key: cord-0495652-nuu2l0y6
authors: Bertram, Wolfgang; Haut, J'er'emy
title: A Functorial Approach to Differential Calculus
date: 2020-06-08
journal: nan
DOI: nan
sha: c8d05002488125f3f9a466dbf6fb1c539d4ee722
doc_id: 495652
cord_uid: nuu2l0y6

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets). To prepare this approach, we define a new, symmetric, presentation of differential calculus, whose main feature is the central r{^o}le played by the anchor map, which we study in detail. Our aim for developing this theory is twofold: (1) define a setting for calculus over any commutative ring, including finite rings; (2) define a setting that can be generalized to categories of graded rings (super differential calculus).

Differential Calculus is a central ingredient of modern mathematics. While the "working mathematician" takes this tool for granted, thinking about its conceptual foundations remains a potentially important topic. In the present work, we continue the line of research started with [BGN04, Be08, BeS14, Be17] , and combine it with what Grothendieck once called the "simple idea of a good functor from rings to sets" (according to W. Lawvere, cf. n-lab) 1 . The "simple idea" mentioned by 1 Here the quote from the n-lab: "The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as its main theme that the 1960 definition of scheme ... should be abandoned AS the FUN-DAMENTAL one and replaced by the simple idea of a good functor from rings to sets. The needed restrictions could be more intuitively and more geometrically stated directly in terms of the topos of such functors, and of course the ingredients from the "baggage" could be extracted when needed as auxiliary explanations of already existing objects, rather than being carried always as core elements of the very definition."

Grothendieck is currently used in algebraic geometry, and in Lie Theory, where one often considers a real "space" -for instance, a Lie group G -as set of "real points" G R of a complex Lie group G C . This is a kind of non-linear analog of the complexification V C = V ⊗ R C of a real vector space (or of a real Lie algebra). Grothendieck's insight was that this idea of "complexification" should not be limited to field extensions, but enlarged to more general ring extensions, in order to incorporate operations belonging to infinitesimal calculus: a K-Lie group G, or a general K-smooth manifold M, should admit "scalar extensions" M A akin to a hypothetic tensor product M ⊗ K A, for certain K-algebras A. The simplest example of such an extension is the one by dual numbers,

where the nilpotent element ε is the class [X] modulo (X 2 ). Grothendieck, following ideas of Weil [We53] , realized that the tangent bundle T M of a "space" M, which is "defined over K", could be understood as something like M ⊗ K K [ε] . This idea has been used by Demazure and Gabriel in their theory of algebraic groups [DG], in differential calculus over general base field and rings [Be08] , and in the approach to natural operations in differential geometry via the so-called Weil functors ( [KMS93] , cf. also [BeS14] ). The most elaborate and systematic development of these ideas leads to what is called nowadays synthetic differential geometry (SDG, see [MR91] ). The approach to be presented here pursues the same goals as SDG, but by different means: we keep closer to the idea of generalizing the algebraic tensor product. In a very direct sense, our problem is to generalize the algebraic scalar extension V A := V ⊗ K A of a K-module V (basic facts are recalled in Appendix A), to more general spaces M, like, e.g., manifolds -where we face the problem that such an operation won't be possible for all K-algebras A, so we have to single out a good class (good category) of algebras for which such an extension is possible. Such a class, called the category of (anchored) tangent algebras, will be defined in this paper. It arises naturally, when questioning the very shape of differential calculus, instead of taking it for granted. Let us briefly explain the main ideas.

0.1. Topological differential calculus. In differential calculus we consider maps f whose domain U and codomain U ′ are locally linear sets -by this we mean U ⊂ V and U ′ ⊂ V ′ are non-empty subsets of linear (or affine, if one prefers) spaces V and V ′ . In this situation, we may define the slope or difference quotient map: when t, s ∈ K are such that t − s is invertible, we look at the difference quotient

To speak of topological calculus, we shall assume that V, V ′ are topological vector spaces or modules over topological fields or rings K, and U, U ′ are open. For the moment, let's consider the "classical case" K = R and V = R n , V ′ = R m . Then the following holds (cf. [BGN04, Be08] ): The map f is of class C 1 if, and only if, the difference quotient map f [1] extends continuously to a map defined on the set (0.3) U [1] := (v 0 , v 1 ; t, s) ∈ V 2 × K 2 v 0 + tv 1 ∈ U v 0 + sv 1 ∈ U .

If this is the case, we denote still by f [1] : U [1] → U ′ the extended map. Then f [1] (v 0 , v 1 ; 0, 0) = df (v 0 )v 1 gives the differential df of f . Now, these conditions make perfectly sense for any "good" topological ring K and for maps defined on open locally linear sets, and thus can serve as definition of differentiability over K -the "topological differential calculus" thus defined has excellent functorial properties allowing to give a "purely algebraic" presentation of certain features of usual calculus (see [BGN04, Be11] ). To understand the structure of formulae like (0.2) and (0.3), the following way of talking turns out to be useful:

• call v = (v 0 , v 1 ) "space variables", with v 0 the "foot point" and v 1 the "direction" (in which we differentiate), • call (t, s) "time variables", and t "target time", and s "source time", • call (t, s) "regular", or "finite", if t − s is invertible in K, and "singular" or "infinitesimal" else, with t − s = 0 being the "most singular value", • call v 0 + sv 1 the "source", and v 0 + tv 1 the "target evaluation point", • for fixed (t, s), call α (v 0 , v 1 ) := v 0 + sv 1 the "source map", and define the "target map" β (v 0 , v 1 ) := v 0 + tv 1 . The slogan summarizing topological calculus is: the slope extends continuously (jointly in space and time variables) from finite to singular times. The notable difference with [BGN04, Be11] is that here we shall use a pair of time parameters (t, s), instead of a single parameter t as in loc. cit. Although the expression (0.2) is of course symmetric under switch of target and source time, it will be important to distinguish "target" and "source". The setting of [BGN04, Be11] is gotten by restricting to s = 0 (we call this "target calculus"); symmetrically, the theory could also be formulated when letting t = 0 ("source calculus"). But now we can take advantage to define a third calculus, the "symmetric calculus", which corresponds to the case t = −s: then v 0 = v 0 +sv 1 +v 0 +tv 1 2 , so the footpoint is the midpoint of target and source evaluation point. 2 This symmetric setting is best suited for a transition of our methods towards super-calculus (cf. Section 4). 0.2. The underlying algebraic structure: anchor. In the second section we shall carve out the algebraic structures underlying topological differential calculus. As in general groupoid theory, the pair (α, β) given by source and target will be called anchor map 3 . We use the same term when considering the pair of time variables (t, s) as a "frozen parameter" (temporarily considered to be fixed); then we write (t, s) as lower index -for instance,

For fixed (t, s), we call again anchor the (linear) map sending the space variables v = (v 0 , v 1 ) to the pair of evaluation points: (0.5)

Of course, a choice is made here: the "first" component of U ×U shall be associated with "source", and the "second" with "target". One of our concerns in the sequel will be to formalize the levels on which such choices are operated. Anyhow, by direct computation, the anchor is seen to be invertible if, and only if, t − s is invertible, and then its inverse is given by

(t,s) ,

The first component is an affine combination v 0 = s s−t x 1 + t t−s x 0 , and the second a "difference quotient". From this, comparing with (0.2), we see that f

The big advantage is that f {1} (t,s) depends functorially on f : the "chain rule" simply

(t,s) , jointly continuous also in the parameter (t, s) ∈ K 2 , such that

In a nutshell, this diagram contains the essential ingredients needed for our approach: our aim is to translate diagram (0.8) into a "categorical" formulation, so that it will make sense in an abstract setting, not requiring topology any more. In a first step, we generalize this diagram at higher order n ∈ N (Theorem 1.7): indeed, differentiability at order n is characterized by a diagram of the same kind, replacing f {1} (t,s) , etc., by higher order maps f {n} (t,s) , etc., where (t, s) = (t 1 , . . . , t n ; s 1 , . . . , s n ) ∈ K 2n . Technically, we work with 2 n -fold direct products, which have to be indexed by elements A of the n-hypercube P(n) (power set of n = {1, . . . , n}). 0.3. The simple idea of a good functor from rings to sets. In order to formalize the idea that the extended domains and maps (U

