The Schwarz-Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds

Informally, ${\mathbb Z}_2^n$-manifolds are 'manifolds' with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ${\mathbb Z}_2^n$-manifolds into a subcategory of contravariant functors from the category of ${\mathbb Z}_2^n$-points to a category of Fr\'echet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ${\mathbb Z}_2^n$-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.


Introduction
Various notions of graded geometry play an important rôle in mathematical physics and can often provide further insight into classical geometric constructions. For example, supermanifolds, as pioneered by Berezin and collaborators, are essential in describing quasi-classical systems with both bosonic and fermionic degrees of freedom. Very loosely, supermanifolds are 'manifolds' for which the structure sheaf is Z 2 -graded. Such geometries are of fundamental importance in perturbative string theory, supergravity, and the BV-formalism, for example. While the theory of supermanifolds is firmly rooted in theoretical physics, it has since become a respectable area of mathematical research. Indeed, supermanifolds allow for an economical description of Lie algebroids, Courant algebroids as well as various related structures, many of which are of direct interest to physics. We will not elaborate any further and urge the reader to consult the everexpanding literature.
Interestingly, Z n 2 -gradings (Z n 2 = Z ×n 2 , n ≥ 2) can be found in the theory of parastatistics, see for example [22,24,25,58], behind an alternative approach to supersymmetry [51], in relation to the symmetries of the Lévy-Lebond equation [2], and behind the theory of mixed symmetry tensors [11]. Generalizations of the super Schrödinger algebra (see [3]) and the super Poincaré algebra (see [10]) have also appeared in the literature. That said, it is unknown if these 'higher gradings' are of the same importance in fundamental physics as Z 2 -gradings. It must also be remarked that the quaternions and more general Clifford algebras can be understood as Z n 2graded Z n 2 -commutative (see below) algebras [4,5]. Thus, one may expect Z n 2 -gradings to be important in studying Clifford algebras and modules, though the implications for classical and quantum field theory remain as of yet unexplored. It should be further mentioned that any 'sign In this paper, we study Grothendieck's functor of points [26] of a Z n 2 -manifold M , which is a contravariant functor M (−) from the category of Z n 2 -manifolds to the category of sets, and restrict it to the category of Z n 2 -points, i.e., trivial Z n 2 -manifolds R 0|q that have no degree zero coordinates. More precisely, we consider the restricted Yoneda functor M → M (−) from the category of Z n 2 -manifolds to the category of contravariant functors from Z n 2 -points to sets. Dual to Z n 2 -points R 0|q are what we will call Z n 2 -Grassmann algebras Λ (see Definition 2.3). The aim of this paper is to carefully prove and generalise the main results of Schwarz & Voronov [44,54] to the 'higher graded' setting. In particular, we show that Z n 2 -points R 0|q Λ are actually sufficient to act as 'probes' when employing the functor of points (see Theorem 3.8). However, not all natural transformations η Λ : M (Λ) → N (Λ) (where Λ is a variable) between the sets M (Λ), N (Λ) of Λ-points correspond to morphisms φ : M → N of the underlying Z n 2manifolds. By carefully analysing the image of the functor of points, we prove that the set M (Λ) of Λ-points of a Z n 2 -manifold M comes with the extra structure of a Fréchet Λ 0 -manifold (see Theorem 3.22; by Λ 0 we mean the subalgebra of degree zero elements of the Z n 2 -Grassmann algebra Λ). Note that we are not trying to define infinite-dimensional Z n 2 -manifolds, yet infinitedimensional manifolds, specifically Fréchet manifolds, are fundamental to our paper. Moreover, we show that natural transformations η Λ between sets of Λ-points arise from morphisms φ of Z n 2 -manifolds if and only if they respect the Fréchet Λ 0 -manifold structures (see Proposition 3.24). By restricting accordingly the natural transformations allowed, we get a full and faithful embedding of the category of Z n 2 -manifolds into the category of contravariant functors from the category of Z n 2 -points to the category of nuclear Fréchet manifolds over nuclear Fréchet algebras. This embedding we refer to as the Schwarz-Voronov embedding (see Definition 3.28). We finally study representability of such contravariant functors and prove that the category of Z n 2 -manifolds is equivalent to the full subcategory of locally trivial functors in the just depicted subcategory of contravariant functors from Z n 2 -points to nuclear Fréchet manifolds (see Theorem 3.34).
Methodology: As Z n 2 -manifolds have well defined local models, we work with Z n 2 -domains and then 'globalize' the results to general Z n 2 -manifolds. We modify the approach of Schwarz & Voronov [44,54] and draw on Balduzzi, Carmeli & Fioresi [7,8] and Konechny & Schwarz [29,30], making all changes necessary to encompass Z n 2 -manifolds. Let us mention that Balduzzi, Carmeli & Fioresi study functors from the category of super Weil algebras and not that of Grassmann algebras. However, if we truly want to build a restricted Yoneda embedding, the source category of the functors of points must be a category of algebras that is opposite to some category of supermanifolds -and super Weil algebras are not the algebras of functions of some class of supermanifolds (unless they are Grassmann algebras). Moreover, the idea behind our restriction of the Yoneda embedding is 'the smaller the class of test algebras, the better'which points again to Grassmann algebras as being the somewhat privileged objects. The most striking difference between supermanifolds and Z n 2 -manifolds (n > 1) is that we are forced, due to the presence of non-zero degree even coordinates, to work with (infinite-dimensional) Fréchet spaces, algebras and manifolds. Interestingly, nuclearity of the values M (Λ) of the functor of points of a Z n 2 -manifold M , i.e., nuclearity of the local models of the Fréchet Λ 0manifolds M (Λ) or of their tangent spaces, does not play a rôle in the proofs of the statements in this paper. More precisely, the functor of points M (−) has values M (Λ) that are nuclear Fréchet Λ 0 -manifolds. Conversely, a functor F(−) whose values F(Λ) are Fréchet Λ 0 -manifolds and which is representable, has nuclear values (nuclearity is encrypted in the representability condition (see Theorem 3.34)). Although nuclearity of the tangent spaces of the manifolds M (Λ) is not explicitly used throughout this work, we do not at all claim that nuclearity is not of importance in the theory of Z n 2 -manifolds. For instance, the function sheaf of a Z n 2 -manifold is a nuclear Fréchet sheaf of Z n 2 -graded Z n 2 -commutative algebras -a fact that is crucial for product Z n 2 -manifolds and Z n 2 -Lie groups [13].
Applications: The functor of points has been used informally in physics as from the very beginning. It is actually of importance in situations where there is no good notion of point (see also Section 2.2), for instance in algebraic geometry and in super-and Z n 2 -geometry. Constructing a set-valued functor and showing that it is representable as a locally ringed space, e.g., a scheme or a Z n 2 -manifold, is often easier than building that scheme or manifold directly. Functors that are not representable can be interpreted as generalised schemes or generalised Z n 2manifolds. Further, the category of functors is better behaved than the corresponding category of supermanifolds or of other types of spaces. Also homotopical algebraic geometry [49,50], as well as its generalisation that goes under the name of homotopical algebraic D-geometry (where D refers to differential operators) [20,21], are fully based on the functor of points approach. Finally, the functor of points turns out to be an indispensable tool when it comes to the investigation of Z n 2 -Lie groups and their actions on Z n 2 -manifolds, of geometric Z n 2 -vector bundles . . . . These concepts are explored in upcoming texts that are currently being written down.
Arrangement: In Section 2, we review the basic tenets of Z n 2 -geometry and the theory of Z n 2 -manifolds. The bulk of this paper is to be found in Section 3. We rely on two appendices: in Appendix A we recall the notion of a generating set of a category, and in Appendix B we review indispensable concepts from the theory of Fréchet spaces, algebras and manifolds.

.1 The category of Z n 2 -manifolds
The locally ringed space approach to Z n 2 -manifolds is presented in a series of papers [15,16,17,18,19,35] by Covolo, Grabowski, Kwok, Ovsienko, and Poncin. We will draw upon these works heavily and not present proofs of any formal statements.
Definition 2.1. A locally Z n 2 -ringed space, n ∈ N, is a pair X := (|X|, O X ), where |X| is a second-countable Hausdorff space, and O X is a sheaf of Z n 2 -graded Z n 2 -commutative associative unital R-algebras, such that the stalks O p , p ∈ |X|, are local rings.
In this context, Z n 2 -commutative means that any two sections a, b ∈ O X (|U |), |U | ⊂ |X| open, of homogeneous degrees deg(a) = a ∈ Z n 2 and deg(b) = b ∈ Z n 2 commute according to the sign rule ab = (−1) a,b ba, where −, − is the standard scalar product on Z n 2 . We will say that a section a is even or odd if a, a ∈ Z 2 is 0 or 1.
Just as in standard supergeometry, which we recover for n = 1, a locally Z n 2 -ringed space is a Z n 2 -manifold if it is locally isomorphic to a specific local model. Given the central rôle of (finite-dimensional) Grassmann algebras in the theory of supermanifolds, we consider here Z n 2 -Grassmann algebras.
