Linear ${\mathbb Z}_2^n$-Manifolds and Linear Actions

We establish the representability of the general linear ${\mathbb Z}_2^n$-group and use the restricted functor of points - whose test category is the category of ${\mathbb Z}_2^n$-manifolds over a single topological point - to define its smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces and linear ${\mathbb Z}_2^n$-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.


Introduction
In order to be able to deal with the technical details of vector bundles and related structures in the category of Z n 2 -manifolds (for n = 1 see [6]), we need some foundational results on Z n 2 -Lie groups and their smooth linear actions on linear Z n 2 -manifolds. However, the proofs of some folklore results, i.e., results that we tended to accept somewhat hands-waving, are often not at all obvious in the Z n 2 -context. The present paper, beyond its supposed applications, intrinsic interest and the beauty of some of its developments, raises the question of the scientific value of "results" that are partially based on speculations.
Loosely speaking, Z n 2 -manifolds (Z n 2 = Z ×n 2 ) are "manifolds" for which the structure sheaf has a Z n 2 -grading and the commutation rules for the local coordinates comes from the standard scalar product (see [11,13,14,15,18,19,20,21,37] for details). This is not just a trivial or straightforward generalization of the notion of a standard supermanifold, as one has to deal with formal coordinates that anticommute with other formal coordinates, but are themselves not nilpotent. Due to the presence of formal variables that are not nilpotent, formal power series are used rather than polynomials. Recall that for standard supermanifolds all functions are polynomial in the Grassmann odd variables. The theory of Z n 2 -geometry is currently being developed and many foundational questions remain. For completeness, we include Appendix B in which the foundations of Z n 2 -geometry are given. In this paper, we examine the relation between Z n 2 -graded vector spaces and linear Z n 2 -manifolds, and then we use this to define linear actions of Z n 2 -Lie groups. In the literature on supergeometry, the symbol R p|q has two distinct, but related meanings. First, we have the notion of a Z 2 -graded, or super, vector space with p even and q odd dimensions, i.e., the real vector space R p|q = R p R q . Secondly, we have the locally ringed space R p|q = R p , C ∞ R p [ξ] , where ξ i (i ∈ {1, . . . , q}) are the generators of a Grassmann algebra. The difference can be highlighted by identifying the points of these objects. The Z 2 -graded vector space has as its underlying topological space R p+q (we just forget the "superstructure"), while for the locally ringed space the topological space is R p . There are several ways of showing that these two notions are deeply tied. In particular, the category of finite dimensional super vector spaces is equivalent to the category of "linear supermanifolds" (see [8,32,33,34,38]).
In this paper, we will show that the categories of finite dimensional Z n 2 -graded vector spaces V and linear Z n 2 -manifolds V are isomorphic. We do this by explicitly constructing a "manifoldification" functor M, which associates a linear Z n 2 -manifold to every finite dimensional Z n 2 -graded vector space, and a "vectorification" functor, which is the inverse of the previous functor. It turns out that working in a coordinate-independent way (V, V ) is much more complex than working with canonical coordinates R p|q , R p|q .
Throughout this article, a special focus is placed on functors of points. The functor of points has been used informally in physics as from the very beginning. It is actually of importance in contexts where there is no good notion of point as in super-and Z n 2 -geometry and in algebraic geometry. For instance, homotopical algebraic geometry [42,43] and its generalisation that goes under the name of homotopical algebraic geometry over differential operators [25,26], are completely based on the functor of points approach. In this paper, we are particularly interested in functors of Λ-points, i.e., functors of points from appropriate locally small categories C to a functor category whose source is not the category C op but the category G of Z n 2 -Grassmann algebras Λ. However, functors of points that are restricted to the very simple test category G are fully faithful only if we replace the target category of the functor category by a subcategory of the category of sets.
More precisely, closely related to the above isomorphism of supervector spaces and linear supermanifolds is the so-called "even rules". Loosely this means including extra odd parameters to render everything even and in doing so one removes copious sign factors (see for example [24,Section 1.7]). We will establish an analogue of the even rules in our higher graded setting which we will refer to as the "zero degree rules" (see Definition 2.1). To address this we will make extensive use of Z n 2 -Grassmann algebras Λ, Λ-points and the Schwarz-Voronov embedding, which is a fully faithful functor of points S from Z n 2 -manifolds to a functor category with source G and the category of Fréchet manifolds (see for example [30]) over commutative Fréchet algebras as target (see [13]). We show that the zero degree rules functor F, understood as an assignment of a functor from G to the category of modules over commutative (Fréchet) algebras, given a (finite dimensional) Z n 2 -graded vector space, is fully faithful (see Theorem 2.2 and Proposition 2.25). The "zero degree rules" allow one to identity a finite dimensional Z n 2 -graded vector space, considered as a functor, with the functor of points of its "manifoldification". In other words, the composite S • M and F can be viewed as functors between the same categories and are naturally isomorphic. This identification is fundamental when describing linear group actions on Z n 2 -graded vector spaces and linear Z n 2 -manifolds. Another important part of this work is the category of Z n 2 -Lie groups and its fully faithful functor of points valued in a functor category with G as source category and Fréchet Lie groups over commutative Fréchet algebras as target category. We define the general linear Z n 2 -group as a functor in this functor category and show that it is representable, i.e., is a genuine Z n 2manifold (see Theorem 3.4). This leads to interesting insights into the computation of the inverse of an invertible degree zero Z n 2 -graded square matrix of dimension p|q with entries in a Z n 2 -commutative algebra. Furthermore, the approach using Λ-points and the zero rules allows us to construct a canonical smooth linear action of the general linear Z n 2 -group on Z n 2 -graded vector spaces and linear Z n 2 -manifolds. All these notions, in particular the equivalence between the definitions of a smooth linear action as natural transformation and as Z n 2 -morphism, are carefully and explicitly explained in the main text.
We remark that many of the statements in this paper are not surprising in themselves. However, due to the subtleties of Z n 2 -geometry, many of the proofs are much more involved than the analogue statements in supergeometry. The main source of difficulty is that one has to deal with formal power series in non-zero degree coordinates, rather than polynomials as one does 2 Z n 2 -graded vector spaces and Linear Z n 2 -manifolds 2.1 Z n 2 -graded vector spaces and the zero degree rules When dealing with linear superalgebra one encounters the so-called even rules (see [16,Section 1.8], [24,Section 1.7] and [45, pp. 123-124], for example). Very informally, the idea is to remove sign factors by allowing extra parameters to render the situation completely even. The idea has been applied in physics since the early days of supersymmetry. More precisely, let be the even part of the extension of scalars in a (real) super vector space V , from the base field R to a supercommutative algebra A ∈ SAlg (in the even rules that we are about to describe, it actually suffices to use supercommutative Grassmann algebras A = R[θ 1 , . . . , θ N ]: the θ i are then the extra parameters mentioned before). The main result in even rules states, roughly, that defining a morphism φ : V ⊗ V → V is equivalent to defining it functorially on the even part of V after extension of scalars, i.e., is equivalent to defining a functorial family of morphisms (indexed by A ∈ SAlg). More precisely, there is a 1 : 1 correspondence between parity respecting R-linear maps φ : V 1 ⊗ · · · ⊗ V n → V and functorial families φ(A) : V 1 (A) × · · · × V n (A) → V (A) (A ∈ SAlg) of A 0 -multilinear maps. We now proceed to generalise this theorem to the Z n 2 -setting. We will work with the category Z n 2 GrAlg of Z n 2 -Grassmann algebras rather than the category Z n 2 Alg of all Z n 2 -commutative algebras.