, we look at the case U = K. From functoriality, it follows that the spaces K {n} (t,s) are in fact K-algebras, which can easily be identified, (1) in terms of polynomial rings: they are polynomial algebras K[X 1 , . . . , X n ], quotiented by the relations (X i − t i )(X i − s i ) = 0, for i = 1, . . . , n, (2) in terms of tensor products: they are n-fold tensor products of "first order algebras" K (t 1 ,s 1 ) ⊗ . . . ⊗ K (tn,sn) . Likewise, the n-fold anchor is identified as the natural evaluation morphism of the polynomial algebra into K P(n) ∼ = K 2 n , or as the n-fold tensor product of first order anchors. This allows us to find an explicit formula for the n-fold anchor (Theorem 2.24), and for its inverse when (t, s) is regular (Theorem 2.25). These are the essential ingredients to define a small category of K-algebras, the category of nth order anchored tangent algebras talg K,n , by an explicit list of its objects and morphisms: its objects are precisely the anchors Υ n (t,s) : K n (t,s) → K P(n) with (t, s) ∈ K 2n , and there is an explicit list of morphisms. Putting these categories together, for all n ∈ N, we get the category of all anchord tangent algebras, talg K . This setting emphasizes the central rôle of the anchor -its status has been "upgraded": it is not a morphism like the others, but is considered as the central object of our theory. Now, the "simple idea of a good functor from rings to sets" is to view "n times K-differentiable spaces" as functors M from the (small) category talg K,n to the category of anchored sets, assigning to an anchor Υ :

The family f n (t,s) then defines a natural transformation of such functors, and which behaves in all respect like a family of "algebraic scalar extensions" f ⊗ K K n (t,s) , thus achieving our goal. In order to fully justify such a functorial approach to differential calculus, one usually requires in SDG that the model be well-adapted, that is, that we obtain a full and faithful imbedding of a "usual" category of differential calculus into the "functorial" one. We show that, for our setting, this is indeed the case (Theorem 3.5). The proof is much easier than the one of analogs in SDG, because, in essence, the whole setting is designed for such a theorem to hold: it is merely the translation of Theorem 1.7 into a more abstract language. 0.4. Towards higher (super) algebra. The present functorial approach is designed to fit best with further developments: on the one hand, higher algebra (by n-fold iteration, the functor categories are in fact n-categories, and they take values in n-fold groupoids), and super-calculus, on the other hand -see Section 4). 0.5. Appendices on linear algebra and on functor categories. In Appendix A we recall some basic facts about scalar extension V ⊗ K A of K-modules V by Kalgebras A, in Appendix B we develop some linear algebra on "2 n -spaces", and in Appendix C we recall the definition of functor categories (cf. e.g., [CWM, MM92] ), with particular attention to categories of functors from anchored K-algebras to anchored sets.

Acknowledgment. Part of these results should have been presented at the CIMPA spring school "Lie gr which had to be cancelled due to the Covid-19 crisis. We thank the organisers for their work, and we hope that the school will take place soon after the end of this crisis.

Notation. We let n = {1, 2, . . . , n}. Categories are denoted in boldface characters: small letters for small categories, such as talg K , and capital letters for large categories, such as Sets (category of sets). The letter Fn stands for "functor category", so Fn(c, Sets) is the category of (covariant) functors from a (small) category c to Sets (cf. Appendix C). Throughout, K is a commutative base ring with unit 1.

In differential calculus, one usually is mostly interested in the morphisms, that is, in maps of class C n . However, let us first say some words about the objects:

1.1. Locally linear sets, and the anchor. A locally linear set is a pair (U, V ), where V is a K-module, and U ⊂ V a non-empty subset. Without fixing time parameters, we define the set U [1] by (0.3), and the anchor by

When time parameters (t, s) ∈ K 2 are fixed, we define U (t,s) := U (t,s) → U × U is given by restricting the map Υ defined above, i.e., it is given by (0.5). Direct computation shows that Υ (t,s) is invertible iff s − t is invertible in K, with inverse given by (0.6). Note that (U (t,s) , V 2 ) is again a locally linear set, and hence the construction can be iterated, with some new parameter (t 2 , s 2 ), and so on. Explicit formulae, describing this, will be given later (restricted iteration, Def. 1.4).

1.2. The topological setting. In the remainder of this section we assume that K is a good topological ring (a topological ring whose unit group K × is open and dense, and inversion is a continuous map), that all K-modules are topological modules, and that all locally linear sets (U, V ), (U ′ , V ′ ), . . . are open inclusions.

Definition 1.1. We say that f : U → V ′ is of class C K 1 if the slope given by (0.2) extends to a continuous map f [1] :

Remark 1.1. Letting s = 0, this clearly implies that f is of class C 1 K in the sense of [BGN04] or [Be08] . Conversely, the map denoted here by f [1] can be expressed by the one denoted f [1] in loc. cit., and hence the C 1 K -notions used there are equivalent to the one given above. We call the calculus obtained by restricting to s = 0 target calculus. Recall from [BGN04] that, in the real or complex finite dimensional case this definition is equivalent to all usual ones, and in the infinite dimensional locally convex case it is equivalent to Keller's definition of differentiability.

Lemma 1.2. For a map f : U → U ′ , the following are equivalent:

(

As we have already seen, when t−s is invertible in K, then f (t,s) is necessarily given by (0.7). Since its second component is the slope f [1] , existence of f (t,s) , jointly continuous in (x, v; t, s), implies existence of a continuous extension of the slope, whence (2) ⇒ (1). To prove the converse, assume (1). Then the second component of f (t,s) (that is, the slope) admits a continuous extension, by definition. Let us show that the first component of f (t,s) also admits a continuous extension. Indeed, let x 0 := f (x + sv) and

showing that this expression extends continuously for all (t, s) if so does f [1] .

Example 1.1. If f : V → V ′ is linear and continuous, then direct computation using (0.7) shows that

then the computation just mentioned shows that f (t,s) • ι (t,s) = ι (t,s) • f :

In this setting, the usual rules of first order calculus hold (chain rule, product rule, linearity of first differential) -for a systematic exposition we refer to [BGN04, Be08, Be11] . Most important for our purposes is the Chain Rule, which we write in functorial form

This follows easily from Lemma 1.2: for invertible t − s, we have functoriality

, and for general (t, s), it follows "by density". [BGN04, Be11] ). However, since full iteration repeats the procedure for all variables together, the number of variables exploses, and it is hard to get control over the structure of the maps f [n] (see [Be15b] ). To reduce the number of variables, in restricted iteration we consider in each step time variables to be frozen, and take difference quotients only with respect to space variables.

Notation. For each k ∈ N, we denote by an upper index {k} a copy of the procedure {1} that has been defined above. An upper index {i, j} (i < j) indicates a double application of the procedure (first {i}, then {j}), etc. E.g., an upper index n := {1, . . . , n} indicates that we first apply {1}, then {2}, etc., and finally {n}.

To abbreviate, in the sequel, we let (t, s) = (t 1 , . . . , t n ; s 1 , . . . , s n ) ∈ K 2n .

Definition 1.4 (Restricted iteration). A map f : U → U ′ is called of class C K,n if: it is of class C 1 K , and, for all (t 1 , s 1 ) ∈ K 2 , the map f 

{n} (tn,sn) . Theorem 1.5. When K = R or C, and V is a locally convex topological vector space, then the conditions C n K and C K,n are both equivalent to the usual (Keller's) definition of C n -maps.

Proof. As already mentioned, C n K clearly implies C K,n , and equivalence of C n K with Keller's definition has been proved in [BGN04] . On the other hand, C K,n obviously implies Keller's C n -definition, which arises simply by taking (t, s) = (0, . . . , 0) in the C K,n -condition. Thus all three conditions are equivalent.

Remark 1.4. For general K, properties C n K and C K,n cease te be equivalent: in positive characteristic, condition C n K appears to be strictly stronger than C K,n (cf. the proof of the general Taylor formula in [BGN04, Be11] , which really uses full iteration; concerning this item, cf. also [Be13] ). It would be interesting to have a criterion allowing to decide when C n K and C n,K are equivalent. Definition 1.6. For all (t, s) ∈ K 2n , the n-th order anchor of U ⊂ V is defined as follows: for all locally linear sets (U, V ), (U ′ , V ′ ), we consider the map

{2,...,n} (t 2 ,...,tn,s 2 ,...,sn)

inductively gives rise to a map Υ n (t,s) : U n (t,s) → U 2 n which we call the n-fold anchor. Remark 1.5. In order to fully formalize this definition, we need an explicit labelling of the 2 n copies of U in U 2 n . For the moment, this is not needed, and will be taken up later (Def. 2.23). Let us, however, give the result for n = 2: space variables have labels 0, 1, 2, 12 corresponding to the subsets of {1, 2}, so we write

(t 1 ,t 2 ,s 1 ,s 2 ) . Then iteration shows that the linear map Υ is given by the (block) matrix (Kronecker product of two first-order anchors)

so we have four "evaluation points" given by the four lines of the (block) matrix:

(1.6)

The inverse matrix of (1.5) is the Kronecker product of the inverses of the respective first order anchors (when these are invertible): writing d Υ :

For the general case, see Theorem 2.25.