Remark 2.2. In the following, we order the elements in Z n 2 lexicographically, and refer to this ordering as the standard ordering. For example, we thus get ] is the Z n 2 -graded Z n 2 -commutative associative unital R-algebra of all formal power series with coefficients in R generated by homogeneous parameters ξ α subject to the commutation relation The Schwarz-Voronov Embedding of Z n 2 -Manifolds 5 where α := deg(ξ α ) ∈ Z n 2 \ 0, 0 = (0, . . . , 0). The tuple q = (q 1 , q 2 , . . . , q N ), N = 2 n − 1, provides the number q i of generators ξ α , which have the i-th degree in Z n 2 \ 0 (endowed with its standard order).
A morphism of Z n 2 -Grassmann algebras, ψ * : Λ q → Λ p , is a map of R-algebras that preserves the Z n 2 -grading and the units. We denote the category of Z n 2 -Grassmann algebras and corresponding morphisms by Z n 2 GrAlg.
Example 2.4. For n = 0, we simply get R considered as an algebra over itself.
Example 2.5. If n = 1, we recover the classical concept of Grassmann algebra with the standard supercommutation rule for generators. In this case, all formal power series truncate to polynomials. In particular, the Grassmann algebra generated by a single odd generator is isomorphic to the algebra of dual numbers.
Within any Z n 2 -Grassmann algebra Λ := Λ q , we have the ideal generated by the generators of Λ, which we will denote asΛ. In particular we have the decomposition Λ = R ⊕Λ, which will be used later on. Moreover, the set of degree 0 elements, Λ 0 ⊂ Λ, is a commutative associative unital R-algebra.
Very informally, a Z n 2 -manifold is a smooth manifold whose structure sheaf has been 'deformed' to now include the generators of a Z n 2 -Grassmann algebra.
for 'small enough' open subsets |U | ⊂ |M |. A Z n 2 -morphism, i.e., a morphism between two Z n 2 -manifolds, say M and N , is a morphism of Z n 2 -ringed spaces, that is, a pair φ = (|φ|, φ * ) : 2 -manifold -we refer to such Z n 2 -manifolds as Z n 2 -domains of dimension p|q. We can employ (natural) coordinates (x a , ξ α ) on any Z n 2 -domain, where the x a form a coordinate system on U p and the ξ α are formal coordinates.
Canonically associated to any Z n 2 -graded algebra A is the homogeneous ideal J of A generated by all homogeneous elements of A having nonzero degree. If f : A → A is a morphism of Z n 2graded algebras, then f (J A ) ⊂ J A . The J-adic topology plays a fundamental rôle in the theory of Z n 2 -manifolds. In particular, these notions can be 'sheafified'. That is, for any Z Many of the standard results from the theory of supermanifolds pass over to Z n 2 -manifolds. For example, the topological space |M | comes with the structure of a smooth manifold of dimension p and the continuous base map of any Z n 2 -morphism is actually smooth. Further, for any Z n 2 -manifold M , there exists a short exact sequence of sheaves of Z n 2 -graded Z n 2 -commutative associative R-algebras The immediate problem with Z n 2 -manifolds is that J M is not nilpotent -for supermanifolds the ideal sheaf is nilpotent and this is a fundamental property that makes the theory of supermanifolds so well-behaved. However, this loss of nilpotency is compensated by Hausdorff completeness of O M with respect to the J M -adic topology. The presence of formal power series in the coordinate rings of Z n 2 -manifolds forces one to rely on the Hausdorff-completeness of the J -adic topology. This completeness replaces the standard fact that supermanifold functions of Grassmann odd variables are always polynomials -a result that is often used in extending results from smooth manifolds to supermanifolds.
What makes Z n 2 -manifolds a very workable form of noncommutative geometry is the fact that we have well-defined local models. Much like the theory of manifolds, one can construct global geometric concepts via the gluing of local geometric concepts. That is, we can consider a Z n 2 -manifold as being covered by Z n 2 -domains together with specified gluing information, i.e., coordinate transformations. Moreover, we have the chart theorem [15,Theorem 7.10 ]), are completely described by the pullbacks of the coordinates (x a , ξ α ). In other words, to define a Z n 2 -morphism valued in a Z n 2 -domain, we only need to provide total sections (s a , s α ) ∈ O M (|M |) of the source structure sheaf, whose degrees coincide with those of the target coordinates (x a , ξ α ). Let us stress the condition (. . . , s a , . . .)(|M |) ⊂ U p , which is often understood in the literature.
A few words about the atlas definition of a Z n 2 -manifold are necessary. Let p|q be as above. A p|q-chart (or p|q-coordinate-system) over a (second-countable Hausdorff) smooth manifold |M | is a Z n 2 -domain over |M |, we set V αβ := |ψ α |(|U αβ |) and V βα := |ψ β |(|U αβ |), where |U αβ | := |U α | ∩ |U β |. We then denote by |ψ βα | the diffeomorphism Whereas in classical differential geometry the coordinate transformations are completely defined by the coordinate systems, in Z n 2 -geometry, they have to be specified separately. A coordinate transformation between two charts, say the ones of (2.1), is an isomorphism of Z n 2 -manifolds where the source and target are the open Z n 2 -submanifolds (note that the underlying diffeomorphism is (2.2)). A p|q-atlas over |M | is a covering U p|q α ,|ψ α | α by charts together with a coordinate transformation (2.3) for each pair of charts, such that the usual cocycle condition ψ βγ ψ γα = ψ βα holds (appropriate restrictions are understood).
Definition 2.10. A (smooth) Z n 2 -manifold of dimension p|q is a (second-countable Hausdorff) smooth manifold |M | together with a preferred p|q-atlas over it.
As in standard supergeometry, the Definitions 2.7 and 2.10 are equivalent [31]. For instance, Therefore, the composite As a matter of some formality, Z n 2 -manifolds and their morphisms form a category. The category of Z n 2 -manifolds we will denote as Z n 2 Man. We remark this category is locally small. Moreover, as shown in [13,Theorem 19], the category of Z n 2 -manifolds admits (finite) products. More precisely, let M i , i ∈ {1, 2}, be Z Remark 2.11. It is known that an analogue of the Batchelor-Gawȩdzki theorem holds in the category of (real) Z n 2 -manifolds, see [16,Theorem 3.2]. That is, any Z n 2 -manifold is noncanonically isomorphic to a Z n 2 \ {0}-graded vector bundle over a smooth manifold. While this result is quite remarkable, we will not exploit it at all in this paper.

The functor of points
Similar to what happens in classical supergeometry, a Z n 2 -manifold M is not fully described by its topological points in |M |. To remedy this defect, we broaden the notion of 'point', as was suggested by Grothendieck in the context of algebraic geometry.
More precisely, set V = {z ∈ C n : P (z) = 0} ∈ Aff, where P denotes a polynomial in n indeterminates with complex coefficients and Aff denotes the category of affine varieties. Grothendieck insisted on solving the equation P (z) = 0 not only in C n , but in A n , for any algebra A in the category CA of commutative (associative unital) algebras (over C). This leads to an arrow which turns out to be a functor where C[V ] is the algebra of polynomial functions of V . The dual of this functor, whose value Sol P (A) is the set of A-points of V , is the functor However, if such M does exist, it is, due to the mentioned injectivity, unique up to isomorphism and it is called 'the' representing Z n 2 -manifold of X. Further, if X, Y ∈ [Z n 2 Man op , Set] are two representable functors, represented by M , N respectively, a morphism or natural transformation between them, provides, due to the mentioned bijectivity, a unique morphism between the representing Z n 2 -manifolds M and N . It follows that, instead of studying the category Z n 2 Man, we can just as well focus on the functor category [Z n 2 Man op , Set] (which has better properties, in particular it has all limits and colimits).
A generalized Z n 2 -manifold is an object in the functor category [Z n 2 Man op , Set] and morphisms of such objects are natural transformations. The category [Z n 2 Man op , Set] of generalised Z n 2manifolds has finite products. Indeed, if F, G are two generalized manifolds, we define the functor F × G, given on objects S, by (F × G)(S) = F (S) × G(S), and on morphisms Ψ : S → T , by It is easily seen that F × G respects compositions and identities. Further, there are canonical natural transformations η 1 : F × G → F and η 2 : F × G → G. If now (H, α 1 , α 2 ) is another functor with natural transformations from it to F and G, respectively, it is straightforwardly checked that there exists a unique natural transformation β : H → F × G, such that α i = η i • β. One passes from the category of Z n 2 -manifolds to the larger category of generalised Z n 2 -manifolds in order to understand, for example, the internal Hom objects. In particular, there always exists a generalised Z n 2 -manifold such that the so-called adjunction formula holds This internal Hom functor is defined on φ ∈ Hom Z n 2 Man (P, S) by In general, a mapping Z n 2 -manifold Hom Z n 2 Man (M, N ) will not be representable. We will refer to 'elements' of a mapping Z n 2 -manifold as maps reserving morphisms for the categorical morphisms of Z n 2 -manifolds. Composition of maps between Z n 2 -manifolds is naturally defined as a natural transformation where ∆ : S −→ S × S is the diagonal of S and 1 S : S −→ S is its identity.