Let V = N i=0 V γ i be a (real) Z n 2 -graded vector space, i.e., a (real) vector space with a direct sum decomposition over i ∈ {0, . . . , N } (we say that the vectors of V γ i are of degree γ i ∈ Z n 2 ).
The category of Z n 2 -graded vector spaces (not necessarily finite dimensional) we denote as Z n 2 Vec. Morphisms in this category are degree preserving linear maps. We denote the category of modules over commutative algebras as AMod (see Appendix A).
To V we associate a functor V (−) ∈ Fun 0 Z n 2 Pts op , AMod in the category of those functors whose value on any Z n 2 -Grassmann algebra Λ ∈ Z n 2 Pts op is a Λ 0 -module, and of those natural transformations that have Λ 0 -linear Λ-components. The functor V (−) is essentially the tensor product functor − ⊗ V . It is built in the following way. First, for every Z n 2 -Grassmann algebra Λ, we define where the tensor product is over R. Secondly, for any Z n 2 -algebra morphism ϕ * : Λ → Λ , we define where the RHS is the restriction of ϕ * ⊗ 1 V to the degree 0 part of Λ ⊗ V , so that V (ϕ * ) is (2.1) whose associated algebra morphism is the restriction (ϕ * ) 0 : Λ 0 → Λ 0 . It is clear that V (−) respects compositions and identities and is thus a functor, as announced. We thus get an assignment The map F is essentially − ⊗ • and is itself a functor. It associates to any grading respecting linear map φ : V → W and any Z n 2 -Grassmann algebra Λ, a Λ 0 -linear map The family F(φ) := φ − is a natural transformation from F(V ) to F(W ). Since F respects compositions and identities, it is actually a functor valued in the restricted functor category Fun 0 Z n 2 Pts op , AMod .
This result is the Z n 2 -counterpart of the 1 : 1 correspondence mentioned above.
Proof . We show first that the map F V,W is injective. Let φ, ψ : V → W be two degree preserving linear maps, and assume that F(φ) = φ − = ψ − = F(ψ), so that, for any Λ ∈ Z n 2 Pts op and any λ ⊗ v ∈ V (Λ), we have Notice now that and let Λ be the Grassmann algebra that has exactly one generator θ j in each non-zero degree γ j ∈ Z n 2 (N = 2 n − 1). For any For v 0 ∈ V 0 := V γ 0 and λ = 1, the same equation shows that φ and ψ coincide also on V 0 .
The vectors w 00 ∈ W 0 (see (2.6)) and w e j ,j ∈ W γ j (j = 0) (see (2.7)) are well-defined and depend obviously linearly on v 0 and v j , respectively. Hence, setting φ(v 0 ) = w 00 and φ(v j ) = w e j ,j (j = 0), we define a degree 0 linear map from V to W . Moreover, since (2.4) is clearly satisfied for Λ 1 and the θ i ⊗ v i (i ∈ {0, . . . , N }), it is satisfied for any Λ, which completes the proof of surjectivity.
Since F: Z n 2 Vec → Fun 0 Z n 2 Pts op , AMod is fully faithful, it is essentially injective, i.e., it is injective on objects up to isomorphism. It follows that Z n 2 Vec can be viewed as a full subcategory of the target category of F. The above considerations lead to the following definition.
is said to be representable, if there exists V ∈ Z n 2 Vec, such that F(V ) is naturally isomorphic to V.
As F is essentially injective, a representing object V , if it exists, is unique up to isomorphism. We therefore refer sometimes to V as "the" representing object.
2.2 Cartesian Z n 2 -graded vector spaces and Cartesian Z n 2 -manifolds In the literature, the space R p|q is viewed, either as the trivial Z n 2 -manifold with canonical Z n 2 -graded formal parameters ξ, or as the Cartesian Z n 2 -graded vector space where R p (resp., R q j ) is the term of degree γ 0 = 0 ∈ Z n 2 (resp., γ j ∈ Z n 2 ). Observe that we use the notation R • (resp., R • ), when R • is viewed as a vector space (resp., as a manifold). It can happen that we write R • for both, the vector space and the manifold, however, in these cases, the meaning is clear from the context. Further, we set q 0 = p, q = (q 0 , q 1 , . . . , q N ), and |q| = i q i . When embedding R q i (i ∈ {0, . . . , N }) into R p|q , we identify each vector of the canonical basis of R q i with the corresponding vector of the canonical basis of R |q| . We denote this basis by and assign of course the degree γ i to every vector e i k . We can now write The dual space of R p|q is defined by where Hom is the internal Hom of Z n 2 Vec, i.e., the Z n 2 -graded vector space of all linear maps, and where Hom γ i is the vector space of all degree γ i linear maps. We sometimes write Hom Z n 2 Vec instead of Hom. The dual basis of e i k i,k is defined as usual by so that ε k i is a linear map of degree γ i and Let us finally mention that any Z n 2 -vector x ∈ R p|q reads x = j, x j e j and that as usual. Notice now that if M is a smooth m-dimensional real manifold and (U, ϕ) is a chart of M , the coordinate map ϕ sends any point x ∈ M to ϕ(x) = (x 1 , . . . , x m ) ∈ R m , so that Hence, what we refer to as coordinate function x i ∈ C ∞ (U ) is actually the function ϕ i . Equations (2.8) and (2.9) suggest to associate to any Z n 2 -graded vector space R p|q a Z n 2 -manifold R p|q with coordinate functions ε k i . In other words, the associated p|q-dimensional Z n 2 -manifold will be the locally Z n 2 -ringed space where C ∞ R p is the standard function sheaf of R p , where the degree γ j linear maps ε 1 j , . . . , ε q j j (j ∈ {1, . . . , N }) are interpreted as coordinate functions or formal parameters of degree γ j , and where the degree 0 linear maps ε 1 0 , . . . , ε p 0 are viewed as coordinates in R p . We often set ξ j := ε j (j = 0) and x := ε 0 . (2.10) Remark 2.4. In the following, we denote the coordinates of R p|q by if we wish to make a distinction between the coordinates of degree 0, γ 1 , . . . , γ N , if we distinguish between zero degree coordinates and non-zero degree ones, or if we consider all coordinates together.
We refer to the category of Z n 2 -graded vector spaces R p|q (p, q 1 , . . . , q N ∈ N) and degree 0 linear maps, as the category Z n 2 CarVec of Cartesian Z n 2 -vector spaces. As just mentioned, the interpretation of the dual basis as coordinates leads naturally to a map where Z n 2 Man is the category of Z n 2 -manifolds and corresponding morphisms. This map can easily be extended to a functor. Indeed, if L : R p|q → R r|s is a morphism in Z n 2 CarVec it is canonically represented by a block diagonal matrix L ∈ gl r|s×p|q, R , its dual (Z n 2 -transpose) L ∨ : R r|s ∨ → R p|q ∨ which is represented by the standard transpose t L ∈ gl p|q × r|s, R is also a degree 0 linear map. If we set where i ∈ I, ∈ {1, . . . , s i } label the row and i ∈ I, k ∈ {1, . . . , q i } label the column, we get where ε i i, is the basis of R r|s ∨ . When using notation (2.10), we obtain . . , s j }).
where the last equality is obvious because of equation (2.10). Hence, the functor M is valued in the subcategory Z n 2 CarMan ⊂ Z n 2 Man of Cartesian Z n 2 -manifolds R p|q (p, q 1 , . . . , q N ∈ N) and Z n 2morphisms whose coordinate pullbacks are global linear functions of the source manifold that have the appropriate degree: The inverse "vectorification functor" V of this "manifoldification functor" M is readily defined: to a Cartesian Z n 2 -manifold R p|q we associate the Cartesian Z n 2 -vector space R p|q , and to a linear Z n 2 -morphism we associate the degree 0 linear map that is defined by the transpose of the block diagonal matrix coming from the morphism's linear pullbacks. It is obvious that Proposition 2.5. We have an isomorphism of categories between the full subcategory Z n 2 CarVec ⊂ Z n 2 Vec of Cartesian Z n 2 -vector spaces R p|q and the subcategory Z n 2 CarMan ⊂ Z n 2 Man of Cartesian Z n 2 -manifolds R p|q and Z n 2 -morphisms with linear coordinate pullbacks.