Theorem 1.7. For a map f : U → U ′ , the following are equivalent:

(

The map f n (t,s) depends functorially on f :

. Proof. By induction: for n = 1, this is Lemma 1.2. Assume the claim holds on level n − 1 and apply it to f replaced by f {1} (t 1 ,s 1 ) . From the inductive definitions, it follows readily that the properties are again equivalent on level n. The (higher order) Chain Rule now also follows by induction.

Example 1.2. Using Formula (1.7), let us give explicit formulae for n = 2:

, the first term is in fact an affine combination of values of f at the four evaluation points, whereas the other three terms are "zero-sum combinations" of these values, and hence correspond to "true" difference quotients. In order to state results at arbitrary order, we need some notation:

1.4. Hypercube notation, and formula for higher order slopes.

Definition 1.8. We call n-hypercube the power set P(n) = P({1, . . . , n}). It serves as index set for space variables, which we write in the form v = (v A ) A∈P(n) . Recall that P(n) is a semigroup for union ∪ and intersection ∩, and a group with respect to the symmetric difference

Definition 1.9. For all t, s ∈ K n and A ∈ P(n), we let t ∅ = 1 = s ∅ , and

The proof will be given in Subsection 2.7. For B = ∅, the component is an affine combination of values of f at the 2 n evalation points, and for all other components it is again a "zero sum combination".

1.5. Categories of locally linear sets and C K,n -maps. To summarize, we have defined a category of locally linear sets and their C K,n -morphisms: Definition 1.11. We denote by Llin K,n the category whose objects are pairs (U, V ), where V is a topological K-module and U ⊂ V a non-empty open subset, and morphisms are C K,n -maps f : U → U ′ . (For n = 0, morphisms are continuous maps, and for n = ∞, these are maps that are C K,n for all n ∈ N.) Definition 1.12. For m ≥ n and (t, s) ∈ K 2n , the (n; t, s)-tangent functor is the functor from Llin K,m to Llin K,m−n given by (U, V ) → (U n (t,s) , V n (t,s) ) and f → f n (t,s) . Remark 1.6 (Manifolds). By the usual glueing procedures, one may now define C K,n -manifolds over K, modelled on locally linear sets -since these methods are independent of the particular form of differential calculus, we do not wish to go here into details (see [Be16] for a formulation of such principles, adapted to most general contexts). The (n; t, s)-tangent functor then carries over to manifolds : for every K-smooth manifold M we have a "generalized higher order tangent bundle" M n (t,s) , depending functorially on M, and coming with an anchor map M n (t,s) → M 2 n .

Our next aim is to understand the (n; t, s)-tangent functor as a functor of scalar extension, from K to a ring K n (t,s) , as a generalisation of the algebraic scalar extension functor −⊗ K K n (t,s) . To this end, let's first restrict attention to the case V = U = K. Since K carries canonical structures, so will, by functoriality, the spaces K n (t,s) ; and the anchor map will be a ring morphism, which we can compute explicitly. In this section, we continue to assume that K is a good topological ring.

2.1. Bilinear maps. We can differentiate bilinear continuous maps β : V ×W → Y in the usual way. Since we think of β as a "product", let us write v • w := β(v, w). Concerning Cartesian products, we use the convention from Def. 1.6, so that β

which by an explicit computation using Formula (0.7) is given by

Moreover, by Lemma 1.2, the anchor Υ (t,s) is a "morphism" from the product β

to β × β (the "direct product algebra").

2.2. First order tangent algebras. Now take V = W = K and β(x, y) = xy.

Definition 2.1 (Normalization: source, target, anchor). When V = W = K, the anchor is a map Υ : K 2 × K 2 → K 2 × K 2 , and when (t, s) ∈ K 2 is fixed, then we have a (linear) map Υ (t,s) : K 2 → K 2 . The standard basis in the domain of Υ (t,s) will be denoted by e ∅ := (1, 0), e 1 := (0, 1), and the standard basis in its range by E ∅ := (1, 0) and E 1 := (0, 1), so v = v ∅ e ∅ + v 1 e 1 . We make the choice to associate E ∅ with "source" and E 1 with "target", that is, we let

In other terms,

With respect to ordered bases, with "usual" ordering (e ∅ , e 1 ), (E ∅ , E 1 ), these linear maps correspond to the matrices

As usual, dual bases are denoted by ((e ∅ ) * , (e 1 ) * ), resp., ((E ∅ ) * , (E 1 ) * ). Thus

The choice appearing in this definition should be seen as an additional structure, akin to the choice of an "orientation".

Lemma 2.2. For all (t, s) ∈ K 2 , the map induced by m :

Proof. The fact that K {1} (t,s) is an associative commutative algebra follows by functoriality from the fact that so is K ([Be08, BeS14]), but it can also be obtained as a consequence of the following computational argument: we compare Formula (2.1), relative to e ∅ , e 1 , with the product in K[X]/(X − t)(X − s), relative to the basis [1], [X]. In both cases, 1 = e ∅ is a neutral element, and we have

showing that, with respect to these bases, the formulae describing the product in

is an algebra morphism into the direct product algebra. It is an isomorphism iff s − t ∈ K × . Source α and target β are characters (morphisms K (t,s) → K).

(P (s), P (t))), the evaluation morphism of the quotient algebra at (s, t).

Theorem 2.5 (Structure of the first order tangent algebra K {1} (t,s) ).

(1) The ideals ker(α) and ker(β) satisfy ker(α) · ker(β) = 0.

(3) The map

Proof.

(1) ker(α) = K(e − s) and ker(β) = K(e − t), and, by the defining relation of the algebra, (e − s)(

(3) Note that κ(1) = 1 + 1 − 1 = 1 and κ(e) = s + t − e, whence κ(κ(e)) = s + t − (s + t − e) = e, so κ 2 = id. Next,

To prove that κ is an automorphism, since κ(1) = 1, it suffices to show that κ(e 2 ) = κ(e) 2 . Indeed, κ(e) 2 = (t + s) 2 − 2(t + s)e + e 2 = (t + s) 2 − ts − (t + s)e and κ(e 2 ) = κ(−ts

(4) If v is invertible, then applying the morphisms α and β, it follows that both α(v) and β(v) are invertible. Conversely, the last formula from (3) shows that under this condition v has an inverse given by v −1 as in the claim.

Remark 2.1. Under Υ, the corresponding "fundamental relation" in K 2 reads

Definition 2.6. We define the following elements in K

Remark 2.2. The pair (1, a) is a basis of K

(t,s) , and so is (1, b). When (t, s) is regular, then (a, b) also is a basis, but when t = s, we have a = b. In the symmetric case (t = −s), we have e = j. This situation is represented by Figure 1 (where t = 0.4 = −s). The arrows indicate the direction of the kernel of the projection α, resp. β, cf. the following lemma.

Lemma 2.7. The elements a, b, j satisfy the relations

and everything follows more or less directly from this. 

By this definition, the elementsẼ 0 ,Ẽ 1 are orthogonal idempotents, and the matrix of Υ with respect to theẼ-basis in its codomain and the E-basis in its domain is the identity matrix. The base change matrix from the e-basis to the theẼ-basis is the "usual" matrix of Υ: indeed,

2.3. The category of first order anchored tangent algebras. Our aim is to define a (small) category of tangent algebras: so, we have to say, what are the objects, and what are the morphisms ?

Definition 2.9. Objects of the category of anchored first order K-tangent algebras are all anchor maps

(t,s) → K 2 for (t, s) ∈ K 2 , as well as the "trivial" anchor id : K → K. Thus, objects can be considered as triples (K (t,s) , K 2 , Υ (t,s) ), resp. (K, K, id) (domain and codomain of the anchor, and its "formula"). For every anchored tangent algebra Υ, there is an opposite anchored tangent algebra Υ ′ := τ • Υ, where τ (x, y) = (y, x) is the exchange automorphism of K 2 .