Similarly to the cases of smooth manifolds and supermanifolds, morphisms between Z n 2manifolds are completely determined by the corresponding maps between the global functions. We remark that this is not, in general, true for complex (super)manifolds. More carefully, we have the following proposition that was proved in [13,Theorem 3.7]. where Z n 2 Alg denotes the category of Z n 2 -graded Z n 2 -commutative associative unital R-algebras, is a bijection. is a homeomorphism, both, when the target space is endowed with its Zariski topology and when it is endowed with its Gel'fand topology. The continuous map |φ| : |M | → |N | that is induced by the morphism ψ is now given by The fact that the functor Hom Z n 2 Man (S, −) respects limits and in particular products directly implies that The latter result is essential in dealing with Z n 2 -Lie groups. A (super) Lie group can be defined as a group object in the category of smooth (super)manifolds. This leads us to the following definition.
Definition 2.13. A Z n 2 -Lie group is a group object in the category of Z n 2 -manifolds.
A convenient fact here is that, if G is a Z n 2 -Lie group, then the set G(S) is a group (see (2.5)). In other words, G(−) is a functor from Z n 2 Man op → Grp.
Remark 2.14. We leave details and examples of Z n 2 -Lie groups for future publications. However, we will remark at this point that the idea of "colour supergroup manifolds" has already appeared in the physics literature, albeit without a proper mathematical definition (see [1,3,36,37], for example). Another approach to Z n 2 -Lie groups is via a generalisation of Harish-Chandra pairs (see [33] for work in this direction).

Z n 2 -points and the functor of points
In view of (2.4), we need to 'probe' a given Z n 2 -manifold M M (−) with all Z n 2 -manifolds. We will show that this is however not the case, since, much like for the category of supermanifolds, we have a rather convenient generating set that we can employ, namely the set of Z n 2 -points.

The category of Z n 2 -points
with vanishing ordinary dimension. We denote by Z n 2 Pts the full subcategory of Z n 2 Man, whose collection of objects is the (countable) set of Z n 2 -points.
Morphisms φ : R 0|m → R 0|n of Z n 2 -points are exactly morphisms φ * : Λ n → Λ m of Z n 2 -Grassmann algebras: There is an isomorphism of categories We can think of Z n 2 -points as formal thickenings of an ordinary point by the non-zero degree generators. The simplest Z n 2 -point is the one with trivial formal thickening, R 0|0 := R 0 , R : Proof . Given any m = (m 1 , m 2 , . . . , m N ) and n = (n 1 , n 2 , . . . , n N ), we write Λ m ≤ Λ n if and only if m i ≤ n i , for all i. This preorder makes the non-empty set of Z n 2 -Grassmann algebras into a directed set, since, any Λ m and Λ n admit Λ p , where p i = sup{m i , n i }, as upper bound.
We will need the following functional analytic result in later sections of this paper. See Definitions B.1 and B.5 for the notion of Fréchet space and Fréchet algebra, respectively.
The proposition is a special case of the fact that the structure sheaf of any Z n 2 -manifold is a nuclear Fréchet sheaf of Z n 2 -graded Z n 2 -commutative algebras [12,Theorem 14]. Moreover, as a direct consequence of [13, Theorem 19, Definition 13], we observe that the category of Z n 2 -points admits all finite categorical products; in particular: R 0|m ×R 0|n R 0|m+n . By restricting attention to elements of degree 0 ∈ Z n 2 , we get the following corollary. See Definition B.7 for the concept of Fréchet module.
Corollary 3.6. The set Λ 0 of degree 0 elements of an arbitrary Z n 2 -Grassmann algebra Λ is a commutative nuclear Fréchet algebra. Moreover, the algebra Λ can canonically be considered as a Fréchet Λ 0 -module.
Remark 3.7. Specialising to the n = 1 case, we recover the standard and well-known facts about superpoints and their relation with Grassmann algebras.

A convenient generating set of Z n 2 Man
It is clear that studying just the underlying topological points of a Z n 2 -manifold is inadequate to probe the graded structure. Much like the category of supermanifolds, where the set of superpoints forms a generating set, the set of Z n 2 -points forms a generating set for the category of Z n 2 -manifolds. For the classical case of standard supermanifolds, see for example [40,Theorem 3.3.3]. For the general notion of a generating set, see Definition A.1.
Theorem 3.8. The set Ob Z n 2 Pts constitutes a generating set for Z n 2 Man. If |φ| = |ψ|, there is at least one point m ∈ |M |, such that |φ|(m) = |ψ|(m). Let now s : R 0|0 → M be the Z n 2 -morphism, which corresponds to the Z n 2 Alg morphism s * : It follows from the reconstruction theorem [13,Theorem 9] that the base morphism |s| : { } → |M | maps to m. Hence, the Z n 2 -morphisms φ • s and ψ • s have distinct base maps.
Assume now that |φ| = |ψ|, so that there exists |V | ⊂ |N |, such that φ * |V | = ψ * |V | , i.e., such whose base map |ι X | is the inclusion and whose pullback ι * X is the obvious restriction. Further, In view of the above, if (U j ) j is a cover of |U i | by coordinate domains, we have for some fixed j. This implies that the Z n More precisely, they have the same base map |φ| = |ψ| : U j → V i , but their pullbacks are distinct. Indeed, these sheaf morphisms' algebra maps at V i are the maps ι * , where y runs through V i and x through U j , and the values of these algebra maps at f | V i are different (see equation (3.1)).
In view of Lemma 3.9, there is a Z n 2 -morphism s : R 0|m → U j , such that However, then the Z n 2 -morphism ι U j • s : R 0|m → M separates φ and ψ, since the algebra maps are fully determined by the pullbacks of the target coordinates, their pullbacks at V i differ for at least one coordinate y b , θ B . It follows from the proof of Lemma 3.9 that the pullback s * s and the similar pullback for ψ differ for the same coordinate. However, the pullback at V i considered is also the algebra map at It remains to prove the following Lemma 3.9. The statement of Theorem 3.8 holds for any two distinct Z n 2 -morphisms between Z n 2 -domains.
Proof . We consider two Z n 2 -domains U p|q and V r|s together with two distinct Z n 2 -morphisms As in the general case above, there are two cases to consider: either |φ| = |ψ|, or |φ| = |ψ| and φ * = ψ * . In the proof of Theorem 3.8, we showed that in the first case, the maps φ and ψ can be separated. In the second case, since a Z n 2 -morphism valued in a Z n 2 -domain is fully defined by the pullbacks of the coordinates, these global Z n where we denoted the coordinates of U p|q by x a , ξ A and used the standard multi-index notation, differ. This means that the functions φ B α (x) and ψ B α (x) differ for at least one α and at least one x ∈ U p , say for α = a and x = x ∈ U p ⊂ R p . From this, we can construct the separating Let us denote the coordinates of R 0|q by χ A . We then define the Z n 2 -morphism s by setting Since we know that |φ| = |ψ|, we can proceed as for In view of Proposition A.3, we get the   Proof . Any Λ-point m * or m = (|m|, m * ) defines a topological point x = |m|( ) ∈ |M |, as well as a Z n 2 Alg-morphism φ x ∈ Hom Z n 2 Alg (O M,x , Λ) between stalks. This morphism is given, for any Conversely, any morphism ψ y ∈ Hom Z n 2 Alg (O M,y , Λ) (y ∈ |M |) between stalks defines a Λpoint µ * ∈ Hom Z n 2 Alg O(|M |), Λ . It suffices to set for all t ∈ O(|M |).
It remains to check that the composites m * → φ x → µ * and ψ y → µ * → φ x are identities. In the first case, for any t ∈ O(|M |), we get µ * t = φ x [t] x = m * t, so that µ * = m * . In the second case, we need the following reconstruction results. Let |U | ⊂ |M | be an open subset and set Then the localization map λ U : , and we identify F s −1 with t U . For the proof of these statements or more details on them, see [13,Proposition 3.5.]. It is further clear from the results of [13, Proposition 3.1.] that x = |µ|( ) is the topological point y.
We now compute the second composite above. For any t U defined in a neighborhood |U | of x, we get where the second equality is part of the reconstruction theorem of Z n 2 -morphisms [13].

Proof . Since it is clear from the definition of a stalk that O
Recall that , and, (ii) for any fixed M ∈ Z n 2 Man, given a morphism ψ = (|ψ|, ψ * ) : R 0|m → R 0|m of Z n 2 -points, or, equivalently, a morphism ψ * : Λ → Λ of Z n 2 -Grassmann algebras, we get the induced Set-map When reading the maps φ Λ and M (ψ * ) through the 1 : 1 correspondence

Restricted Yoneda functor and fullness
The Yoneda functor from any locally small category C into the category of Set-valued contravariant functors on C, is fully faithful. This holds in particular for C = Z n 2 Man. When we restrict the contravariant functors to the generating set Z n 2 Pts, the resulting restricted Yoneda functor is automatically faithful. In the following, we show that it is not full, i.e., that not all natural transformations are induced by a Z n 2 -morphism. Naturality of any transformation φ : M (−) → N (−) between Set-valued contravariant (resp., covariant) functors on Z n 2 Pts (resp., Z n 2 GrAlg), means that the diagram If the manifold is a Z n 2domain U p|q , we use the notation x * or x = (|x|, x * ). If x a , ξ A are the coordinates of U p|q , a Λ-point x * in U p|q is completely determined by the degree-respecting pullbacks and where γ 1 , . . . , γ N denote the non-zero Z n 2 -degrees in standard order. Here thex a Λ (resp., the ξ A Λ ) are formal power series containing at least 2 (resp., at least 1) of the generators θ C of the Z n 2 -Grassmann algebra Λ. As mentioned above, any Z n 2 -morphism, in particular any morphism φ : U p|q → V r|s between Z n 2 -domains, naturally induces a natural transformation, with Λ-component If y b , η B are the coordinates of V r|s , the morphism φ reads where the right-hand sides have the appropriate degrees and where φ 0 Let us recall that there is no convergence issue with terms in x || [15]. Thus the components of a natural transformation implemented by a Z n 2 -morphism between Z n 2 -domains, are very particular formal power series in the formal variablesx a Λ and ξ A Λ , which are themselves formal power series in the generators θ C of Λ.