Remark 2.6. Let us stress that the Z n 2 -vector space of linear Z n 2 -functions is of course not an algebra. In the case p = 0, we get ] is the Z n 2 -Grassmann algebra that corresponds to R 0|q , and where Λ lin is the Z n 2 -vector space of homogeneous degree 1 polynomials in the θ 1 1 , . . . , θ q N N (with vanishing term Λ lin 0 of Z n 2 -degree zero).
We close this subsection with some observations regarding the functor of points. The Yoneda functor of points of the category Z n 2 Man is the fully faithful embedding In [13], we showed that Y remains fully faithful for appropriate restrictions of the source and target of the functor category, as well as of the resulting functor category. More precisely, we proved that the functor is fully faithful. The category A(N)FM is the category of (nuclear) Fréchet manifolds over a (nuclear) Fréchet algebra, and the functor category is the category of those functors that send a Z n 2 -Grassmann algebra Λ to a (nuclear) Fréchet Λ 0 -manifold, and of those natural transformations that have Λ 0 -smooth Λ-components. For any M ∈ Z n 2 Man and any R 0|m Λ ∈ Z n 2 Pts op , we have On the other hand, the Yoneda functor of points of the category Z n 2 CarVec is the fully faithful embedding The value of this functor on R 0|m R 0|m Λ, is the subset More precisely, Remark that, if we denote the coordinates of R p|q compactly by (u a ), the bijection in equation (2.17) sends a degree 0 linear map L to the linear pullbacks L * (u a ) of the corresponding Z n 2 -morphism L = M(L).
Remark 2.7. If we restrict the functor R p|q resp., the functor F R p|q from Z n 2 CarVec op Z n 2 CarMan op (resp., from Z n 2 Pts op ) to the joint subcategory Z n 2 CarPts op of Z n 2 -points and Z n 2morphisms with linear coordinate pullbacks, the restricted Hom functor R p|q is actually a subfunctor of the restricted tensor product functor F R p|q . This observation clarifies the relationship between the fully faithful "functor of points" the morphisms R p|q (L) and F R p|q (L) are defined on R p|q (Λ) and its supset F R p|q (Λ), respectively. When interpreting an element K of the first as an element of the second, we use the identifications Similar identifications are of course required when Λ is replaced by Λ . We thus get On the other hand, we have This completes the proof of the subfunctor-statement.

Finite dimensional Z n 2 -graded vector spaces and linear Z n 2 -manifolds
In this subsection, we extend equivalence (2.14) in a coordinate-free way.
2.3.1 Finite dimensional Z n 2 -graded vector spaces We focus on the full subcategory Z n 2 FinVec ⊂ Z n 2 Vec of finite dimensional Z n 2 -graded vector spaces, i.e., of Z n 2 -vector spaces V of finite dimension p|q p ∈ N, q = (q 1 , . . . , q N ) ∈ N ×N .
Clearly Z n 2 CarVec ⊂ Z n 2 FinVec is a full subcategory.
Above, we already used the canonical basis of R p|q , i.e., the basis maps a basis to a basis and is thus an isomorphism of Z n 2 -vector spaces. We already discussed extensively the functor of points F = F(•)(−) = •(−) of Z n 2 Vec. Since Z n 2 FinVec is a full subcategory of Z n 2 Vec, the functor F remains fully faithful when restricted to Z n 2 FinVec: Proposition 2.8. The functor of points F: Z n 2 FinVec → Fun 0 Z n 2 Pts op , AMod of the category Z n 2 FinVec is fully faithful. Remark 2.9. Later on, we consider linear Z n 2 -manifolds and denote them sometimes using the same letter V as for Z n 2 -vector spaces. We often disambiguate the concept considered by writing V in the vector space case.
Linear Z n 2 -Manifolds and Linear Actions 13 2.3.2 Linear Z n 2 -manifolds In this subsection, we investigate the category of linear Z n 2 -manifolds, linear Z n 2 -functions of its objects, as well as its functor of points.
Linear Z n 2 -manifolds and their morphisms. A Z n 2 -manifold of dimension p|q is a locally Z n 2 -ringed space M := |M |, O M that is locally isomorphic to R p|q .
The diffeomorphism h is referred to as a linear coordinate map or a linear one-chart-atlas.
We now mimic classical differential geometry and say that two linear one-chart-atlases are linearly compatible, if their union is a "linear two-chart-atlas". In other words: Definition 2.11. Two linear coordinate maps h 1 , h 2 : L → R p|q are said to be linearly compatible, if the Z n 2 -morphisms have linear coordinate pullbacks, i.e., if they are Z n 2 CarMan-morphisms.
Linear compatibility is an equivalence relation on linear one-chart-atlases. There is a 1 : 1 correspondence between equivalence classes of linear one-chart-atlases and maximal linear atlases, i.e., the unions of all linear one-chart-atlases of an equivalence class. For simplicity, we refer to a maximal linear atlas as a linear atlas.
Just as a classical smooth manifold is a set that admits an atlas, or, better, a set endowed with an equivalence class of atlases, a linear Z n 2 -manifold is a locally Z n 2 -ringed space L equipped with a linear atlas (L, h α ) α .
We continue working in analogy with differential geometry and define a linear Z n 2 -morphism between linear Z n 2 -manifolds as a locally Z n 2 -ringed space morphism, or, equivalently, a Z n 2morphism, with linear coordinate form: has linear coordinate pullbacks.
It follows that any linear coordinate map h of the linear atlas of a linear Z n 2 -manifold L, is a linear Z n 2 -morphism between the linear Z n 2 -manifolds L and R p|q . This justifies the name "linear coordinate map". Further, the inverse h −1 of h is a linear Z n 2 -morphism.
Proposition 2.13. If φ : L → L is a linear Z n 2 -morphism, then, for any linear coordinate maps (L, h ) and (L , k ) of the linear atlases of L and L , respectively, the Z n 2 -morphism k • φ • h −1 has linear coordinate pullbacks.
Proof . We use the notations of Definition 2.12 and Proposition 2.13. Since and each parenthesis of the RHS has linear pullbacks, their composite has linear pullbacks as well.
Proposition 2.14. Linear Z n 2 -manifolds and linear Z n 2 -morphisms form a subcategory Z n 2 LinMan ⊂ Z n 2 Man of the category of Z n 2 -manifolds. Further, Cartesian Z n 2 -manifolds and Z n 2 -morphisms with linear coordinate pullbacks form a full subcategory Z n 2 CarMan ⊂ Z n 2 LinMan. Proof . If φ : L → L and ψ : L → L are linear Z n 2 -morphisms, the composite Z n 2 -morphism is linear as well. Indeed, if k•φ•h −1 and q•ψ •p −1 have linear pullbacks, then q•ψ •k −1 has linear pullbacks and so has q The second statement is obvious.