Definition 2.10. A pair of algebra morphisms

is called anchor-compatible if

We say that Φ is direct if the pair (Φ, id K 2 ) is anchor-compatible, and indirect if the

We shall require that morphisms are anchor-compatible, either direct or indirect. When (t, s) is regular and Φ is direct, then necessarily Φ = (Υ (t,s) ) −1 • Υ (t ′ ,s ′ ) , so there is exactly one direct morphism from K (t ′ ,s ′ ) to K (t,s) . However, when (t, s) is singular, then we cannot always "classify" direct morphisms, and therefore we will proceed to give an explicit list of morphisms that are admitted in our small category. The case of indirect morphisms is reduced to the case of direct ones:

Lemma 2.11. For every (t, s) ∈ K 2 , the automorphism κ : K (t,s) → K (t,s) defined in Theorem 2.5 is indirect. There is a bijection between the sets of direct and indirect morphisms.

Proof. We have seen in Theorem 2.5 that α • κ = β and β • κ = α, so J is indirect. Clearly, Φ is direct iff Φ • κ is indirect, whence the claimed bijection.

Remark 2.3. The map κ can be interpreted as the inversion map of the groupoid U (t,s) , see Lemma 4.1. As such, it exchanges source and target of the groupoid.

Proof. Any affine map φ induces an algebra-morphism K[X] → K[X], P → P • φ. By our assumption on φ, this map passes to the quotient and thus defines a morphism of quotient algebras. It is direct, Υ(φ * ([P ])) = Υ([P • φ]) = (P (φ(t)), P (φ(s))) = (P (t ′ ), P (s ′ )) = Υ([P ]).

Definition 2.13. A morphism defined by the preceding lemma is called of affine type. We distinguish the following cases:

(1) morphism of translation type: (t ′ , s ′ ) = (t + µ, s + µ), φ(x) = x − µ,

(2) morphism of scaling type: (t, s) = (λt ′ , λs ′ ), φ(x) = λx,

As algebra morphism, it is the same as κ; but κ acts from the label (t, s) to itself, and thus is considered as indirect, whereas φ * acts from the label to the reversed label, and thus is considered as direct.

Remark 2.5. By the lemma, each tangent algebra K (t,s) is isomorphic to an algebra K (r,0) with r belonging to a system of representatives of the K × -orbits in K. Thus, when K is a field, there are only two isomorphism classes, corresponding to r = 1 (direct product algebra K 2 ) and r = 0 (dual numbers T K).

Definition 2.14. Morphisms of the category of anchored first order K-tangent algebras are all morphisms of affine type (direct), as well as their compositions with κ (indirect), and the canonical injection of the trivial anchor morphism, given by

Remark 2.6. It may turn out to be useful, or necessary, to add more morphisms to this list when developing the theory further (for instance, for the moment we refrain from adding α and β to our list, although they give rise to anchor-compatible morphisms). In any case, the list should be explicit, in order to control the behaviour of calculus with respect to morphisms, see the following proposition. Note also that the anchor "morphism" does not appear in our list of morphisms, since it is considered as "object"! Proposition 2.15. Let U ⊂ V be open, (t, s) ∈ K 2 , λ, µ ∈ K. Then the maps

are well-defined, and, for every map f : U → U ′ of class C 1 K , they commute with tangent maps, in the sense that

Proof. Direct computation using (0.3) and (0.7).

2.4. Higher order tangent algebras. Now we apply restricted iteration to the construction of the tangent algebra. Recall that K

which we call "of k-th generation". Now, each iteration step doubles the Kdimension of our K-algebra, so that in degree n we get an algebra of dimension 2 n . We will index basis elements by the hypercube P(n) of n = {1, . . . , n}, which is in keeping with notation for restricted iteration introduced above.

Definition 2.16. For any K-module V , the canonical isomorphism

(tn,sn) . The canonical basis e ∅ , e k in each factor K {k} (t k ,s k ) gives rise to a canonical basis in the K-module K n (t,s) , indexed by elements A ∈ P(n), and given by e A = ⊗ n k=1 e {k}∩A . Lemma 2.17. Applying restricted iteration n-times, with parameter (t, s) ∈ K 2n , to the product map of K, we obtain an algebra structure on K n (t,s) , isomorphic to (1) the n-fold tensor product of algebras K

equivalently, to the quotient algebra

The canonical basis (e A ) A∈P(n) of the tensor product corresponds to the canonical K-basis of the polynomial algebra,

If V is a (topological) K-module, then applying n-fold restricted iteration to the scalar multiplication map K × V → V gives the scalar action of the ring K n (t,s) on the algebraic scalar extension V ⊗ K K n (t,s) . Proof. Formula (2.1) shows that, if µ : A × A → A, (x, y) → x • y is any (finitedimensional) algebra (associative or not), then µ {k} (t k ,s k ) is, as algebra, isomorphic to the tensor product of algebras A ⊗ K {k} (t k ,s k ) , that is, the algebraic scalar extension of the algebra A by the ring K {k} (t k ,s k ) . If A is an associative commutative algebra, then this algebraic scalar extension is also isomorphic to A[X k ]/((X k − t k )(X k − s k )). Applying this remark n times, starting with A = K, the lemma follows. Similarly for the bilinear salar multiplication map K × V → V .

Note that, since e ∅ is the unit element in K (t,s) , we have, e.g., e {1,2} = e 1 ⊗ e 2 = (e 1 ⊗ e ∅ ) · (e ∅ ⊗ e 2 ) = e 1 · e 2 , by identifying x ⊗ 1 with x and 1 ⊗ y with y. And so on: we may write

We also define the a-basis, resp. b-basis, of K n (t,s) by a

again indexed by A ∈ P(n). If 2 is invertible in K, we define the j-basis by

Lemma 2.19. We have the product rules, for all A, B ∈ P(n),

Proof. The first product rule comes from a 2 k = (s k − t k )a k and a i a j = a {i,j} when i = j, and similarly for the other two, with j 2 k = 1 4 (t k − s k ) 2 = 1 4 (t k − s k ) 2 j ∅ . Corollary 2.20. When t = 0 and s = (1, . . . , 1), then K n (0,1) with a-basis is the semigroup algebra of (P(n), ∪), and when s = −t = (1, . . . , 1), then K n (1,−1) with j-basis is the group algebra of (P(n), ∆).

Proof. In the first case, a A a B = a A∪B , which are the defining relations for the semigroup algebra K[P(n)] (with ∪), and in the second, j A j B = j A∆B , which defines the group algebra of K[P(n)] with ∆.

2.5. The n-fold anchor and its inverse. By restricted iteration, the n-fold anchor, taken for U = V = K, has been defined, in Definition 1.6, to be a (linear) map Υ n (t,s) : K n (t,s) → K 2 n . By induction, it will be an algebra morphism, but in order to put hands on it, we now have to fix a basis in K 2 n , in such a way that it is compatible with iteration procedures. Forgetting the algebra structure, we get a linear space, called hypercubic space, see Appendix B.

Definition 2.21. For any subset N ⊂ N of finite cardinal n, the 2 n -fold direct product algebra of K, in standard order, is the algebra

, and product defined on basis elements by

When N is fixed, the notation E A suffices. In particular, this notation is in keeping with our preceding one for the standard basis in K 2 :

{1} . We simplify notation by writing, e.g., k instead of {k}, and ij instead of {i, j}, etc.

Lemma 2.22. The algebra K P(N ) is an associative algebra canonically isomorphic to the algebra of functions from P(N) to K with pointwise product, with E N A corresponding to the function having value 1 at A and 0 else. For two disjoint finite subsets N 1 , N 2 ⊂ N, we have an isomorphism of algebras

For the first statement, just recall that the algebra K S of K-valued functions on a finite set S has canonical basis (1 x ) x∈S , where 1 x (y) = δ x,y , so E A corresponds to the element 1 A . For the second statement, recall that for finite sets S, T ,

defines an isomorphism of algebras sending 1 x ⊗ 1 y to 1 (x,y) , and combine this with the canonical bijection

E.g., using the lemma, for n = 2, the algebra K P({1,2}) has K-basis

In general, the neutral element of K P(N ) is the function that is 1 everywhere, that is, the sum of all basis elements:

Definition 2.23. The n-fold anchor is the tensor product of n copies of the first order anchor, with respect to bases defined as above: it is the algebra morphism

is a copy of the first order anchor. Thus, by definition,

Recall Formula (1.5) for the matrix of the second order anchor. Note that, when s 1 = 1 = s 2 , then this matrix is a symmetric matrix, whereas for t 1 = 1 = t 2 , this is not the case. Using notation introduced above, we generalize:

Theorem 2.24. Fix n ∈ N, and (t, s) ∈ K 2n . With respect to the bases (e A ) A∈P(n) in its domain and (E A ) A∈P(n) in its range, the n-fold anchor is given by

In other terms, it is characterized by the following equivalent conditions:

the matrix of Υ with respect to these bases has coefficients

In particular, in the symmetric case s = −t, we have Υ (B,A) = (−1) |A∩B| s A :

Proof. This is the special case of Theorem B.3 for a = 1 = c, b = s, d = t.