We are now able to prove that not all natural transformations between the restricted functors From this data we obtain another natural transformation The natural transformation α is not implemented by a morphism ψ : R p → R p . Indeed, otherwise α Λ = ψ Λ , for all Λ. This means that for all Λ and all Λ-points.
Take now any β : |β| = 1, so that β a = 1, for some fixed a ∈ {1, . . . , p}. As we can choose Λ and The latter observation is a contradiction, since the Jacobian determinant of φ does not vanish anywhere in R p .
We now generalise a technical result [54,Theorem 1] be the Z n 2 -graded Z n 2 -commutative associative unital R-algebra of formal power series in p parameters X a of Z n 2 -degree 0 and q 1 , . . . , q N parameters Ξ A of non-zero Z n 2 -degree γ 1 , . . . , γ N , and with coefficients in arbitrary R-valued functions on U p , i.e., we do not ask that these functions be continuous let alone smooth. Following [43,44,54], we will refer to this algebra as a Z n 2 -Berezin algebra. Any element of this algebra is of the form where the x a are coordinates in U p .
Theorem 3.16. For any Z n 2 -domains U p|q and V r|s , there is a 1 : 1 correspondence Nat U p,q , V r,s → B p|q U p r|s between -the set of natural transformations in [Z n 2 Pts op , Set] between U p|q (−) and V r|s (−), and -the set of 'vectors' F with r (resp., with s 1 , . . . , s N ) components F b of degree 0 (resp., components F B of degrees γ 1 , . . . , γ N ) of the type (3.5), such that the r-tuple F b 00 made of the coefficients F b 00 (x) of the r series F b satisfies Proof . Let F be such a 'vector'. For any Λ, we define the map Sincex a Λ , ξ A Λ have the same degrees as X a , Ξ A , the right-hand sides of (3.3) have the same degrees as F b , F B , hence, y b Λ , η B Λ have the degrees required to be a Λ-point in V r|s . Moreover, we have so that y || ∈ V r . The target of the map β Λ is thus actually V r|s (Λ). The naturality of β under morphisms of Z n 2 -Grassmann algebras is obvious: β is a natural transformation in [Z n 2 Pts op , Set] between U p|q (−) and V r|s (−). Finally, we defined a map I : B p|q U p r|s → Nat U p,q , V r,s .
We will explain now that any natural transformation β : U p|q (−) −→ V r|s (−) is the image by I of a unique 'vector' F. We first show that, for any Λ R 0|m , the image β Λ (x * ) ∈ V r|s (Λ) of any Λ-point Step 1. We prove that any Λ-point in U p|q is the image by a Z n 2 -Grassmann algebra map ϕ * : Λ → Λ of a Λ -point in U p|q , some of whose defining series are series in formal pairings.
Let θ C be the generators of Λ. The Λ-point x * then reads where the degree of K a κλ ∈ Λ is the sum of the degrees of θ λ and θ κ . Recall that a (resp., A) runs through {1, . . . , p} (resp., through {1, . . . , |q|}), and that λ, κ run through {1, . . . , |m|}. Consider now the set S of generators where b has the same range as a, and define their (non-zero) Z n 2 -degrees by Let Λ be the Z n 2 -Grassmann algebra defined by S, and set (no sum over a in the formal pairings λ η aλ ζ a λ ). The degree-respecting equalities Indeed, any term of the right-hand side is a series in θ whose terms contain at least |ε| generators. Hence, for any , only the terms |ε| ≤ | | can contribute to θ , and therefore there is no convergence issue with the coefficient of θ . Since the Λ-point ϕ * • x * in U p|q reads naturality of the transformation β : U p|q (−) −→ V r|s (−) implies that where y b Λ and η B Λ are series in the generators of Λ .
Step 2. We define formal rotations under which the formal pairings are invariant. Moreover, we show that any formal series that is invariant under the formal rotations is a series in the formal pairings.
The formal part of each degree 0 component of x * can be viewed as a formal pairing η a · ζ a = λ η aλ ζ a λ , which is stable under formal rotations R * . More precisely, we set where O a and O bt are any (m 1 + · · · + m N ) × (m 1 + · · · + m N ) block-diagonal matrices with entries in R that satisfy Since, for any fixed a (resp., b), the components η aλ (resp., ζ b κ ) are ordered such that the m 1 first components have degree γ 1 , the next m 2 degree γ 2 , and so on, these equalities are degreepreserving. Hence, they define a Z n 2 -Grassmann algebra morphism R * : Λ → Λ via Since the images R * (η aλ ) (resp., R * (ζ b κ )) are linear in the η aκ (resp., ζ b λ ) (of the same degree), the term indexed by αβγ is a homogeneous polynomial of order |α|+|β|+|γ| in the generators θ . Hence, for any , only the terms |α| + |β| + |γ| = | | can contribute to θ , so that no convergence problems arise. In view of (3.7), it is clear that, as mentioned above, the formal pairing η a · ζ a = λ η aλ ζ a λ is invariant under R * . As any Z n 2 -Grassmann algebra morphism, the formal rotation R * induces maps U p|q (R * ) and V r|s (R * ), and due to naturality of β, we find We are now prepared to continue the computation (3.6). Since is invariant under the rotations R * , the seriesẙ b Λ , η B Λ in the generators θ are invariant. More explicitly, for each series, we have an equality of the type γ k, |α|=k, |β|= which is equivalent to |α|=k, |β|= F αβγ · · · η aλ η bµ ζ c ν · · · = |α|=k, |β|= and holds for all (!) formal rotations. This is only possible, if the power series considered, i.e., the seriesẙ b Λ and η B Λ , are series in pairings η a · ζ a = λ η aλ ζ a λ . In the classical setting, the result is known under the name of first fundamental theorem of invariant theory for the orthogonal group [23,57]. It has been extended to the graded situation in [7,Proposition 4.13]. In view of (3.6), we thus get where any image by ϕ * is of the type (α,β) =(0,0) It is clear from (3.8) and (3.3) that the coefficients F b αβ , F B αβ ((α, β) = (0, 0)), and F b 00 := y b || depend (only) on x || ∈ U p . Hence, the image is actually of the type (3.3). Since β Λ (x * ) is a Λ-point in V r|s , the r series F b (x || ,x Λ , ξ Λ ) and the s i series F B (x || ,x Λ , ξ Λ ) are of degree 0 and degree γ i , respectively, i.e., the r series F b (x, X, Ξ) and the s i series F B (x, X, Ξ) are of degree 0 and degree γ i , respectively. For the same reason, we have F 00 (x || ) ∈ V r , for all x || ∈ U p , so that we constructed a 'vector' F ∈ (B p|q (U p )) r|s , whose image by I is obviously β.
Step 3. We show that F is unique (which concludes the proof). If there is another 'vector' F , such that I(F ) = β, we have for all b ∈ {b, B}, all Λ, and all x * . Notice first that anyx a Λ (resp., any ξ A Λ ) is a series of degree 0 (resp., of degree deg ξ A = γ A ) in the θ-s that contains at least two parameters θ C θ C (resp., at least one parameter θ C ). Hence, both sides are series in θ, and the left-hand side and righthand side coefficients of any monomial θ ε coincide. A term (α, β) = (0, 0) cannot contribute to the independent term θ 0 .
, for an arbitrarily fixed (α, β) = (0, 0). Since Λ is arbitrary, we can choose as many different generators θ in each non-zero degree as necessary, and, since x * is arbitrary, we can choose x || arbitrarily in U p and we can choose the coefficients of the seriesx a Λ and ξ A Λ arbitrarily (except that we have to observe that the coefficient of a monomial θ ε , which does not have the required degree, must be zero). Let now α 1 , . . . , α µ and β 1 , . . . , β ν be the non-zero components in the fixed α and β. For each factor ξ A i Λ of we choose a monomial in one generator θ C i of degree γ A i , set its coefficient r C i to 1, and all the other coefficients in the series ξ A i Λ to zero. Further, for different ξ A i Λ , we choose different generators θ C i . Similarly, for each factorx we choose monomials θ D jk θ E jk (k ∈ {1, . . . , β j }) in two generators of the same odd degree (for all Z n 2 -manifolds with n ≥ 1, there is at least one odd degree), set their coefficient r D jk E jk to 1, and all the other coefficients in the seriesx a j Λ to zero. Further, we choose the generators so that all generators θ C i , θ D jk , and θ E jk are different. When setting the terms indexed by (the fixed) (α, β) in both sides of (3.9), read For any term (α , β ) = (α, β), we either get a new series ξ A Λ orx a Λ (i.e., a series that is not present in ξ α Λ orx β Λ ), or we get an old series a different number of times. In the second case, the term (α , β ) does not contribute to the coefficient of θ ω ; in the first, we set all the coefficients of the new series to 0, so that the term (α , β ) vanishes. Finally, we obtain F b αβ (x || ) = F b αβ (x || ), for any x || ∈ U p . We now show that R p|q (Λ) is a Fréchet space and that U p|q (Λ) is an open subset of R p|q (Λ). This means that we have a notion of directional derivative, as well as a notion of smoothness of continuous maps between the Λ-points of Z n 2 -domains. For more details on Fréchet objects, we refer the reader to Appendix B.