, then for any chart (L, h ) of the linear atlas of L, we have . This follows from the equation and the compatibility of the two charts. As is also a vector subspace. We thus get vector subspaces O lin is an isomorphism of Z n 2 -vector spaces of dimension p|q. (ii) The restriction maps and the gluing property of O L endow O lin L with a sheaf of Z n 2 -vector spaces structure.
It is straightforward to check the first two statements. For the third one, let L (resp., L ) be of dimension p|q (resp., r|s) and denote the coordinates of the corresponding Cartesian Z n 2manifold by u a = x a , ξ A resp., v b = y b , η B . The morphism φ is linear, if and only if there exist linear coordinate maps (L, h) and (L , k), such that k • φ • h −1 has linear coordinate pullbacks, i.e., such that On the other hand, in view of the first item of the previous remark, the condition of the third item is equivalent to asking that The conditions (2.18) and (2.19) are visibly equivalent.
Functor of points of Z n 2 LinMan. We start with the following Proposition 2.17. For any linear Z n 2 -manifold L (of dimension p|q) and any Z n 2 -Grassmann algebra Λ R 0|m , the set of Λ-points of L admits a unique Fréchet Λ 0 -module structure, such that, for any chart h : L → R p|q of the linear atlas of L, the induced map The definition of the category FAMod of Fréchet modules over Fréchet algebras can be found in Appendix A. In the preceding proposition, it is implicit that the (unital) Fréchet algebra morphism that is associated to h Λ is id Λ 0 .
Proof . Let Λ ∈ Z n 2 GrAlg. In view of the fundamental theorem of Z n 2 -morphisms, there is a 1 : 1 correspondence between the Λ-points x * of R p|q and the (p + |q|)-tuples (we used this correspondence already in equation (2.16)). Indeed, the algebra Λ is the Z n 2commutative nuclear Fréchet R-algebra of global Z n 2 -functions of some R 0|m (in particular, the degree zero term Λ 0 of Λ is a commutative nuclear Fréchet algebra). 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 a nuclear Fréchet space. The latter statements can be found in [14]. The Fréchet Λ 0 -module structure on R p|q (Λ) is then defined by Since this action (which is compatible with addition in Λ 0 and addition in R p|q (Λ)) is defined using the continuous associative multiplication · : We now define the Λ 0 -module structure on L(Λ). Observe first that, for any chart map h : L R p|q : h −1 of the linear atlas of L, the induced maps h Λ : This defines a Λ 0 -module structure on L(Λ) that makes h Λ a Λ 0 -module isomorphism. The Λ 0 -module structures L(Λ) h and L(Λ) k that are implemented by h and another chart k of the linear atlas, respectively, are related by the Λ 0 -module isomorphism Hence, the Λ 0 -module structure on L(Λ) is well-defined. In order to get a Fréchet structure on the real vector space L(Λ) that we just defined, we need a countable and separating family of seminorms (p n ) n∈N , such that any sequence in L(Λ) that is Cauchy for every p n , converges for every p n to a fixed vector (i.e., a vector that does not depend on n). We define this family (of course) by transferring to L(Λ) the analogous family (ρ n ) n∈N of the Fréchet vector space R p|q (Λ) (see [14,Theorem 14]). In other words, for each y * ∈ L(Λ), we set It is straightforwardly checked that (p n ) n∈N is a countable family of seminorms that has the required properties. Moreover, the vector space isomorphism h Λ is an isomorphism of Fréchet vector spaces, i.e., a continuous linear map with a continuous inverse. We show that h Λ is continuous for the seminorm topologies implemented by the p n and the ρ n , i.e., that, for all n ∈ N, there exist m ∈ N and C > 0, such that for all y * ∈ L(Λ). This requirement is of course satisfied. Hence, the composite k −1 Λ • h Λ of isomorphisms of Fréchet spaces is an isomorphism of Fréchet spaces, so that the Fréchet space structure on L(Λ) is well-defined.
The Λ 0 -module structure and the Fréchet vector space structure on L(Λ) combine into a Fréchet Λ 0 -module structure, if they are compatible, i.e., if the Λ 0 -action is continuous. The condition is obviously satisfied as this action is the composite of the continuous maps id ×h Λ , and h −1 Λ . Further, the map h Λ is clearly a Fréchet Λ 0 -module isomorphism, for any h in the linear atlas of L.
There is obviously no other Fréchet Λ 0 -module structure on L(Λ) with that property. Indeed, if there were, it would be isomorphic to the Fréchet Λ 0 -module structure on R p|q (Λ), hence isomorphic to the Fréchet Λ 0 -module structure that we just constructed.
In the following, we denote the Λ 0 -action on L(Λ) by simple juxtaposition, i.e., we write ay * instead of a y * .
To proceed, we need some preparation. Let Fun 0 Z n 2 Pts op , FAMod be the category of functors F , whose values F (Λ) are Fréchet Λ 0modules, and of natural transformations β, whose Λ-components β Λ are continuous Λ 0 -linear maps. We already used above the category Fun 0 Z n 2 Pts op , AFM of functors, whose values are Fréchet Λ 0 -manifolds, and of natural transformations, whose components are Λ 0 -smooth maps. Proof . Observe first that composition of natural transformations (resp., identities of functors) is (resp., are) induced by composition (resp., identities) in the target category of the functors considered, which is (resp., are) in both target categories the standard set-theoretical composition (resp., identities). Hence, composition and identities are the same in both functor categories. However, we still have to show that objects (resp., morphisms) of the first functor category are objects (resp., morphisms) of the second. Let F be a functor with target FAMod. Since a Fréchet Λ 0 -module (i.e., a Fréchet vector space with a (compatible) continuous Λ 0 -action) is clearly a Fréchet Λ 0 -manifold, the functor F sends Z n 2 -Grassmann algebras Λ to Fréchet Λ 0 -manifolds F (Λ). Let now ϕ * : Λ → Λ be a morphism of Z n 2 -algebras. As F (ϕ * ) : F (Λ) → F (Λ ) is a morphism between Fréchet modules over the Fréchet algebras Λ 0 and Λ 0 , respectively, it is continuous and it has an associated continuous (unital, R-) algebra morphism ψ : Λ 0 → Λ 0 , such that for all a, a ∈ Λ 0 and all v, v ∈ F (Λ). We must show that F (ϕ * ) is a morphism between Fréchet manifolds over Λ 0 and Λ 0 , respectively, i.e., we must show that F (ϕ * ) is smooth and has first order derivatives that are linear in the sense of (2.22) (see [13]). Since, for any t ∈ R, we have for any x, v, v 1 , . . . , v k+1 ∈ F (Λ) and any k ≥ 1. Hence, all derivatives exist everywhere and are (jointly) continuous. This implies that F (ϕ * ) has the required properties, so that F is a functor with target AFM.
As for morphisms, let η : F → G be a natural transformation between functors valued in FAMod. Its Λ-components η Λ : F (Λ) → G(Λ) are continuous and Λ 0 -linear maps. Repeating the proof given in the preceding paragraph for F (ϕ * ), we obtain that η Λ is Λ 0 -smooth, i.e., is smooth and has Λ 0 -linear first order derivatives. Therefore, the morphism η of the functor category with target FAMod is a morphism of the functor category with target AFM.
are subcategories, we expect that: Proof . We have to explain why S sends linear Z n 2 -manifolds and linear Z n 2 -morphisms to objects and morphisms, respectively, of the target subcategory.