Next, to compute the inverse of the anchor, in the regular case, recall Formula (1.7) concerning the case n = 2. This generalizes as follows:

Theorem 2.25. Fix n ∈ N and (t, s) ∈ K 2n . Recall the notation (t − s) n = n k=1 (t k − s k ). The anchor map Υ = Υ n (t,s) is invertible if, and only if, t k − s k is invertible for all k = 1, . . . , n, and then its inverse map is given by the formula

Equivalently,

In particular, in case s = −t, we get (using

Proof. This is a special case of Theorem B.4.

2.6. The idempotent basis. Just like in case n = 1, we define, for regular (t, s) the basis (Ẽ A ) A∈P(n) of K n (t,s) byẼ A := Υ −1 (E A ). Thus, by definition, the matrix M Ẽ E (Υ) of Υ for these bases is the identity matrix, whereas Υ := M E e (Υ) is its "usual" matrix, computed in Theorem 2.24. It follows that the base change matrix fromẼ to e is given by

Thus the idempotent basis is given by base change with the coeffiencts of the matrix Υ −1 computed in Theorem 2.25.

Remark 2.7. For A ∈ P(n), the linear form E * A : K P(n) → K is the A-projection, which is a character, i.e., an algebra morphism into the base ring. Thus

s) → K also is a character (generalising α, β from n = 1). Seen this way, the anchor Υ can be considered as an analog of the Fourier transform for the algebra K n (t,s) . 2.7. The n-th order restriced slope map. Having established the explicit formulae for Υ and Υ −1 , we can prove the already anounced formula from Theorem

with the 2 n evaluation points given by

2.8. The category of anchored n-th order tangent algebras. We generalize the concepts defined in 2.3 to the case of any order n ∈ N:

Definition 2.26. Objects of the category talg K,n of anchored n-th order K-tangent algebras are all anchor morphisms Υ n (t,s) : K n (t,s) → K P(n) for (t, s) ∈ K 2n , along with the trivial anchor morphism id : K → K.

It is called anchor preserving if n = n ′ and Ψ = id.

Note that, when (t ′ , s ′ ) is regular, then this condition is automatically satisfied, by taking Ψ = Υ n (t,s) • Φ • (Υ n ′ (t ′ ,s ′ ) ) −1 . From Lemma 2.12 we directly get, by taking tensor products, the following anchor preserving morphisms:

Definition 2.28. Let φ i : K → K affine maps, i = 1, . . . , n, and φ = φ 1 × . . . × φ n : K n → K n . Then φ induces an anchor-preserving morphism, called of affine type, φ * : K n (t ′ ,s ′ ) → K n (t,s) . In particular, we say that φ * is (1) of translation type if φ * is the tensor product of morphisms of translation type, parametrized by µ = (µ 1 , . . . , µ n ) ∈ K n , t ′ i = t i + µ, s ′ i = s i + µ, (2) of scaling type if φ * is the tensor product of morphisms of scaling type, parametrized by λ = (λ 1 , . . . , λ n ) ∈ K n , t ′ i = λ i t t , s ′ i = λ i s i . Remark 2.8. When defined, the composition of morphisms of affine type is again of affine type. In more elaborated language, we get a small category which is the action category associated to the affine monoid.

On the most basic level of the theory, it would be enough to consider only anchorpreserving morphisms of affine type. However, certain other anchor-compatible morphisms will soon become relevant, which are the higher order analogs of the indirect morphisms defined for n = 1.

Definition 2.29. We denote by B n the hyperoctahedral group, which is defined as follows: it is the group acting on P(n) generated by (1) the symmetric group S n acting on P(n) in the natural way by σ.A = σ(A),

(2) the action of the abelian group (P(n), ∆) ∼ = (Z/(2)) n by translations on itself:

Remark 2.9. The group B n is a semidirect product of S n with the normal subgroup (P(n), ∆), whence is of cardinal n!2 n . It is a Coxeter group of type B n . It has a non-trivial center: the element n acts by the complement map A → n∆A = A c , which belongs to the center of B n .

Definition 2.30. We call admissible automorphism of K P(n) the algebra automorphisms induced by elements of τ ∈ B n , via Ψ(E A ) = E τ (A) . In particular, (1) Ψ is of σ-permutation type if τ = σ belongs to the subgroup S n of B n (in particular, we say Ψ is a flip if τ is a transposition), (2) Ψ is of B-inversion type if τ (A) = B∆A belongs to the subgroup (P(n, ∆)) of B n . In particular, we say that Ψ is an elementary inversion if B is a singleton, B = {i}, and Ψ is the central inversion, if B = n.

We use the same terminology for the corresponding pairs of anchor-compatible morphisms, having the corresponding type as base-map Ψ.

Lemma 2.31. The action of the hyperoctahedral group on K P(n) lifts to an action by anchor compatible isomorphisms on the objects of the category of anchored n-th order tangent algebras. More precisely, for every B ∈ P(n), there is an automorphism of B-inversion type Φ : K n (t,s) → K n (t,s) , namely

(tensor product of inversions for each k ∈ B), and for every σ ∈ S n , there is a morphism of σ-permutation type (with t ′ = σ(t), s ′ = σ(s)), namely

. Proof. Concerning inversions, this follows by taking tensor products of the first order inversions, as in the definition of κ B .

The symmetric group S n acts canonically on K[X 1 , . . . , X n ]. This action passes to the quotient and defines morphisms

Definition 2.32. Morphisms of the category of anchored n-th order K-tangent algebras are all morphisms of affine type (direct), as well as their compositions with all morphisms coming from the action of the hyperoctahedral group, along with the canonical injection of the trivial anchor morphism, given by the canonical maps

In particular, the central inversion belongs to the central element of B n (exchanging a i and b i for all i, and acting by the complement map on K P(n) ).

The higher order analog of Proposition 2.15 holds: morphisms of scaling and translation type induce maps that are compatible with tangent maps f n (t,s) : Proposition 2.33. Let V be a K-module and φ : K n (t ′ ,s ′ ) → K n (t,s) be a morphism of affine type, or permutation type, or of inversion type. Then, for all open subsets U ⊂ V , the (linear) map Φ = id ⊗ φ :

Concerning morphisms of affine type, and of inversion type, this follows by iteration from Proposition 2.15. Concerning morphisms of permutation type, for n = 2 and φ the "flip" map corresponding to the transposition (12), the claim amounts to "Schwarz's Theorem", which holds in topological calculus (see [BGN04, Be11] ). By induction, the claim then follows for all morphisms of permutation type.

Remark 2.10. Let us call diagonal case the case where t i = t and s i = s for all i = 1, . . . , n. In this case, the permutation group S n acts by automorphisms on K n (t,s) . This is in particular the case for t = 0 = s, giving the canonical action of S n on higher order tangent bundles, extensively used in [Be08] .

In this section, K is an abstract (i.e., without topology) unital commutative ring.

3.1. The category talg K of anchored K-tangent algebras. So far, we have defined for each n ∈ N a category of n-th order anchored tangent algebras. Now, we shall put them together to define a category comprising all of them. The following diagram may help to "visualize" the objects of the category:

On the left, we have the "analytic leg" (algebras of type K n (t,s) ), on the right, the "synthetic leg" (algebras of type K P(n) ), and the arrow from left to right represents the anchor. The left hand side takes account of the "analytic" (local, chart dependent) aspects of calculus, and the right of "synthetic" aspects (independent of the language of charts and local affinizations).