Proposition 3.17. For any Λ ∈ Z n 2 GrAlg, the set R p|q (Λ) is a nuclear Fréchet space and a Fréchet Λ 0 -module. Moreover, the set U p|q (Λ) is an open subset of R p|q (Λ).
Proof . Let Λ ∈ Z n 2 GrAlg. As explained above, there is a 1 : 1 correspondence between the Λ-points x * of R p|q (resp., of U p|q ) and the (p + |q|)-tuples (resp., the same (p + |q|)-tuples, but with the additional requirement that the p-tuple (x a || ) made of the independent terms of (x a Λ ) be a point in U p ⊂ R p ). Note now that Λ is the Z n 2 -graded Z n 2 -commutative nuclear Fréchet R-algebra of global Z n 2 -functions of some R 0|m . Hence, all its homogeneous subspaces Λ γ i (i ∈ 0, . . . , N , γ 0 = 0) are nuclear Fréchet vector spaces. Since any product (resp., any countable product) of nuclear (resp., Fréchet) spaces is nuclear (resp., Fréchet), the set R p|q (Λ) of Λ-points of R p|q is nuclear Fréchet. The latter statements can be found in [12].
As for the second claim in Proposition 3.17, recall that Λ 0 is a (commutative) Fréchet algebra, see Corollary 3.6. The Fréchet Λ 0 -module structure on R p|q (Λ) is then defined by Since this action is defined using the continuous associative multiplication · : Λ γ i × Λ γ j → Λ γ i +γ j of the Fréchet algebra Λ, it is (jointly) continuous. As any closed subspace of a Fréchet space is itself a Fréchet space, the space is Fréchet. We thus see that is open.
Remark 3.18. In the following, we will use the isomorphisms (3.11) (and similar ones) without further reference.
The just described Λ 0 -module structure is vital in understanding the structure of the Λ-points of any Z n 2 -manifold. In particular, morphisms between Z n 2 -domains induce natural transformations between the associated functors that respect this module structure. The converse is also true, that is, any natural transformation between the associated functors that respects the Λ 0module structure comes from a morphism between the underlying Z n 2 -domains. More carefully, we have the following proposition.
Proof . Part I. Let β : U p|q (−) −→ V r|s (−) be a natural transformation with Λ 0 -smooth components β Λ , Λ ∈ Z n 2 GrAlg. From Theorem 3.16, we know that β Λ is completely specified by the systems where the coefficients F b αβ (b ∈ {b, B}) are set-theoretical maps from U p to R. Part Ia. Smoothness of β Λ implies that these coefficients are smooth. Indeed, we will show that F b αβ ∈ C 0 (U p ) and that, if is continuous, any of its components R is continuous. For simplicity, we wrote y B Λ instead of η B Λ , and we will continue doing so. Moreover, the target space are the formal power series in θ with coefficients in R, all whose terms have the degree γ i(b) of y b , and this space is identified with the corresponding space of families of reals. For any ω such that θ ω has the degree γ i(b) , the corresponding real coefficient gives rise to a continuous map Since this joint continuity implies separate continuity with respect to x || ∈ U p , for any fixed (x Λ , ξ Λ ) and any Λ, we can proceed as at the end of the proof of Theorem 3.16. More precisely, select any (α, β) and select (for an appropriate Λ) the pair (x Λ , ξ Λ ) such thatx β Λ ξ α Λ = β!θ ω , where θ ω is now the degree γ i(b) monomial defined in the proof just mentioned. The real coefficient of this monomial is β!F b αβ (x || ), which, as said, is an R-valued continuous map on U p , so that F b αβ ∈ C 0 (U p ), for all b and all (α, β).
is an open subset of a product of two Fréchet spaces, smoothness of β Λ implies (via an iterated application of Proposition B.4) that, for any b ∈ {b, B}, any ∈ N and any γ ∈ N p (|γ| = ), the partial derivative R is continuous. Assume now that F b αβ ∈ C k (U p ) (k ≥ 0), for any b and any (α, β), as well as that, for any γ ∈ N p (|γ| = k) and any b, the continuous partial Gâteaux derivative is given by Observe that for k = 0, this condition is automatically satisfied. We will now show that, under these assumptions, the same statements hold at order k + 1. In view of (3.13), any order k + 1 continuous partial Gâteaux derivative (3.14) When proceeding as in Step 1, we get that the limit is an R-valued continuous function in U p . In other words, the partial derivative ∂ x a ∂ γ x F b αβ exists and is continuous in U p , i.e., F b αβ ∈ C k+1 (U p ). Moreover, formula (3.13) pertaining to order k derivatives, extends to the order k +1 derivatives, see (3.14).
Part Ib. We examine the further consequences of Λ 0 -smoothness, in particular those of Λ 0linearity. Since β Λ is of class C 1 , its components y b is an open subset of a product of two Fréchet spaces, the partial Gâteaux derivative R is continuous. It is given by AsΛ 0 is a commutative algebra, it follows from the binomial formula that where (e a ) a is the canonical basis of R p . Observe now that, in view of (3.10), the Λ 0 -linearity of the total Gâteaux derivative of y b Λ with respect to x * is equivalent to the Λ 0 -linearity of all its partial Gâteaux derivatives with respect to the x a Λ = x a || ,x a Λ and the ξ A Λ . For a = 0 +v Λ ∈ Λ 0 and v = 1 + 0 ∈ R +Λ 0 = Λ 0 , this implies that Since Λ ∈ Z n 2 GrAlg,v Λ ∈Λ 0 , and x * ∈ U p|q (Λ) are arbitrary, we can repeat the θ ω -argument used above. More precisely, we select (α, β), select (x Λ , ξ Λ ) such thatx β Λ ξ α Λ = β!θ ω , and select v Λ = θ D θ E ∈Λ 0 such that θ D θ E θ ω = 0. The coefficients of the latter monomial in the left and right hand sides do coincide, which means that or, equivalently, for all b, α, a, all γ : γ a = 0, and all x || ∈ U p . For any b, α, and x || , we now set An iterated application of (3.15) shows that Hence, the y b Λ have the form (3.4a) and (3.4b). This means that the natural transformation β is implemented by the φ b α , which define actually a Z n 2 -morphism from U p|q to V r|s . Indeed, the property φ b 0 (U p ) ⊂ V r follows from the similar property of F b 00 . On the other hand, the pullback must have the same degree as y b . However, if deg(ξ α ) = deg y b , then deg(ξ α Λ ) = deg y b Λ , whatever ξ Λ . It follows therefore from (3.12) that φ b α = F b α0 = 0. Part II. The proof of the converse implication is less demanding. Let β : U p|q (−) → V r|s (−) be a natural transformation that is induced by a Z n 2 -morphism φ : U p|q → V r|s , i.e., that is of the form (3.4a) and (3.4b). For any Λ ∈ Z n 2 GrAlg, the map β Λ is smooth and its derivative is Λ 0 -linear if and only if its components y b Λ have these properties. The total derivative of y b Λ with respect to x * exists, is continuous, and is Λ 0 -linear if and only if its partial derivatives with respect to the x a Λ and the ξ A Λ exist, are continuous, and are Λ 0 -linear. When computing the derivative y b Λ with respect to ξ A i Λ ∈ Λ γ i at x * ∈ U p|q (Λ) in the direction of w Λ ∈ Λ γ i , we get If γ i is odd, the exponent α i is 0 or 1. In the first (resp., the second) case, the limit vanishes (resp., is w Λ ). If γ i is even, the multiplication of vectors in Λ γ i is commutative and the binomial formula shows that the limit is The derivative considered does therefore exist, is continuous, and is Λ 0 -linear (note that it is essential that the derivative is the series over αβ multiplied by v Λ -as a ∈ Λ 0 does not act on v || ).
Remark 3.20. The Λ 0 -linearity is a strong constraint that takes us from the category of generalized Z n 2 -manifolds to the one of Z n 2 -manifolds. A similar phenomenon exists in complex analysis. Indeed, for any real differentiable function f = u + iv : Ω ⊂ C R 2 → C R 2 , the Jacobian is an R-linear map J f : R 2 → R 2 . However, if we further insist that the Jacobian be C-linear, then we see that f must be holomorphic, that is, it must satisfy the Cauchy-Riemann equations on Ω. Imposing C-linearity thus greatly restricts class of functions and takes us from real analysis to complex analysis.
It will also be important to understand what happens to the Λ-points of a given Z n 2 -domain under morphisms of Z n 2 -Grassmann algebras.
is a smooth map from the open subset U p|q (Λ) of the Fréchet space and Fréchet Λ 0 -module R p|q (Λ) to the open subset U p|q (Λ ) of the Fréchet space and Fréchet Λ 0 -module R p|q (Λ ), such that for all x * ∈ U p|q (Λ), v ∈ R p|q (Λ) and a ∈ Λ 0 .