Let L ∈ Z n 2 LinMan. The functor S(L) is an object of the functor category with target AFM. Since composition and identities are the same in both target categories, it suffices to show that, for any Z n 2 -Grassmann algebra Λ, the value S(L)(Λ) = L(Λ) is a Fréchet Λ 0 -module and that, for any Z n 2 -algebra morphism ϕ * : Λ → Λ , the morphism is a morphism of the category FAMod. The first of the preceding conditions holds in view of Proposition 2.17. We start proving the second condition for L = R p|q . Since R p|q (ϕ * ) is a morphism of AFM, it is smooth, hence, continuous. Further, omitting the summation symbols and using our standard notation, we get It now suffices to recall that the Z n 2 -algebra morphism ϕ * is the pullback ϕ * over the whole base manifold { } of a Z n 2 -morphism ϕ : R 0|m → R 0|m , and that all pullbacks of Z n 2 -morphisms are continuous, so that the restriction ϕ * : Λ 0 → Λ 0 is a continuous algebra morphism. We are now able to prove that the second condition holds also for an arbitrary linear Z n 2 -manifold L. Indeed, since ϕ * : Λ → Λ is a morphism of Z n 2 -algebras, the map L(ϕ * ) : and, due to invertibility, that (we used our standard notation). Hence, the functor S(L) is an object of the functor category with target FAMod.
As for morphisms, we consider a linear Z n 2 -morphism φ : L → L and will prove that S(φ), which is a natural transformation φ − of the functor category with target AFM, i.e., a natural transformation with Λ 0 -smooth Λ-components φ Λ , has actually continuous (but this results from Λ 0 -smoothness) Λ 0 -linear components.
Let p|q (resp., r|s) be the dimension of L (resp., of L ). We first discuss the case of a linear between the corresponding Cartesian Z n 2 -manifolds with canonical coordinates x a , ξ A and y b , η B , respectively. We know from [13] that, if the Z n 2 -morphism (resp., the linear Z n 2morphism) Φ reads (where the right-hand sides have the appropriate degree and where the coefficients L * * are real numbers), then the Λ-component Φ Λ associates to the Λ-point Here, we used the obvious decomposition Λ = R ×Λ and wrote x a Λ = (x a || ,x a Λ ). The particular linear versions of equations (2.24a) and (2.24b) (in parentheses), show that the component Φ Λ is Λ 0 -linear, as needed.
In the general case of a linear Z n 2 -morphism φ : L → L , the Z n 2 -morphism Φ := k • φ • h −1 : R p|q → R r|s has linear coordinate pullbacks Φ * y b and Φ * η B (and is thus a linear Z n 2morphism), for any charts h and k of L and L , respectively. Proof . We need to prove that the map is bijective, for all linear Z n 2 -manifolds L, L . Since S is the restriction of the fully faithful functor S: Z n 2 Man → Fun 0 Z n 2 Pts op , AFM , the map S L,L is injective.
To prove that S L,L is also surjective, it actually suffices to show that the property holds for Cartesian Z n 2 -manifolds. Indeed, in this case, if η : − is a natural transformation in the same category from R p|q (−) to R r|s (−), and this transformation is implemented by a linear Z n 2 -morphism ϕ : R p|q → R r|s . It follows that where the latter composite is a linear Z n 2 -morphism. Let now H : R p|q (−) → R r|s (−) be a natural transformation of Fun 0 Z n 2 Pts op , FAMod , hence, a natural transformation of Fun 0 Z n 2 Pts op , AFM . We know from [13] that H is implemented by a Z n 2 -morphism Φ : R p|q → R r|s , but we still have to prove that this morphism is linear. It follows from equations (2.24a) and (2.24b) that H Λ = Φ Λ is given by where we set are the coefficients of the coordinate pullbacks by Φ, see equations (2.23a) and (2.23b)), and where the RHS-s have of course the same Z n 2 -degree as the corresponding coordinates of R r|s . Since H Λ is Λ 0 -linear, we have i.e., for any r ∈ R >0 ⊂ Λ 0 , any α, β and for any x || ∈ R p . When deriving with respect to r, we obtain so that setting r = 1 yields again for all α, β and all x || ∈ R p .
Recall now that Euler's homogeneous function theorem states that, if F ∈ C 1 (R p \{0}), then, for any ν ∈ R, we have is equivalent to In view of (2.27), we thus get where we could extend the equality from R p \ {0} to R p due to continuity. If r tends to 0 + , the limit of the LHS is F * αβ (0) ∈ R and, for n = 0 (resp., n = 1; resp., n ≥ 2), the limit of the RHS is 0 resp., F * αβ x || ; resp., +∞ · F * αβ x || . In the case n ≥ 2, we conclude that For n = 0, we get Observe that α = β = 0 is only possible in equation (2.25a). Differentiating (2.28), in the case n = 0, with respect to any component x a || of x || and simplifying by r, we obtain and taking the limit r → 0 + , we get Integration yields as F b 00 (0) = 0. In the remaining case n = |α| + |β| = 1, we have necessarily α = 0 and β = e a , or α = e A and β = 0 (the e * are of course the vectors of the canonical basis of R p and R |q| , respectively). For Z n 2 -degree reasons, the first (resp., second) possibility is incompatible with equation (2.25b) (resp., equation (2.25a)). Hence, the only terms in (2.25a) that still need being investigated are the terms (α, β) = (0, e a ). It follows from equation (2.28) and its limit r → 0 + (see above) that However, equations (2.26) and (2.30) imply that In equation (2.25b), the only terms that still need being investigated are the terms (α, β) = (e A , 0). Using again the limit r → 0 + of equation (2.28), we find and that the Z n 2 -morphism Φ that induces the natural transformation H is defined by the coordinate pullbacks i.e., that Φ is linear (see (2.23a), (2.23b), (2.24a), and (2.24b)). Z n 2 -symmetric tensor algebra. We start with some remarks on tensor and Z n 2 -symmetric tensor algebras over a (finite dimensional) Z n 2 -vector space (see [36] and [9]). Let

Isomorphism between finite dimensional
be of dimension p|q. The Z n 2 -symmetric tensor algebra of V is defined exactly as in the nongraded case, as the quotient of the Z n 2 -graded associative unital tensor algebra of V by the homogeneous ideal More precisely, for k ≥ 2, we have where Perm is the set of all permutations of i 1 ≤ · · · ≤ i k . For instance, if n = 1, i.e., in the standard super case, the space V ⊗3 is the direct sum of the tensor products whose three factors have the subscripts 000, 001, 010, 011, 100, 101, 110, 111. The notation we just introduced means that we write where we used the lexicographical order and where Further, as we are dealing with formal power series in this paper, we define the Z n 2 -graded tensor algebra of V by where Π k means that we consider not only finite sums of tensors of different tensor degrees, but full sequences of such tensors. The vector space structure on such sequences is obvious and the algebra structure is defined exactly as in the standard case. Indeed, for T k ∈ V ⊗k and U ∈ V ⊗ , we have T k ⊗U ∈ V ⊗(k+ ) and we just extend this tensor product by linearity. In other words, if It is clear that the just defined tensor multiplication endows T V with a Z n 2 -graded algebra structure. Indeed, since , which shows that T V is a Z n 2 -graded (associative unital) algebra (over R), as announced.