Definition 3.1. Objects of the category talg K are all objects of talg K,n , as n ranges over N. Morphisms of talg K are:

(1) for fixed n, all morphisms of talg K,n , (2) for two objects

Remark 3.1. The category talg K is stable under taking tensor products: if Υ n (t,s) and Υ k (t ′ ,s ′ ) are objects, then Υ n (t,s) ⊗ Υ k (t ′ ,s ′ ) is naturally identified with the anchor at order n + k given by parameters (t, t ′ ; s, s ′ ) ∈ K 2(n+k) .

3.2. The good functor category. Recall from Appendix C definition and some basic facts on functor categories.

Definition 3.2. The functor category Fn(talg K,n , Sets 2 ) is called the category of (K, n)-space laws: its objects are called (K, n)-space laws M, and its morphisms f are called (K, n)-mapping laws. A (K, ∞)-space law (or just: K-space law) is a functor in Fn(talg K , Sets 2 ), and a (K, ∞)-mapping law is a morphism in this category.

We demand explicitly that the category of locally linear sets together with their (restrictions) of linear maps be considered as subcategory of this functor category in the way given by the following Example 3.1.

Example 3.1 (Locally linear sets). Let (U, V ) be a locally linear set, and Υ : A → A ′ an object of talg K . We define V A := V ⊗ K A (algebraic scalar extension), and

(1) for A = K n (t,s) , the set U A := U n (t,s) is defined as in Definition 1.4, (2) for A ′ = K P(n) , we let U K P(n) := × A∈P(n) U ⊂ V K P(n) , (3) the anchor Υ n (t,s) : U n (t,s) → U K P(n) then is defined by the algebraic formula from Theorem 2.24.

We have seen in Subsection 2.8 that morphisms of tangent algebras induce maps on the level of sets. For a K-linear map f : V → V ′ we define f A = f ⊗ id A to be its algebraic scalar extension, restricted to U A . In this way, in virtue of Proposition 2.15, we get a functor U from talg K to Sets 2 , and a natural transformation f from U to U ′ .

Remark 3.2. Note that the preceding example cannot be generalized to "arbitrary" functor categories from K-algebras to sets, since for general algebras A there is no good definition of domain U A . Already for K = R and A = C this fails, in general (the "complexification" of a real set cannot be defined in a purely algebraic way).

Remark 3.3 (Category of extensions of a given domain). For any locally linear set (U, V ), the collection Υ n (t,s) : U n (t,s) → U K P(n) of all scalar extensions (for n ∈ N, (t, s) ∈ K 2n ), is a category (image of the functor U ), with morphisms id U ⊗ Ψ where Ψ is a morphism of talg K . Call it the category of tangent extensions of U.

Example 3.2 (The scaloid). In the preceding remark, take (U, V ) = (0, 0) =: 0, the zero set in the zero-module. The category of tangent extensions of 0 is called the scaloid and denoted by scal K : since 0 ⊗ K Υ = 0, the underlying map of each object is a trivial map on a trivial K-module 0; however, each of these maps and sets 0 still carries a label -for instance, K n (t,s) ⊗ 0 = 0 as set, but still keeping its label (n; t, s). Therefore the scaloid scal K is the small category of labels belonging to objects and morphisms of talg K , and the functor 0 can be seen as the forgetful functor * : talg K → scal K associating to each object Υ n (t,s) from talg K its label (t, s; n), and to each morphism a label characterizing it. Note also that 0 is a terminal object in the category of C K,∞ -laws: every law M admits a unique morphism f : M → 0, attaching to M A the label of A.

Example 3.3 (The line). Next, consider the case U = K (the line). Then, since A ⊗ K = A, id ⊗ Υ = Υ, the category of tangent extensions of K gives us back the category talg K . Since K carries an associative bilinear product, so do the objects of talg K .

Example 3.4 (Scalar function laws). A scalar law is a natural transformation f : M → K. Such laws can be turned into a ring O M by letting

Thus M gives rise to a functor O M : A → (O M ) A , which is contravariant, that is, a functor from (talg K ) opp to Sets 2 (and to Ring 2 , the category of anchored rings).

3.3. Full imbedding of topological calculus into functor categories. Let's return to the topological case, and recall from Definition 1.11 the category Llin K,n of locally linear sets with C K,n -maps as morphisms.

Theorem 3.3. Let K be a good topological ring. There is an imbedding of the category Llin K,n into the functor categories Fn(talg K,n , Sets 2 ), resp. Fn(talg K , Sets 2 ), given by associating to an object (U, V ) the functor U defined in Example 3.1, and to a C K,n -map f : U → U ′ the natural transformation s) . This imbedding has the following properties:

(1) functors U respect direct products, (2) for locally linear spaces and linear maps, functors are given by algebraic scalar extension, as in Example 3.1, (3) functors take values in topological spaces, morphisms are continuous and jointly continuous with respect to parameters.

Proof. First of all, U is indeed a functor: all morphisms between tangent algebras belonging to our explicit list do indeed induce set-maps, in a functorial way. Next, C K,n -maps indeed induce natural transformations (for this issue, it was necessary to limit morphisms between algebras to an explicit list: there is no general theorem ensuring that these natural transformations commute with maps induced by "arbitrary" algebra morphisms). Finally, the underlying functor gives us back the original objects and morphisms, U K = U, f K = f , so we have indeed an imbedding. Properties (1), (2), (3), which we re-state formally in the following definition, are clearly satisfied by the constructions that have been described in the preceding sections.

Definition 3.4. Let K be a good topological ring. A functor category Fn(c K , Sets 2 ) may have (or not) the following properties:

(1) it respects direct products (cf. Definition C.3),

(2) it satisfies the axiom of algebraic scalar extension stated in (2) above, (3) it is a continuous functor category: the scaloid scal K (cf. example 3.2) is a topological space, functors M take values in topological spaces, morphisms f take values in continuous set-maps f A that are jointly continuous in the scaloid, i.e.: for all locally linear sets (U, V ) and morphisms f , the following map is continuous (where V n (t,s) ∼ = V 2 n via the e-basis, and likewise for W 2 n ):

. We denote by a superscript "top" the functor categories satisfying (3).

Theorem 3.5. The imbedding from Theorem 3.3 defines a full imbedding of the category Llin K,n into Fn top (talg K,n , Sets 2 ) and a full imbedding of the category Llin K,∞ into Fn top (talg K , Sets 2 ).

Proof. Let f : U → U ′ be a continuous morphism of laws. We have to show that f is induced by a map of class C K,n ; more precisely, we show that the underlying map f = f K : U K = U → U ′ = (U ′ ) K is of class C K,n , and that it induces f . By definition of the functor category, the anchor Υ : K n (t,s) → K P(n) induces a map Υ U : U K n (t,s) → U K P(n) . This anchor map is indeed the same as the one considered in the preceding chapters, since Υ U is the restriction to U K n (t,s) of id V ⊗ Υ K . Since f is a natural transformation, it commutes with the anchor in the sense that Υ • f K n (t,s) = f K P(n) • Υ. By the continuity property (3) from Definition 3.4, these maps are continuous and jointly continuous also in (t, s), whence satisfy the condition from Theorem 1.7, showing that the base map f = f K is of class C K,n , with the components of f given the construction from topological differential calculus; thus f K induces the natural transformation f .

Remark 3.4 (Manifold laws). As already mentioned (Remark 1.6), one can define C K,n -manifolds over a good topological ring K (n ∈ N ∪ {∞}). Then every such manifold defines a (K, n)-space law M , and every C K,n -map f : M → M ′ a (K, n)law f , such that properties (1), (2), (3) from Definition 3.4 are satisfied (where (3) applies, by definition, only to locally linear sets, that is, to chart domains of the manifold). Then the preceding arguments apply on each chart domain, showing that morphisms between manifold laws are induced by smooth maps, and hence we get a full imbedding of the category of C K,n -manifolds into Fn top (talg K,n , Sets 2 ).

Remark 3.5 (Infinitesimal vs. local and global). A remark on comparison with the case of Weil laws as defined in [Be14] is in order here. Taking for c K the category of Weil algebras, we get a formally quite similar theory, leading to an imbedding of topological differential calculus into the category of Weil spaces and their morphisms, which, essentially, is the so-called the Dubuc topos, [Du79] . As shown by Dubuc, loc. cit., the Dubuc topos contains morphisms that are not induced by smooth maps. Roughly speaking, the reason for this is that Weil algebras are by nature infinitesimal objects (because of the nilpotency condition), and the link with the local and global theory is not encoded on the algebra side, whereas in our approach it is. To ensure fullness, this link has to be encoded in some wayeither by using more sophisticated topoi (see [MR91] ), or (as done here) by taking account of anchors and allowing regular parameters (t, s), thus taking account of non-infinitesimal mathematics.