] endowed with their standard locally convex topologies [12], and so are its restrictions ψ * | Λγ i from Λ γ i to Λ γ i . We thus see that the induced map At x * (x Λ , ξ Λ ) =: u Λ ∈ U p|q (Λ) and v v Λ ∈ R p|q (Λ), the derivative is defined as where a is the label a ∈ {1, . . . , p} or A ∈ {1, . . . , |q|} of any coordinate in R p|q (Λ), and where we used the R-linearity of the Z n 2 -algebra morphism ψ * : Λ → Λ . Hence, for any a ∈ Λ 0 , we get Since the higher order derivatives of Ψ vanish, all its derivatives exist and are continuous, hence, the map Ψ is actually smooth.  (3.16) and that, for any fixed M , a Z n 2 -Grassmann algebra morphism ψ * : Λ → Λ induces a map

The manifold structure on the set of Λ-points
, where α varies in some A and where |U α | ⊂ |M | is open, provide an atlas on M (see paragraph below Definition 2.10). As recalled above, the Z n 2 -isomorphisms (3.17) whose inverses are the similar maps defined using The family (U α (Λ), h α,Λ ) (α ∈ A) is an atlas that endows M (Λ) with a nuclear Fréchet Λ 0manifold structure. Indeed: where the target is the open Z n 2 -subdomain U Further, we have Hence, the image h α,Λ (U α (Λ) ∩ U β (Λ)) = U In view of equations (3.16) and (3.17), we get (ii) The statement of part (ii) is purely local, see Appendix B. Let (x, m * ) ∈ M (Λ), let (U α (Λ), h α,Λ ) be a chart of M (Λ) around (x, m * ), and let (U β (Λ ), h β,Λ ) be a chart of M (Λ ), such that M (ψ * )(U α (Λ)) ⊂ U β (Λ ). We must show that the local form Actually, we can choose (U α (Λ ), h α,Λ ) as second chart, since the image by M (ψ * ) of a point (y, n * ) in U α (Λ), i.e., a point (y, n * ) ∈ Hom Z n 2 Alg (O M,y , Λ) with y ∈ |U α |, is the point i.e., in U α (Λ ). From here, we omit subscript α. Since h : U (−) → U p|q (−) is a natural transformation, the diagram commutes. Since h is in fact a natural isomorphism, we get that From Proposition 3.21 we conclude that this local form is indeed ψ * -smooth.
In view of (3.3), in general, the local model R p|q (Λ) of M (Λ) is infinite-dimensional, due to the non-zero degree even coordinates of Λ. If the particular Z n 2 -Grassmann algebra has no non-zero degree even coordinates, then it is a polynomial algebra and the resulting local model R p|q (Λ) will, of course, be finite-dimensional. Further, we have the where the target category is either the category AFMan of Fréchet manifolds over a Fréchet algebra or the category ANFMan of nuclear Fréchet manifolds over a nuclear Fréchet algebra, see Appendix B. Therefore, the faithful restricted Yoneda functor Y Z n 2 Pts , see Corollary 3.10, can be viewed as a faithful functor The latter statement requires that the natural transformation φ : M (−) → N (−) induced by a Z n 2 -morphism φ : M → N have components φ Λ : M (Λ) → N (Λ) that are morphisms in A(N)FMan between the Fréchet Λ 0 -manifolds M (Λ) and N (Λ), i.e., that the φ Λ be ρ-smooth for some morphism ρ : Λ 0 → Λ 0 of Fréchet algebras. We will show in the next subsection that this condition is satisfied for ρ = id Λ 0 , i.e., we will show that: (i) The nuclear Fréchet Λ 0 -manifold M (Λ) is a fiber bundle in the category ANFMan. Its base is the nuclear Fréchet R-manifold M (R), i.e., the smooth manifold |M |, and its typical fiber is the nuclear Fréchet Λ 0 -manifold (ii) The topology of M (Λ), which is defined, as in the case of smooth manifolds, by the atlas providing the nuclear Fréchet Λ 0 -structure, is a Hausdorff topology, so that M (Λ) is a genuine Fréchet manifold.
Proof . (i) We think of fiber bundles in ANFMan exactly as of fiber bundles in the category of smooth manifolds. Of course, in such a fiber bundle, all objects and arrows are ANFMan-objects and ANFMan-morphisms. Let p * : Λ → R be, as above, the canonical Z n 2 GrAlg-morphism. The induced map is p * -smooth, i.e., is a morphism in the category ANFMan.
We will show that π is surjective and that the local triviality condition is satisfied. Let z ∈ |M |. There is a Z n 2 -chart (U, h) of M , such that |U | ⊂ |M | is a neighborhood of z. The Z n 2 -isomorphism h : U → U p|q induces a natural isomorphism h, whose Λ-components are Λ 0 -diffeomorphisms, i.e., Λ 0 -smooth maps that have a Λ 0 -smooth inverse. We have the following commutative diagram: where prj 1 is the canonical projection. Let us explain that U p|q (p * ) prj 1 , when read through : U p × Λ p|q ↔ U p|q (Λ). We need a more explicit description of the equivalent views on Λpoints of a Z n 2 -domain, see beginning of Section 3.4. As elsewhere in this text, we denote a Z n 2 -morphism R 0|m → U p|q by x = (|x|, x * ) and we denote the morphism it induces between the stalks O U p|q ,|x|( ) → Λ by x * . The morphism is the succession of identifications (3.18) where the components of the base morphism |x| are obtained (see [15]) by applying the base projection ε : Λ → R of R 0|m , i.e., the canonical morphism p * , to the components x a Λ = (x a || ,x a Λ ) ∈ Λ 0 . Hence, we get Therefore, we actually obtain that Since π| U (Λ) = |h| −1 • prj 1 •h Λ , the local projection π| U (Λ) is surjective, so that z is in the image of π, which is thus surjective as well.
As just mentioned, we started from z ∈ |M | and found a neighborhood |U | of z and a Λ 0diffeomorphism h Λ . When identifying |U | with U p via |h| (which then becomes id), we get the Λ 0 -diffeomorphism Observe that in equation (3.20) we used −1 defined in equations (3.18) and (3.19), thus identifying with h • m ∈ Hom Z n 2 Man R 0|m , U p|q , and then with y, pr 2 (m * (h * (x))), m * (h * (ξ)) ∈ |U | × Λ p|q , where we denoted the projection of Λ 0 ontoΛ 0 by pr 2 . Notice also that the conclusion that Λ p|q is a nuclear Fréchet Λ 0 -manifold comes from the facts that any subspace (resp., any closed subspace) of a nuclear (resp., a Fréchet) space is a nuclear (resp., a Fréchet) space. Hence, the trivialization condition is satisfied as well, and M (Λ) is a fiber bundle in ANFMan, as announced.
(ii) Now consider two different Λ-points m * = (x, m * ) and n * = (y, n * ) in M (Λ). If x = y, then, as |M | is Hausdorff, there exist open neighborhoods |U | of x and |V | of y, such that |U |∩|V | = ∅. When denoting the corresponding open Z n 2 -submanifolds by U and V , respectively, we get open neighborhoods U (Λ) and V (Λ) of m * and n * , such that U (Λ) ∩ V (Λ) = ∅. We have of course to check that, for any Z n 2 -chart (U α , h α ), the image is open in R p|q (Λ), and similarly for V (Λ). To see this, it suffices to proceed as in the proof of Theorem 3.22.
Since prj 2 (h Λ (m * )) = prj 2 (h Λ (n * )) are points in the Hausdorff space Λ p|q , there are open neighborhoods V m * and V n * of these projections that do not intersect. The preimages U m * and U n * of V m * and V n * by the continuous map are then open neighborhoods of m * and n * that do not intersect.
Finally, the space M (Λ) is indeed a Hausdorff topological space.

The Schwarz-Voronov embedding
In order to get a fully faithful functor, hence, to embed the category Z n 2 Man as full subcategory into a functor category, we need to replace the target category [Z (i) whose objects are the functors F, such that, for any Λ ∈ Z n 2 Pts op , the value F(Λ) is a (nuclear) Fréchet Λ 0 -manifold, and (ii) whose morphisms are natural transformations η : F → G, such that, for any Λ, the component η Λ : The proof of Λ 0 -smoothness uses the following construction, which we will also need later on. Let M, N ∈ Z n 2 Man be manifolds of dimension p|q and r|s, respectively, let |φ| ∈ C ∞ (|M |, |N |), and let (|V β |) β be an open cover of |N | by Z n 2 -charts The open subsets |U β | := |φ| −1 (|V β |) ⊂ |M | cover |M |, and each |U β | can be covered by Z n 2 -charts The Z n 2 -morphism φ : M → N restricts to a Z n 2 -morphism φ| U βα : U βα → V β . In particular, the composite is a Z n 2 -morphism. We now show that φ Λ is Λ 0 -smooth. Therefore, let (x, m * ) ∈ M (Λ). There is a Z n 2 -chart (V β , g β ) of N such that |φ|(x) ∈ |V β |, and there is a Z n 2 -chart (U βα , h βα ) of M such that x ∈ |U βα |. These charts (we omit in the following the subscripts β and α) induce charts (U (Λ), h Λ ) of M (Λ) around (x, m * ), and (V (Λ), g Λ ) of N (Λ) such that φ Λ (U (Λ)) ⊂ V (Λ). It suffices to show (see Appendix B) that the local form is Λ 0 -smooth. This is the case in view of Theorem 3.19. Finally, the faithfulness is established in Corollary 3.10. This completes the proof.