The idealĪ is homogeneous with respect to the decomposition Therefore, the Z n 2 -symmetric tensor algebra of V is given bȳ see [9]. We denote by the Z n 2 -commutative multiplication that is induced onSV by the multiplication ⊗ of T V . By definition, we have, for [T ] ∈ (SV ) γ i and [U ] ∈ (SV ) γ j (obvious notation), (2. 35) Notice further that, if i < j, the linear map ∩Ī and is therefore a finite sum of generators ofĪ: It follows that the LHS of equation (2.37) vanishes; hence, the first term of the RHS vanishes, due to the isomorphism V i ⊗ V j V j ⊗ V i , and thus T vanishes as well. In order to show that ι is also surjective, consider an arbitrary vector in V i V j . It reads is the corresponding class. This class coincides with [T ], since the difference of the representatives is a vector ofĪ. It follows that, for n = 2 for instance, we have in particular where ∨ (resp., ∧) is the symmetric (resp., antisymmetric) tensor product. Moreover, if the (finite dimensional) vector space V has dimension q 0 |q 1 , q 2 , q 3 , we denote the vectors of its basis (in accordance with the notation we adopted earlier in this text) by b i j , where i ∈ {0, 1, 2, 3} refers to the degrees 00, 01, 10, 11 and where j ∈ {1, . . . , q i }. The basis of the Z n 2 -symmetric tensor product (2.38) is then made of the tensors (j 1 ≤ j 2 and j 4 < j 5 < j 6 ), which can also be written b 0 (j 1 ≤ j 2 and j 4 < j 5 < j 6 ) (see (2.36)). More generally, the basis of V i 1 · · · V i k (i 1 ≤ · · · ≤ i k ) is made of the tensors Linear Z n 2 -Manifolds and Linear Actions 25 (j ≤ j +1 (resp., <), if i = i +1 and γ i , γ i +1 even (resp., odd)). To refer to the previous condition regarding the j-s, we write in the following j 1 ¡ · · · ¡ j k . Observe also that as well as that, in order to define a linear map on V i 1 · · · V i k (see (2.38)), it suffices to define a k-linear map on V i 1 × · · · × V i k that is Z n 2 -commutative in the variables i = · · · = i m . Manifoldification functor. If V is a Z n 2 -graded vector space, its dual V ∨ is defined by More explicitly, we consider the space of R-linear maps from V to R of any Z n 2 -degree. It is clear that the linear maps of degree γ i are the linear maps from V i to R (that vanish in any other degree). Hence, It follows that, if V is finite dimensional of dimension p|q, its dual V ∨ has the same dimension. Moreover, any basis Proposition 2.21. If V is a Z n 2 -graded vector space of dimension p|q, there is a non-canonical isomorphism of Z n 2 -commutative associative unital R-algebras where R[[ξ]] is the global function algebra of R 0|q .
On the other hand, it follows from equations (2.34) and (2.39) that, choosing a basis b j j, of V * (defined similarly as V ∨ * ) and denoting its dual basis by (β j ) j, , leads tō where α j ∈ N (resp., α j ∈ {0, 1}), if γ j , γ j is even (resp., odd). In view of (2.33) and (2.35), the multiplications of R[[ξ]] andS V ∨ * are exactly the same, so that the two Z n 2 -commutative algebras are canonically isomorphic, once a basis of V * has been chosen.
Remark 2.22. We denoted the isomorphism by to remind us of its dependence on the basis b j j, . We are now prepared to define the linear Z n 2 -manifold associated to a finite dimensional Z n 2vector space. From here we denote the vector space by V instead of V and reserve the notation V for the manifold V := M(V).
Hence, let V ∈ Z n 2 FinVec be of dimension p|q. The p-dimensional vector space V 0 of degree 0 is of course a smooth manifold of dimension p, as well as a linear Z n 2 -manifold V 0 of dimension p|0. On the other hand, the algebraS V ∨ * is a sheaf of Z n 2 -commutative associative unital Ralgebras over { }, i.e., it is a Z n 2 -ringed space with underlying topological space { }, and, in view of Proposition 2.21, this space is (non-canonically) globally isomorphic to (since Ω and { } are Z n 2 -chart domains; for more information about the problem with the function sheaf of product Z n 2 -manifolds, we refer the reader to [15]). In particular, the Z n 2 -algebras O V (V 0 ) and O R p|q (R p ) are isomorphic (see also Definition 13 of product Z n 2 -manifolds in [15]), so that the Z n 2 -manifolds V and R p|q are diffeomorphic (given what has been said above, the diffeomorphism is implemented by the choice of a basis of V). Finally V ∈ Z n 2 LinMan dim V = p|q . We define the manifoldification functor M on objects by We now define M on morphisms. A degree zero linear map L : V → W between finite dimensional vector spaces (of dimensions p|q and r|s, respectively) is a family of linear maps L i : V i → W i (i ∈ {0, . . . , N }). We denote the transpose maps by t L i : W ∨ i → V ∨ i . The linear map L 0 : V 0 → W 0 is of course a smooth map L 0 : V 0 → W 0 , where V 0 , W 0 are the vector spaces V 0 , W 0 viewed as smooth manifolds. The map L 0 can also be interpreted as Z n 2 -morphism L 0 : V 0 → W 0 between the Z n 2 -manifolds V 0 , W 0 (which are of dimension zero in all non-zero degrees). The base morphism of L 0 is L 0 itself and, for any open subset Ω ⊂ W 0 , the pullback (L 0 ) * Ω is the (unital) algebra morphism − • L 0 | ω : The linear maps t L j : Observe first that to define such a map, it suffices to define a linear map in each tensor degree k, hence, it suffices to define a linear map for any j 1 ≤ · · · ≤ j k (j a ∈ {1, . . . , N }). Since the k-linear maps are Z n 2 -commutative in the variables j = · · · = j m , they define the degree zero linear maps t L k j 1 ...j k we set t L 0 = id R and thus the degree zero linear mapS t L that we are looking Linear Z n 2 -Manifolds and Linear Actions 27 for. In view of our definitions, the latter is a (unital) Z n 2 -algebra morphism between the global function algebras of the Z n 2 -manifolds W > and V > , and it therefore defines a unique Z n 2 -morphism L > : V > → W > . The base morphism of L > is the identity c : { } → { }.
We thus get a Z n 2 -morphism We must now prove that the Z n 2 -morphism M(L) = L is a morphism of Z n 2 LinMan, i.e., that in linear coordinates it has linear coordinate pullbacks. As said above, the linear coordinate map k : W → R r|s is the product of the linear coordinate maps k 0 : W 0 → R r|0 and k > : W > → R 0|s . The first of these coordinate maps is implemented by a basis b W of W 0 and its global pullback where β j j, is the dual of a basis of W * . Based on what we just said and on the statement (2.43), we get that the coordinate pullbacks in the linear coordinate expression of L are LinMan, whose construction has been described above. It is almost obvious from the penultimate paragraph that the diffeomorphism h coincides with the diffeomorphism b. Indeed, the diffeomorphism h is the product of two Z n 2 -diffeomorphisms h 0 : V 0 → R p|0 and h > : V > → R 0|q (see k in the penultimate paragraph). The same holds for b, which is defined as where b 0 : V 0 → R p|0 and b > : V > → R 0|q (see (2.42) and (2.43)). The map h 0 is canonically induced by the basis (b 0 k ) k of V 0 , and so is b 0 ; hence h 0 = b 0 . The Z n 2 -diffeomorphism b > is defined by the corresponding Z n 2 -algebra isomorphism where the source algebra is Π α R ε α = R[[ξ]]. As seen above, this algebra morphism is fully defined by the transposes t b j : (R q j ) ∨ → V ∨ j and their action on the basis (ε j ) . The action is since the image of any v j = k v k j b j k ∈ V j by the two maps is v j . It follows that This yields b > = h > . Finally, we get Vectorification functor. In this subsection, we define the vectorification functor V: Z n 2 LinMan → Z n 2 FinVec.