With Theorem 3.5, we have shown that the functor category Fn(talg K , Sets 2 ) can be considered as a "well adapted model" for general differential calculus. In subsequent work, we will develop the theory further: on the one hand, comparing with SDG, we will investigate categorical questions, on the other hand, by enriching the structure of our category of algebras, the theory naturally offers links with higher algebra and with super-calculus. Let us briefly describe some basic ingredients.

4.1. Groupoids, and higher algebra. In topological calculus, the extended domains U n (t,s) carry a natural structure of n-fold groupoid (see [Be15a, Be15b, Be17] , for the case of target calculus). Indeed, this follows by iteration from 

Proof. The defining properties of a groupoid are checked by direct computation (cf. loc. cit.).

In order to implement this aspect in functorial calculus, we observe that the structure maps of this groupoid are induced by algebra morphisms:

Theorem 4.2. (K (t,s) , α, β, K, * ) is a groupoid object in the category of algebras, i.e., the groupoid structure from the preceding lemma is given by algebras and algebra morphisms: source and target projections and unit section are algebra morphisms, the subset

is a subalgebra of the direct product algebra K (t,s) × K (t,s) , the groupoid law * :

is a morphism of algebras, and inversion J is an algebra automorphism.

Proof. Source α and target β are algebra morphisms, hence the subset K (t,s) × α,β K (t,s) of K (t,s) × K (t,s) is stable under addition + and product ·, and it contains the unit (1, 1), so it is a (unital) subalgebra. To see that * is a morphism, we use the "fundamental relation": for all v, w ∈ K (t,s) ,

Thus if α(x) = β(z) and α(x ′ ) = β(z ′ ),

Finally, we have already seen that κ is an algebra automorphism (Theorem 2.5).

We will discuss in subsequent work how to implement this groupoid aspect into a functorial calculus. In this context, it will be important to have an analog of the sequence T T M → T M × M T M → T M playing an important rôle for second and higher order tangent bundles (see [Be08, Be14] ). This analog reads as follows:

Theorem 4.3. The product map

is an algebra morphism, which factorizes over the groupoid law from the preceding theorem via µ = * • ψ, where ψ is the morphism

Proof. For any commutative algebra A, the product map A⊗A → A is a morphism.

by the "fundamental relation" from Theorem 2.5.

Iterating these morphisms and constructions, higher algebra naturally enters into the picture (cf. loc. cit).

A main motivation to develop the present functorial approach is that it is perfectly in keeping with known functorial approaches to super-calculus. For this purpose, it was necessary to introduce the source parameter s into differential calculus: the transition from usual to graded calculus is most natural in case t = −s (symmetric calculus), since only in this case the groupoid inversion κ (which becomes the grading automorphism of superalgebras) is given by the simple formula κ(v 0 + ev 1 ) = v 0 − ev 1 (cf. Theorem 2.5). In this context, tensor products have to be replaced by graded tensor products.

Coalgebras, and duality. The reader may have noticed that our algebras carry more structure than has been used so far: they are coalgebras, and this structure should play a significant rôle in the theory, related to duality aspects, and, possibly, to the preceding item. 4.4. Full iteration, and simplicial calculus. As mentioned in Remark 1.3, full iteration leads to higher order "tangent maps" f {1,...,n} having a very complicated structure. In principle, this structure can also be interpreted in terms of higher groupoids (see [Be15b] ). In this setting, the analog of the tangent algebra category talg K will be some small higher order category, whose structure remains to be understood yet. However, restricting again variables to certain subspaces, one can obtain a suffiently simple calculus, called simplicial in [Be13] , and corresponding to the classical concept of divided differences. It is certainly possible to put this simplicial calculus into a categorical form, essentially as done in this work for restricted iteration. The advantage should be a better compatibility of calculus with algebra in positive characteristic, but the drawback is that the close link with the tensor product, featured in the present approach, gets lost: iteration is no longer given by subsequent tensor products. Corollary A.3. Let V, W be K-modules, and let X A := Hom A (W A , V A ) for a Kalgebra A. Then A → X A is functorial in the sense that, for every algebra morphism φ : A → B, there is a canonical morphism X φ :

Proposition A.4. Assume A is commutative. Then, for all K-modules V, W , we have a canonical isomorphism

and likewise for iterated finite tensor products

In one direction, the morphism is given by

in the other, using commutativity of A,

Clearly, both are inverse to each other.

Concerning polynomials and forms, see Appendix A of [BGN04] . Polynomials do admit scalar extensions (see also [Ro63, Lo75] ): Since B is not required to be symmetric, f is not uniquely determined by B. 

By the preceding proposition, this map can be considered as A-linear

which is a polynomial over A extending B. For general m, the claim follows by induction.

Remark A.2. We may define, as usual: a polynomial map between V and W is a finite sum of homogeneous polynomial maps. Then the theorem can be stated for polynomial maps; however, we have to mention explicitly the degrees of the components, and require that the functor B → B A be applied in each degree.

In other terms,

Proof. When the cardinality n of N is equal to one, then the claim is true, directly by definition of the matrix coefficients. For n = 2, the matrix of f 1 ⊗ f 2 is

in keeping with the claim. In the general case, we expand the expression

by distributivity: we get a sum of 4 n terms, which correspond exactly to the 4 n terms in the last formula of the claim. (E.g., for n = 2, there are 16 terms, corresponding to expanding the product (a 1 +b 1 +c 1 +d 1 )(a 2 +b 2 +c 2 +d 2 ) by distributivity, giving the 16 matrix coefficients shown above. The first column contains the 4 terms from expanding (a 1 + c 1 )(a 2 + c 2 ), etc.)

To memorise the formula: for 2 × 2-matrices and indices, the correspondence is

Next, we give a formula for the inverse of f , when its determinant is invertible. From well-known properties of the Kronecker product it follows that

whence the first statement of the following theorem:

Theorem B.4. Let N and f = ⊗ n i=1 f i be as in the preceding theorem. Then f is invertible if, and only if, all f i are invertible, and then its inverse is given by the matrix coeffients, for (A, B) ∈ P(N) 2 (recall A∆B is the symmetric difference)

Proof. Assume each f i is invertible. For n = 1, N = {k}, the inverse is

For n = 2, the matrix of the inverse is the Kronecker product of the inverses 1 det(f 1 ) det(f 2 )

which is in keeping with the formula announced in the claim. To put this computation into a conceptual framework, note that the inverse in (B.1) is obtained by first taking the adjugate matrix, and then dividing by the determinant. The adjugate X ♯ of a 2 × 2-matrix X, in turn, is given by i.e., J sends E ∅ → E 1 , E 1 → −E ∅ (so X ♯ is the adjoint of X with respect to the canonical symplectic form on K 2 ; call it "symplectic adjoint"). For each 2×2-matrix M let M n = ⊗ n i=1 M : K P(n) → K P(n) . Then, for the matrices I, J, K defined by (B.2), the effect on E A is

. Using this, we compute

which together with |A| + |B| ≡ |A∆B| mod (2), so (−1) |A| (−1) |B| = (−1) |A∆B| , gives us the adjugate and the claim.

Remark B.2. In the same way, it follows that, even if f is not invertible, we have

Appendix C. Functor categories C.1. Categories of anchored sets or algebras. A category C is a pair (C obj , C mor ) of collections of objects and of morphisms, together with certain structure operations, see [CWM, MM92] .

Definition C.1. We denote by Sets the (large) category of sets and set-maps, and (following notation from [MM92] , p. 25) by Sets 2 the (large) category of anchored sets, that is, objects (M, Υ, M ′ ) are maps Υ : M → M ′ (we call Υ the anchor map, and often denote, by abuse of notation, the triple just by Υ or by M, according to context), and morphisms are anchor-compatible pairs of maps Φ : M → N, Φ ′ : M ′ → N ′ , i.e., they commute with anchors:

We fix once and for all a commutative unital ring K, and denote by Alg K the (large) category of all (associative, unital) K-algebras, and by Alg 2 K the category all anchored (associative, unital) K-algebras, that is, objects (A, Υ, A ′ ) are algebra morphisms Υ : A → A ′ (called anchor), and morphisms are pairs of anchor-compatible algebra morphisms. The identity morphism id K : K → K is called the trivial anchor. It is an initial object in Alg 2 K . We denote by suitable supercripts certain subcategories of Alg K such as: all commutative associative algebras; in particular, we denote by Alg fin K the subcategory of all K-algebras that are commutative, free and finite-dimensional as K-modules. The corresponding subcategories of anchored algebras are then defined in the obvious way.