We will prove that the functor S is fully faithful, hence, injective (up to isomorphism) on objects. Therefore, it embeds the category Z n 2 Man of Z n 2 -manifolds as full subcategory into the larger functor category [[Z n 2 Pts op , A(N)FMan]].
Definition 3.28. We refer to the faithful functor as the Schwarz-Voronov embedding. Proof . Notice first that it follows from the results of [13] and Lemma 3.13 that there is a 1 : 1 correspondence which is given by is a smooth map between the reduced manifolds. As in the proof of Proposition 3.27, let (V β , g β ) β be an open cover of |N | by Z n 2 -charts, and, for any β, let (U βα , h βα ) α be an open cover of |U β | := |φ| −1 (|V β |) by Z n 2 -charts. When denoting the canonical Z n 2 -Grassmann algebra morphism Λ → R by p * , we get the commutative diagram |φ| which shows that, for any β, α, we get the Λ 0 -smooth map Indeed, if, for (x, m * ) ∈ U βα (Λ), we set η Λ (x, m * ) = (y, n * ), the commutativity of the diagram implies that Therefore, the restriction is a natural transformation with Λ 0 -smooth components. Note that are Z n 2 -isomorphisms and induce natural isomorphisms, also denoted by h βα and g β , whose components are chart diffeomorphisms of nuclear Fréchet Λ 0 -manifolds. The local form of η Λ is thus Λ 0 -smooth. In other words, any Λ-component of the natural transformation between functors associated to Z n 2 -domains, is Λ 0 -smooth. It therefore follows from Theorem 3.19 that ϕ βα is implemented by a Z n 2 -morphism so that the composite is a Z n 2 -morphism that is defined on an open Z n 2 -submanifold of M . The question is whether we can patch together these locally defined Z n 2 -morphisms, which are inherited from η, and get a globally defined Z n 2 -morphism φ : M → N that induces η. Let φ βα | U βα,νµ and φ νµ | U βα,νµ be the Z n 2 -morphisms obtained by restriction to the open Z n 2submanifold U βα,νµ with base manifold |U βα,νµ | := |U βα |∩|U νµ |. They coincide as Z n 2 -morphisms, if they do as associated natural transformations, i.e., if all Λ-components of those transformations coincide. This is the case since both Λ-components are equal to η Λ | U βα,νµ (Λ) . It follows that the Z n 2 -algebra morphisms coincide. This implies that we can glue the Z n 2 -algebra morphisms φ * βα : O N (|N |) → O M (|U βα |) and get a Z n 2 -algebra morphism For any Λ ∈ Z n 2 GrAlg, let be the canonical projection, let be the corresponding smooth map. The preimage is an open (nuclear) Fréchet Λ 0 -submanifold of F(Λ).
Consider now a morphism ϕ * : Λ −→ Λ in Z n 2 GrAlg. As p * Λ • ϕ * = p * Λ , we get the restriction which is a morphism in A(N)FMan. and any open subset |U | ⊂ F(R), the restriction of F to |U | is the functor that is defined by equations (3.25) and (3.26). since both maps are given by The Z n 2 -ringed space where h * α is the identity map (a composite of direct images is the direct image by the composite). In other words, we have an isomorphism of Z n 2 -manifolds Consider now an overlap |U αβ | := |U α | ∩ |U β | = ∅. Omitting restrictions, we get that between functors corresponding to Z n 2 -domains (defined as usual). In view of Theorem 3.19, the natural isomorphism k βα is implemented by a Z n 2 -isomorphism It follows that . The Z n 2 -manifolds U α can thus be glued and provide then a Z n 2 -manifold M over |M | = F(R), such that there are Z n 2 -isomorphisms (|U α |, O M | |Uα| ) → U α , if the ψ βα satisfy the cocycle condition.
Since the Schwarz-Voronov embedding is fully faithful, we have that ψ γβ ψ βα = ψ γα as Z n 2morphisms if and only if the induced natural transformations coincide. However, for any Λ, we get It remains to show that M actually represents F, i.e., that we can find a natural isomorphism η : M (−) → F in the category [[Z n 2 Pts op , A(N)FMan]], i.e., that, for any Λ ∈ Z n 2 GrAlg, there is a Λ 0 -diffeomorphism η Λ : M (Λ) → F(Λ) that is natural in Λ. As (|U α |) α is an open cover of |M |, the source decomposes as the U α (Λ) being open (nuclear) Fréchet Λ 0 -submanifolds. On any U α (Λ), we define η Λ by setting These restrictions provide a well-defined map if and only if ψ βα,Λ (x, m * ) = (x, m * ).
However, since we glued M from the U α , the gluing Z n 2 -isomorphisms ψ βα became identities and so did the induced natural isomorphisms. The definition of η −1 Λ is similar. The source F(Λ) decomposes as the F| |Uα| (Λ) being open (nuclear) Fréchet Λ 0 -submanifolds. On any F| |Uα| (Λ), we define η −1 Λ by setting The condition for these restrictions to give a well-defined map is equivalent to the condition for η Λ . Clearly, the maps η Λ and η −1 Λ are inverses. Naturality and Λ 0 -smoothness are local questions and are therefore consequences of the naturality and the Λ 0 -smoothness of (k α,Λ ) −1 h α,Λ and of (h α,Λ ) −1 k α,Λ .
We are now prepared to show that the category Z n 2 Man is equivalent to a functor category. In other words, the category Z n 2 Man is equivalent to the category of locally trivial functors in the subcategory of the functor category [Z n 2 Pts op , A(N)FMan], whose objects F have values F(Λ) in (nuclear) Fréchet Λ 0 -manifolds and whose morphisms are the natural transformations with Λ 0 -smooth components.

A Generating sets of categories
We will freely use Mac Lane's book [32] as our source of categorical notions and proofs of general statements. For completeness, we recall the concept of generating set of a category.
are distinct, i.e., φ • s = ψ • s. In this case, we say that the object S i separates the morphisms φ and ψ, and that the set S generates the category C.
Example A.2. The set {R} is a generating set of the category of finite-dimensional real vector spaces. This is easily seen, as, if we have two distinct linear maps φ, ψ : V → W , then there exists a vector v ∈ V (v = 0), such that φ(v) = ψ(v). Thus, the two linear maps differ on the one-dimensional subspace generated by v. Now let z be a basis of R. Then, the linear map s : R → V given by s(z) = v, keeps φ and ψ separate.
For any locally small category C, a set S ⊂ Ob(C) generates C if and only if the restricted Yoneda embedding where S is viewed as full subcategory of C, is faithful.
Proof . The restricted embedding is defined on objects by and on morphisms by The embedding Y S is faithful if and only if, for any different φ, ψ : A → B, the corresponding natural transformations are distinct, i.e., there is at least one i ∈ I and one s ∈ Hom C (S i , A), such that φ • s = ψ • s.

B Fréchet spaces, modules and manifolds
Manifolds over algebras A, also known as A-manifolds, are manifolds for which the tangent spaces are endowed with a module structure over a given finite-dimensional commutative algebra. For details, the reader may consult Shurygin [46,47,48], and for a discussion of the specific case of (the even part of) Grassmann algebras one may consult Azarmi [6]. A comprehensive introduction to the subject can be found in the book (in Russian) by Vishnevskiȋ, Shirokov, and Shurygin [53]. The concept needed in this paper is a infinite-dimensional generalisation of an A-manifold to the category of Fréchet algebras and Fréchet manifolds. For an introduction to locally convex spaces, including Fréchet vector spaces, we refer the reader to Conway [ There exist a few other, equivalent, definitions of Fréchet spaces. The topology on a locally convex space is metrizable if and only if it can be derived from a countable family of seminorms || − || k , k ∈ N. The topology is Hausdorff if and only if the family of semi-norms is separating, i.e., if ||x|| k = 0, for all k, implies x = 0. Given such a family of semi-norms, one defines a translationally invariant metric that induces the topology by setting for all x and y. where is the projection ε : O(U ) → C ∞ (U ) of Z n 2 -functions to base functions, where C is any compact subset of U , and where D is any Z n 2 -differential operator over U . Details on the construction of a countable family of semi-norms that is equivalent to (|| − || C,D ) C,D , can be found in the proof of the last lemma in [12].
Given two Fréchet spaces F, || − || F k k∈N and G, || − || G k k∈N , a linear map is continuous if and only if, for every semi-norm || − || G k , there exists a semi-norm || − || F l and a positive real number C > 0, such that for every x ∈ F . A similar result holds for continuous bilinear maps The morphisms of Fréchet spaces are the continuous linear maps, so that the category of Fréchet spaces is a full subcategory of the category of topological vector spaces. What makes Fréchet spaces interesting, is the fact that they have just enough structure to define a derivative of a mapping between such spaces. This leads to a meaningful notion of a smooth map between Fréchet spaces, and so much of finite-dimensional differential geometry can be transferred to the infinite-dimensional setting, using Fréchet spaces as local models. The well known Gâteaux (directional) derivative is defined as follows.