If L ∈ Z n 2 LinMan has dimension p|q, we set where L i has dimension q i (q 0 = p). Further, in view of item (iii) of Remark 2.16, if φ : L → L is a morphism of Z n 2 LinMan, then t φ * is a degree preserving linear map The definition of V(φ) implies that V is a functor.

Compositions of the manifoldification and the vectorification functors.
(i) We first turn our attention to V • M. If is the product of the linear Z n 2 -manifolds V 0 and V > . Let b i i, be a basis of V with dual β i i, and induced Z n 2 -vector space isomorphism b : V → R p|q (we denote the induced diffeomorphism from V 0 to R p by b 0 ). As explained above, it defines a linear coordinate map (see (2.45), (2.43) and (2.44)). Using equation (2.13), denoting the basis of R p|q ∨ as usual by ε i i, , and remembering the identifications (2.10), we thus get (see (2.41)). If we choose a basis β j j, of L ∨ * , we havē where α k j ∈ N (resp., α k j ∈ {0, 1}), if γ j , γ j is even (resp., odd) (see (2.40)). Just as Remark 2.23. Let us mention that L and L denote a priori different linear Z n 2 -manifolds and that our goal is to show that they do coincide.
Recall first that, for any Z n 2 -manifold M , there is a projection of |M |-sheaves of Z n 2 -algebras and that M commutes with pullbacks. In particular, if h : L → R p|q is a linear coordinate map of L (a (linear) Z n 2 -diffeomorphism), its pullback is, for any open subset |U | ⊂ |L|, a Z n 2 -algebra isomorphism and it restricts to a Z n 2 -vector space isomorphism Further, as just said, we have on O R p|q (|h|(|U |)). Taking |U | = |L| and restricting the equality to degree zero linear functions (see (2.13)), we obtain or, equivalently, where (h * ) −1 is a vector space isomorphism from (L 0 ) ∨ = O lin L,γ 0 (|L|) to (R p ) ∨ and where − • |h| is an algebra isomorphism from C ∞ (R p ) to C ∞ (|L|). In view of the diffeomorphism |h| : |L| → R p , the smooth manifold |L| is linear. Hence, it is a finite dimensional vector space also denoted |L| and |h| is a vector space isomorphism, whose dual t |h| = − • |h| is a vector space isomorphism from (R p ) ∨ ⊂ C ∞ (R p ) to |L| ∨ . It follows (see also equation (2.50)) that the canonical map L is a vector space isomorphism from (L 0 ) ∨ to |L| ∨ . When identifying these vector spaces, we get L = id and |L| = L 0 , hence the corresponding linear manifolds do also coincide: |L| = L 0 .
Linear Z n 2 -Manifolds and Linear Actions 31 (see (2.48)). The Z n 2 -algebra morphism we get this way (notice that the targets of the arrows (2.53) and (2.54) are different) is visibly an isomorphism. Indeed, it is obviously injective, and it is surjective due to (2.52). It follows from (2.53) and ( Comparison of the functors of points. Since Z n 2 FinVec Z n 2 LinMan, the fully faithful functors of points F (see Proposition 2.8) and S (see Theorem 2.20) of these categories should coincide. However, up till now, the functor F is valued in Fun 0 Z n 2 Pts op , AMod , whereas the functor S is valued in Fun 0 Z n 2 Pts op , FAMod . Since FAMod is a subcategory of AMod, the latter functor category is a subcategory of the former. Hence, if we show that the image F(V) of any object V of Z n 2 FinVec is a functor of Fun 0 Z n 2 Pts op , FAMod ( ) and that the image F(φ) of any morphism φ : V → W of Z n 2 FinVec is a natural transforation of Fun 0 Z n 2 Pts op , FAMod ( * ), we can conclude that F is a functor F: Z n 2 FinVec → Fun 0 Z n 2 Pts op , FAMod .
We start proving ( ). Since FAMod is a subcategory of AMod, we just have to show that the image of any object Λ of Z n 2 Pts op is a Fréchet Λ 0 -module (•) and that the image Alg is a morphism of FAMod (•). To prove (•), we consider a basis of V (dim V = p|q), i.e., an isomorphism b : of Λ 0 -modules. We use this isomorphism to transfer to V(Λ) the Fréchet vector space structure of  16)), thus obtaining a well-defined Fréchet structure and making b Λ a Fréchet vector space isomorphism, i.e., a continuous linear map with continuous inverse. Since b Λ is Λ 0 -linear, the action · of Λ 0 on V(Λ) is related to its action on R p|q (Λ) by for any a ∈ Λ 0 and any v ∈ V(Λ). The action · is thus the composite of the continuous maps id × b Λ , , and b −1 Λ , hence, it is itself continuous. The Λ 0 -module and the Fréchet vector space structures on V(Λ) therefore define a Fréchet Λ 0 -module structure on V(Λ) and b Λ becomes an isomorphism of Fréchet Λ 0 -modules (for any basis b of V).
As concerns (•), recall that V(ϕ * ) is a (ϕ * ) 0 -linear map, where the algebra morphism (ϕ * ) 0 : Λ 0 → Λ 0 is the restriction of ϕ * . Observe now that, in view of (2.55), we have so that R p|q (ϕ * ) is continuous as product of continuous maps (indeed, the Z n 2 Alg-morphism ϕ * is continuous as pullback of the associated Z n 2 -morphism). As b − is a natural transformation of Fun 0 Z n 2 Pts op , AMod , we have so that V(ϕ * ) is continuous (and (ϕ * ) 0 -linear), hence, is a morphism of FAMod.
It remains to show that ( * ) holds. We know that F(φ) = φ − is a natural transformation of Fun 0 Z n 2 Pts op , AMod , i.e., its Λ-components φ Λ are Λ 0 -linear maps and the naturality condition is satisfied. It thus suffices to explain that φ Λ = (1 ⊗ φ) 0 is continuous. Since Λ is a Fréchet algebra, it is a locally convex topological vector space (LCTVS) and 1: Λ → Λ is a degree zero continuous linear map. Further, since V and W are finite dimensional Z n 2 -vector spaces, the degree zero linear map φ : V → W is automatically continuous for the canonical LCTVS structures on its source and target. It follows that 1 ⊗ φ and (1 ⊗ φ) 0 are continuous linear maps. Proof . The result is obvious in view of Proposition 2.8, since Fun 0 Z n 2 Pts op , FAMod is a subcategory of Fun 0 Z n 2 Pts op , AMod .