C.2. Functor categories. A functor F from a category C to a category D is a pair (F obj , F mor ) of arrows (C obj → D obj , C mor → D mor ), all of this satisfying the usual axioms (see [CWM, MM92] ). A natural transformation τ between two functors S, T : C → B is an arrow τ : C obj → B mor , again satisfying the usual axioms. Functors from a category C to a category B, together with their natural transformations, form a functor category Fn(C, B) = B C (see e.g. [CWM] , II.4, or [MM92] ). Specifically, for the case of categories we are interested in:

Definition C.2. Let c K be some (small) subcategory of Alg 2 K (containing at least the trivial anchor as object, and the unique morphism of the trivial anchor to any other object as morphism). A c K -space law is a functor 

The map f K : M K → N K is called the underlying set-map of f . The functor category of c K -space laws and their morphisms, denoted by

is the (large) category whose objects are functors M and morphisms natural transformations f , which are composed "pointwise", i.e., for two laws f : M → N , g : N → P and all Υ :

Remark C.1. By definition, for each object Υ : A → A ′ of c K , evaluation at level Υ,

is a functor from Fn(c K , Sets 2 ) to Sets 2 . The natural morphism ι A : K → A induces a map M ι A : M K → M A .

Remark C.2. One can define the direct product of c K -space laws M i , i ∈ I, by

Definition C.3. An algebraic K-space law is a functor M from the whole category Alg 2 K to Sets 2 . Example C.1 (Linear sets). Every K-module V gives rise to a functor V from Alg 2 K to Sets 2 (algebraic space law), via algebraic scalar extension (Appendix A)

A . Natural transformations are given, e.g. by linear maps f : V → W : letting f A = f ⊗ id A : V A → W A , we get a morphism f : V → W . We get other natural transformations if we restrict our functors to categories of commutative algebras: then every m-multilinear map f : V m → W gives rise to a natural transformation f A (Theorem A.6). Thus multilinear maps give rise to polynomial laws in the sense of Roby [Ro63] (cf. Appendix of [Lo75] ).

Example C.2. For every m ∈ N, there is an algebraic space law M = Gl(m, K), by letting M A = Gl(m, A), M Υ = id ⊗ Υ. For instance, one can define its "tangent group" by T M := M K[ε] = Gl(m, K[ε]), and use this to develop a kind of algebraic differential calculus -this approach is used in [DG] , see also [Be08, Be14] .

The following is an "axiom" which is characteristic to our setting, compared with the general theory of functor categories. It rules out that "model spaces" give rise to trivial functors (like the functor sending every algebra to a point), and will force that all constructions from functorial differential calculus reduce to familiar linear algebra constructions, when applied to linear spaces and linear maps.

Definition C.4. By definition, the c K -law V of a K-module V shall be given by algebraic scalar extension (as in Example C.1), and likewise for the mapping law corresponding to a K-linear map. In other words, the category of K-linear spaces shall be imbedded into the functor category Fn(c K , Sets 2 ) in the natural way.

C.3. Further properties. It is not our aim here to develop the general theory of the functor categories in question. In practice, the categories c K will have further properties; in particular, the most interesting additional property is the one of a (strict) monoidal categories with respect to the tensor product of algebras.

Definition C.5. We say that some subcategory C of Alg 2 K is (1) closed under direct products if it contains, along with objects Υ A , Υ B , also the object Υ A ⊕ Υ B : A × B → A ′ × B ′ (we write also A ⊕ B for the direct product algebra A × B, with product (a, a ′ ) · (b, b ′ ) = (ab, a ′ b ′ )), and likewise for morphisms;

(2) closed under tensor products if it contains, along with objects Υ A , Υ B , also their tensor product

(recall the tensor product algebra A⊗B has product (a⊗b)·(a ′ ⊗b ′ ) = aa ′ ⊗bb ′ ), and likewise for morphisms;

When M = V is a K-module, then, by linear algebra,

These rules should serve as model, when generalizing functor categories beyond the realm of linear algebra.

Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, Memoirs of the AMS 192

Calcul différential topologiqueélémentaire, Calvage et Mounet

Simplicial differential calculus, divided differences, and construction of Weil functors

Weil Spaces and Weil-Lie Groups

Conceptual Differential Calculus. I : First order local linear algebra

Conceptual Differential Calculus. II : Cubic higher order calculus

A precise and general notion of manifold

Lie Calculus

Differential Calculus over general base fields and rings

A general construction of Weil functors

Categories for the Working Mathematician

Groupes Algébriques. I, Masson

Sur les modèles de la géométrie différentielle synthétique

Synthetic Differential Geometry

Synthetic Geometry of Manifolds

Natural Operations in Differential Geometry

Sheaves in Geometry and Logic

Models for Smooth Infinitesimal Analysis

Lois polynomes et lois formelles en théorie des modules

Théorie des points proches sur les variétés différentiables

For convenience, we recall some basic facts concerning linear algebra over rings (cf. e.g., [Bou] , see also on the n-lab, or here or there.) Definition A.1. Let K be a commutative ring, A an associative unital K-algebra, and V a K-module. Then we denote by V A the K-module V ⊗ K A, together with its right A-module structure given byand if φ : A → B is an algebra morphism, we define the K-linear map.This relation can be interpreted in terms of functor categories, see Appendix C: V is a functor from Alg K to Sets, and linear maps define natural transformations between such functors.Remark A.1. More precisely, V A is in fact an A-bimodule: it is also a left A-module, with left action given by µ.(v ⊗ λ) = v ⊗ µλ, such that left and right A-actions commute. Then f A is a morphism of A-bimodules.For all K-modules V , there is a canonical K-linear mapand for all A-(right) modules V , there is also a canonical A-linear mapIf V is an A-module, then we write V K = V as set and abelian group, but with scalar action by K only ("scalar restriction"). The following lemma formalizes that scalar restriction and scalar extension are adjoint functors:Lemma A.2. Assume V is a right A-module and W a K-module. Then the following two maps are mutually inverse K-linear isomorphisms:Replacing V by V A for some K-module V , we get a canonical isomorphismConcerning the other direction,whence, for w ⊗ λ ∈ W A , and using A-linearity of F ,

In this appendix, "linear spaces" are modules over a commutative ring K.Definition B.1. A hypercubic K-linear space, based on P(N), is a free K-module V together with a basis (E A ) A∈P(N ) indexed by the hypercube P(N) (power set of N, where N ⊂ N is a set, supposed to be finite in this appendix, and of cardinality n. If necessary, we write E N A instead of E A .) In other words, V ∼ = K P(N ) together with its canonical basis. When N = n = {1, . . . , n}, then we just speak of a n-hypercubic space, or of a 2 n -space. For n = 1, N = {k}, the two basis elements are also denoted by (E ∅ , E i ) or (E i ∅ , E i i ). Remark B.1. A hypercubic space carries several algebra structures: for instance, it is a Clifford algebra, or an exterior algebra, or a commutative algebra, in rather canonical ways. However, in this appendix, we forget about such algebra structures. Therefore, everything will apply to the underlying K-module of any such algebra.Lemma B.2. When N and M are disjoint finite subsets of N, then there is a canonical isomorphism of hypercubic spaces,Proof. See proof of Lemma 2.22.When f : V → W is linear, for bases (b j ) j∈J in V and (c i ) i∈I in W , we denote by f i,j := c * i (f (b j )) its matrix coefficients (where (c * i ) i∈I is the dual basis of c). We write also ( Writing a matrix in the usual way as rectangular number array, we use the natural total order on the index set -that is, the lexicographic order; for instance,In the following, for an n-tuple a = (a i ) i∈N ∈ K n , we use the notation a N := i∈N a i , in the same way as we do for t, s ∈ K n in the main text. When N is considered to be fixed, and A ⊂ N, we denote by A c = N \ A its complement. Then the matrix of the linear map f := ⊗ n i=1 f i : K P(N ) → K P(N ) is given by the matrix coefficients, for (A, B) ∈ P(N) 2 ,