

## 8 The category of Colombeau algebras

In this section, we want to show that the Colombeau AG algebra, the related derivation of generalized functions and the embedding of distributions are all functorial constructions with respect to the change of AG. Although in this section we work on an arbitrary set of indices, we restrict our study only to embeddings of Schwartz distributions defined through a mollifier. Therefore, we are going to deal with mollifiers with null positive moments, namely with functions \(\rho \in \mathcal {S}(\mathbb {R}^{n})\) such that \(\int \rho (x)x^{k}\,\mathrm{d}x=0\) for every \(k\in \mathbb {N}^{n}\), \(|k|\ge 1\). We call *Colombeau mollifier* any such function.

*b*is a generator of \(\mathcal {Z}\). Note that (8.1) is only slightly stronger than the usual universal property mentioned in (v) at Definition 40, i.e.

### Definition 39

Let \(n\in \mathbb {N}_{>0}\) be a fixed natural number. Then \(\textsc {pAG}\) denotes the *category of principal AG*s, whose objects are pairs \((b,\mathcal {B})\), where \(\mathcal {B}\) is a principal AG on a set of indices \(\mathbb {I}\) and \(b\in \mathcal {B}\) is a generator of \(\mathcal {B}\). Arrows \(f\in \textsc {pAG}((b_{1},\mathcal {B}_{1}),(b_{2},\mathcal {B}_{2}))\) are morphisms \(f\in \textsc {AG}_{1}(\mathcal {B}_{1},\mathcal {B}_{2})\) of AGs that preserve the generator, i.e. such that \(b_{1}\circ f=b_{2}\). Let us note that if \(f\in \textsc {pAG}((b_{1},\mathcal {B}_{1}),(b_{2},\mathcal {B}_{2}))\) and \(g\in \textsc {pAG}((b_{2},\mathcal {B}_{2}),(b_{3},\mathcal {B}_{3}))\), then the composition in \(\textsc {pAG}\) is given as in \(\textsc {AG}_{1}\), i.e. by \(f\circ g\) because \(f:I_{2}\longrightarrow I_{1}\), \(g:I_{3}\longrightarrow I_{2}\).

We observe that there exist non isomorphic pairs \((b,\mathbb {R}_{M}(b)),\,(c,\mathbb {R}_{M}(c))\) defined on the set of indices \(\mathbb {I}^{\mathrm{s}}\). In fact, let \(b_{\varepsilon }=\varepsilon ^{-1}\) and \(c_{\varepsilon }=\lceil \varepsilon ^{-1}\rceil \) for every \(\varepsilon \in (0,1]\). Then \((c_{\varepsilon })\) assumes only a countable amount of values, whilst \((b_{\varepsilon })\) assumes a continuum of values. Hence, there can not be a morphism *f* such that \(b_{\varepsilon }=c_{f(\varepsilon )}\) for every \(\varepsilon \in (0,1]\). The existence of non-isomorphic principal AGs also implies that the isomorphism stated in [21, 22, 24] is not an isomorphism of principal AGs, i.e. it is not an arrow of the category \(\textsc {pAG}\).

Whilst the category \(\textsc {pAG}\) acts as domain in the Colombeau construction, the following *category of Colombeau algebras* acts as codomain.

### Definition 40

- (i)
\((G,\partial ):(\mathcal {T}\mathbb {R}^{n})^{\text {op}}\longrightarrow \textsc {DAlg}_{\mathbb {R}}\) is a functor (i.e. it is a presheaf of differential real algebras). In particular, \(\partial _{\Omega }^{\alpha }:G(\Omega )\longrightarrow G(\Omega )\) is a derivation for all \(\alpha \in \mathbb {N}^{n}\).

- (ii)
If we think at both functors \(\mathcal {D}'\), \(G:(\mathcal {T}\mathbb {R}^{n})^{\text {op}}\longrightarrow \textsc {Vect}_{\mathbb {R}}\) with values in the category of real vector spaces, then \(\iota :\mathcal {D}'\longrightarrow G\) is a natural transformation such that \(\text {ker}(\iota _{\Omega })=\{0\}\) for every \(\Omega \in \mathcal {T}\mathbb {R}^{n}\).

- (iii)
\({\mathcal {C}}^{\infty }(\Omega )\) is a subalgebra of \(G(\Omega )\);

- (iv)
\(\iota _{\Omega }(S_{\Omega }(f))=f\) for all \(f\in {\mathcal {C}}^{\infty }(\Omega )\);

- (v)Let \(D_{\Omega }^{\alpha }:\mathcal {D}'(\Omega )\longrightarrow \mathcal {D}'(\Omega )\) be the \(\alpha \in \mathbb {N}^{n}\) derivation of distributions, then the following diagram commutes(8.2)

The following results prove the goal of the present section:

### Lemma 41

\(\textsc {Col}_{n}\) is a category.

### Proof

### Theorem 42

*b*and the fixed mollifier \(\rho \) (see [12, Sec. 4] for details). For \(f\in \textsc {pAG}((b_{1},\mathcal {B}_{1}),(b_{2},\mathcal {B}_{2}))\) and \(\Omega \in \mathcal {T}\mathbb {R}^{n}\), set

### Proof

The property \(\mathcal {C}\!{{o}}\ell _{n}^{\rho }(b,\mathcal {B})\in \textsc {Col}_{n}\) for every \((b,\mathcal {B})\in \textsc {pAG}\) is a consequence of the results about Colombeau principal AG-algebras and embeddings of distributions proved in [12, Sec. 3 and 4].

Properties (i)–(v) of Definition 40 we started from have been taken as defining attributes for the category \(\textsc {Col}_{n}\) of Colombeau algebras. Although this gives an important role to these properties, it clearly does not aim to be an axiomatic characterization. For results in this direction see [21, 22], where an axiomatic characterization is given but whose consistency depends on the generalized continuum hypothesis.

## 9 Conclusions

If we consider the most studied Colombeau algebra \(\mathcal {G}^s(\Omega )=\mathcal {G}(\mathcal {B}_{\text {pol}},\Omega )\) and if we have to deal with particular differential problems whose solutions grow more than polynomially in \(\varepsilon \), then we are forced to consider a different algebra. Since our framework of Colombeau AG-algebras includes all known algebras of this type, this means that we are forced to consider a different AG. It is therefore natural to search for the correct notion of morphism of AGs, and to see whether Colombeau AG constructions behave in the correct way with respect to these morphisms. The results of Sects. 5, 7, 8 show that both the construction of the differential algebra and that of the embedding by means of a mollifier are functorial with respect to a natural notion of morphism of AGs. As shown in Sects. 6, 7, this permits to relate differential problems solved for different AGs.

## Footnotes

- 1.
In the naturals \(\mathbb {N}=\{0,1,2,3\ldots \}\) we include zero.

## Notes

### Acknowledgments

Open access funding provided by University of Vienna.

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