Definition B.3. Let F and G be Fréchet spaces and U ⊂ F be open, and let φ : U → G be a (nonlinear) continuous map. Then the derivative of φ in the direction of v ∈ F at x ∈ U is defined as provided the limit exists. We say that φ is continuously differentiable, if the limit exists for all x ∈ U and v ∈ F , and if the mapping Higher order derivatives are defined inductively, i.e., A continuous map φ : U → G is then said to be k times continuously differentiable or to be of class C k , if is continuous (or, more explicitly, if all its derivatives of order ≤ k exist everywhere and are continuous). If φ is of class C k , its derivative d k x φ(v 1 , v 2 , . . . , v k ) is multilinear and symmetric in F ×k [45]. Furthermore, we say that φ is smooth, if it is of class C k , for all k.
Proposition B.4. Let F 1 , F 2 be Fréchet spaces and let U ⊂ F 1 × F 2 be an open subset. A continuous map φ : U → G valued in a Fréchet space G is of class C 1 if and only if its (total) derivative is continuous, which is the case if and only if the naturally defined partial derivatives are continuous. In this case, we have The Gâteaux or Fréchet-Gâteaux derivative gives a rather weak notion of differentiation, however, most of the standard results from calculus in the finite-dimensional setting remain true. Specifically, the fundamental theorem of calculus and the chain rule still hold. However, the inverse function theorem is in general lost. For a special class of Fréchet spaces, known as 'tame' Fréchet spaces, there is an analogue of the inverse function theorem known as the Nash-Moser inverse function theorem, see Hamilton [27] for details.
A nuclear space is a locally convex topological vector space F , such that, for any locally convex topological vector space G, the natural map F ⊗ π G −→ F ⊗ ι G from the projective to the injective tensor product of F and G is an isomorphism of locally convex topological vector spaces. In particular, a nuclear Fréchet space is a locally convex topological vector space that is a nuclear space and a Fréchet space. Loosely, if a space F is nuclear, then, for any locally convex space G, the complete topological vector space F ⊗G is independent of the locally convex topology considered on F ⊗ G. Because of this, and their nice dual properties, nuclear spaces provide a reasonable setting for infinite-dimensional analysis. All the Fréchet spaces we encounter in this paper are in fact nuclear.
The following definition is standard.
Definition B.5. A Fréchet algebra is a Fréchet vector space A, which is equipped with an associative bilinear and (jointly) continuous multiplication · : A × A → A. If (p i ) i∈I is a family of semi-norms that induces the topology on A, (joint) continuity is equivalent to the existence, for any i ∈ I, of j ∈ I, k ∈ I, and C > 0, such that We can always consider an equivalent increasing countable family of semi-norms (|| − || n ) n∈N . The preceding condition then requires that, for any n ∈ N, there is r ∈ N (r ≥ n) and C > 0, such that ||x · y|| n ≤ C||x|| r ||y|| r , ∀ x, y ∈ A.
In particular, the topology can be induced by a countable family of submultiplicative semi-norms, i.e., by a family (q n ) n∈N that satisfies q n (x · y) ≤ q n (x)q n (y), ∀ n ∈ N, ∀ x, y ∈ A.
Note that many authors define a Fréchet algebra simply as a Fréchet vector space, which carries an associative bilinear multiplication, and whose topology can be induced by a countable family of submultiplicative semi-norms. This latter definition is equivalent to the former.
In general, a Fréchet algebra need not be unital, and, if it is, one does not require p i (1 A ) = 1, in contrast to what is usually required for Banach algebras.
Example B.6 (formal power series). Consider the space R[[z 1 , z 2 , . . . , z q ]] of formal power series in q parameters and coefficients in reals. We set j := (j 1 , j 2 , . . . , j q ) ∈ N ×q and |j| := j 1 + j 2 + · · · + j q . A general series x now reads x = j z j a j = j z j 1 1 z j 2 2 · · · z jq q a jq...j 2 j 1 , with no question on the convergence. The algebra structure is the standard multiplication of formal power series. The topology of coordinate-wise convergence is metrizable and given by the family of semi-norms This algebra is unital with the obvious unit, and it is submultiplicative.
Let us denote the category of Fréchet algebras (resp., commutative Fréchet algebras) as FAlg (resp., CFAlg). Morphisms in this category are defined to be continuous algebra morphisms. If we restrict attention to nuclear Fréchet algebras (resp., commutative nuclear Fréchet algebras), then we work in the full subcategory NFAlg (resp., CNFAlg).
Definition B.7. Fix A ∈ FAlg. A Fréchet A-module is a Fréchet vector space F , together with a continuous action which we will write as µ(a, v) := a · v (and which is of course compatible with the multiplication in A).
We give a short survey on Fréchet manifolds.
A new F -chart (U, φ) on M is compatible with a given smooth F -atlas, if and only if their union is again a smooth F -atlas, i.e., the subsets φ(U ∩ U α ) ⊂ F and φ α (U α ∩ U ) ⊂ F are open, and the transition maps are smooth (for every α ∈ A). Similarly, two smooth F -atlases are compatible provided their union is also a smooth F -atlas. Compatibility is an equivalence relation on all possible smooth F -atlases on M.
Definition B.10. A smooth F -structure on a set M is a choice of an equivalence class of smooth F -atlases on M. We say that M is a Fréchet manifold modelled on the Fréchet space F , if M comes equipped with a smooth F -structure. If the model vector space F is nuclear, we speak of a nuclear Fréchet manifold.
A smooth F -atlas on a Fréchet manifold M allows us to define in the obvious way a topology on M, which is independent of the atlas considered in the chosen equivalence class. The domain U of an F -chart (U, φ) is open in this topology and the bijective map φ : U ⊂ M → φ(U ) ⊂ F is a homeomorphism for the induced topologies. Most authors confine themselves to Fréchet manifolds, whose topology is Hausdorff.
Morphisms between two Fréchet manifolds are the smooth maps between them, where smoothness is defined, just as in the finite-dimensional case of smooth manifolds, in terms of charts and smoothness of local representatives of the maps. We denote the category of Fréchet manifolds and the morphisms between them by FMan.
Further, the tangent space T f M to a Fréchet manifold M at a point f ∈ M can be defined as usual, using the tangency equivalence relation for the smooth curves of M that pass through f at time 0. One can easily see that T f M is a Fréchet space. The concept of Fréchet vector bundle is the natural generalization of the notion of smooth vector bundle to the category of Fréchet manifolds. The tangent bundle TM of a Fréchet manifold M is an example of a Fréchet vector bundle.
In general, we must make a distinction between the (kinematic) tangent bundle as defined here and the operational tangent bundle defined in terms of derivations of the algebra of functions of a Fréchet manifold. Indeed, the two notions do not, in general, coincide, there are derivations that do not correspond to tangent vectors. However, it is known that for nuclear Fréchet manifolds the two concepts do coincide.
Let F : M → N be a smooth map between Fréchet manifolds modelled on Fréchet spaces F and G, respectively. There is a tangent map TF of F, which is a smooth map TF : TM → TN , and restricts, for any f ∈ M, to a linear map As in the finite-dimensional case, the local representative of T f F is the derivative d φ(f ) ψFφ −1 of the corresponding local representative ψFφ −1 : φ(U ) ⊂ F → G of F at the point φ(f ).
Fundamental to the work in this paper are Fréchet manifolds with a further module structure on their tangent bundle.
Definition B.11. Let M be a Fréchet manifold, whose model Fréchet space F is a module over a Fréchet algebra A. We say that M is a Fréchet A-manifold, if and only if all transition maps are A-linear, i.e., d φα(f ) φ βα (a · v) = a · d φα(f ) φ βα (v), for all f ∈ U α ∩ U β , a ∈ A, and v ∈ F .
Morphisms between Fréchet A-manifolds M and N are the A-smooth maps between them, i.e., are the smooth maps F : M → N that are A-linear at every point. This means that, for any point f ∈ M, there is an M-chart (U, φ) around f and an N -chart (V, ψ) around F(f ) that contains F(U ), such that the local representative of the derivative T f F is an A-linear endomorphism of the A-module F . The requirement actually means that the derivative T f F must be A-linear at any point f ∈ M. In this way, we obtain the category of Fréchet A-manifolds, which we denote as AFMan.
In this paper, we will use the category AFMan, whose objects are the Fréchet A-manifolds, where A is not a fixed Fréchet algebra, but any Fréchet algebra. The definition of AFManmorphisms generalizes the definition of AFMan-morphisms. Suppose that M is a Fréchet Amanifold modelled on an A-module F and N is a Fréchet B-manifold modelled on a B-module G. The AFMan-morphisms from M to N are the A-smooth maps between them, i.e., those smooth maps F : M → N that are at any point compatible with the module structures of F and G. This means that there is a Fréchet algebra morphism ρ : A → B, and, for any f ∈ M, there exist charts (U, φ) and (V, ψ) as above, such that for any a ∈ A and v ∈ F . This requirement actually means that, for any f , the derivative T f F is compatible with the induced actions on the tangent spaces. We will refer to an A-smooth map with associated Fréchet algebra morphism ρ, as a ρ-smooth map. If we restrict our attention to nuclear objects, i.e., the model Fréchet vector space and the Fréchet algebra are both nuclear, then we denote the corresponding category as ANFMan.