We are now ready to refine the idea expressed at the beginning of this subsection that the (fully faithful) functors of points Proof . In order to construct a natural isomorphism I : S• M → F, we must define, for any R p|q , a natural isomorphism of Fun 0 Z n 2 Pts op , FAMod that is natural in R p|q . To build I R p|q , we have to define, for each Λ, an isomorphism of Fréchet Λ 0 -modules that is natural in Λ. Recalling that the source and target of this arrow are respectively, we set where u k i = x k , ξ k j are the coordinates of R p|q and where e i k i,k is the canonical basis of R p|q . Since we actually used this 1 : 1 correspondence to transfer the Fréchet Λ 0 -module structure from R p|q (Λ) to R p|q (Λ) (see (2.20)), the bijection I R p|q ,Λ is an isomorphism of Fréchet Λ 0 -modules. This isomorphism is natural with respect to Λ. Indeed, if ϕ * : Λ → Λ is a Z n 2 -algebra map (with corresponding Z n 2 -morphism ϕ), we have It now suffices to check that I R p|q is natural with respect to R p|q . Hence, let L : R p|q → R r|s be a degree zero linear map and let L : R p|q → R r|s be the corresponding linear Z n 2 -morphism M(L). In order to prove that we have to show that the Λ-components of these natural transformations coincide. To find that these Fréchet Λ 0 -module morphisms coincide, we must explain that they associate the same image to every x ∈ R p|q (Λ). When denoting the coordinates of R r|s by u i = x , ξ j , we obtain in view of (2.11) and (2.12). It follows that where e i i, is the basis of R r|s . We are now able to prove Theorem 2.26.
Proof . For simplicity, we set In order to build a natural isomorphism I: S• M → F, we must define, for any V ∈ Z n 2 FinVec, a natural isomorphism and let b be a basis of V, or, equivalently, a Z n 2 -vector space isomorphism b : V → R p|q . In view of (2.47), the morphism M(b) : V → R p|q is a linear Z n 2 -diffeomorphism. Using Proposition 2.25 and Theorem 2.20, we obtain that is a natural isomorphism of T as well, the transformation is a natural isomorphism as requested. In view of equation (2.56), the transformation I V is well-defined, i.e., is independent of the basis chosen. It remains to show that I V is natural in V, i.e., that, for any degree zero linear map φ : V → W (dim W = r|s) and for any basis b (resp., c) of V (resp., W), we have or, equivalently, Since L := c • φ • b −1 is a degree zero linear map L : R p|q → R r|s , equation (2.56) allows once more to conclude.
Internal Homs. A topological property is a property of topological spaces that is invariant under homeomorphisms (isomorphisms of topological spaces). More intuitively, a "topological property" is a property that only depends on the topological structure, or, equivalently, that can be expressed by means of open subsets. Similarly, equivalences of categories ("isomorphisms" of categories) preserve all "categorical properties and concepts". Hence, an equivalence should preserve products. It turns out that this statement is actually correct. More precisely, if E: S → T is part of an equivalence of categories, then a functor D: I → S has limit s if and only if the functor E • D: I → T has limit E(s). Applying the statement to the discrete index category I with two objects {1, 2} and setting D(i) = s i (i ∈ {1, 2}), we get that s 1 and s 2 have product s if and only if E(s 1 ) and E(s 2 ) have product E(s). Now, the category Z n 2 FinVec has the obvious binary product ×. It follows that, for any vector spaces V, W ∈ Z n 2 FinVec, the manifolds M(V), M(W) ∈ Z n 2 LinMan have product If L, L ∈ Z n 2 LinMan, the categorical isomorphism implies that L = M( V(L)) and similarly for L , so that the product L × L exists and is (2.57) Hence, the category Z n 2 LinMan has finite products. Equation (2.57) shows that we got the product of Z n 2 LinMan by transferring to T := Z n 2 LinMan the product of S := Z n 2 FinVec. We can similarly transfer to T the closed symmetric monoidal structure of S. Indeed, the category Z n 2 Vec is closed symmetric monoidal for the standard tensor product − ⊗ Z n 2 Vec − of Z n 2 -vector spaces and the standard internal Hom Hom Z n 2 Vec (−, −) of Z n 2 -vector spaces, which is defined, on objects for instance, by for any V, W ∈ Z n 2 Vec. Of course, if V, W ∈ S, then Hom Z n 2 Vec (V, W) ∈ S, and the same holds for V ⊗ Z n 2 Vec W. It follows that S = Z n 2 FinVec is also a closed symmetric monoidal category. If we set now and similarly for morphisms, we get a closed symmetric monoidal structure on T = Z n 2 LinMan: Proposition 2.28. The category Z n 2 LinMan is closed symmetric monoidal for the structure (2.58).
Alternatively, we could have defined Hom T (L, L ) ∈ T using the fully faithful functor of points To shed some light on our more abstract definition above, we now compute Hom T R p|q , R r|s (Λ) ( ) assuming some familiarity with Z n 2 -graded matrices gl(r|s × p|q, Λ) with entries in Λ ∈ Z n 2 Alg. Details can be found in Section 3.1 which we leave in its natural place. However, we highly recommend reading it before working though the end of this section.
We observe first that Hom Z n 2 Vec,γ k R p|q , R r|s = gl γ k r|s × p|q, R ∈ Vec.
In order to understand the gist here, we consider the case n = 2, so that a matrix X ∈ gl γ k r|s× p|q, R has the block format where the degree x ij of the block X ij is Since the entries of the X ij are real numbers and so of degree γ 0 , all the blocks with non-vanishing x ij do vanish. For instance, if γ k = 01 ∈ Z n 2 (resp., γ k = 11) (do not confuse with the rowcolumn index 01 in X 01 (resp., 11 in X 11 )), the degree x ij = 0 if and only if ij ∈ {01, 10, 23, 32} (resp., ij ∈ {03, 12, 21, 30}) (as in most of the other cases in this text, the Z n 2 -degrees are lexicographically ordered), so that only these X ij do not vanish. It follows that is made of the matrices (2.59), where no block X ij vanishes a priori. The canonical basis of this Z n 2 -vector space are the obvious matrices E ik,j (i, j ∈ {0, . . . , N }, k ∈ {1, . . . , s i }, ∈ {1, . . . , q j }) with all entries equal to 0 except the entry kl in X ij which is 1. In view of equation (2.60), the vectors of this basis have the degrees γ i + γ j . We can of course identify (up to renumbering) this Z n 2 -vector space with R t|u , where u n (n ∈ {0, . . . , N }) is equal to (we set s 0 = r, q 0 = p, u 0 := t). Hence: Combining (2.58) and (2.62), we get We now come back to ( ). Setting as usual R 0|m Λ, we get the isomorphism On the other hand, the vector space gl 0 r|s × p|q, Λ is a Λ 0 -module and this module "coincides" obviously with gl 0 r|s × p|q, Λ = Π N n=0 Λ ×un γn .
By transferring the Fréchet structure, we get an "equality" of Fréchet Λ 0 -modules. Hence, the Fréchet Λ 0 -module isomorphism There is a natural upgrade that is independent of the internal Homs and makes G := gl 0 (r|s× p|q, −) a functor G ∈ Fun 0 Z n 2 Pts op , FAMod . Indeed, it suffices to define G on a Z n 2 Alg-morphism ϕ * : Λ → Λ as where ϕ * (X) is defined entry-wise. The morphism G(ϕ * ) is clearly (ϕ * ) 0 -linear. It is also continuous, as it can be viewed as a product of copies of ϕ * . Since G respects compositions and identities it is actually a functor of the functor category mentioned. The functors G and R t|u (−) = S(R t|u ) are of course naturally isomorphic. Since S is a fully faithful functor S: Z n 2 LinMan → Fun 0 Z n 2 Pts op , FAMod , the functor G can be viewed as represented by the linear Z n 2 -manifold R t|u . 3 Z n 2 -Lie groups and linear actions 3.1 Z n 2 -matrices We will consider matrices that are valued in some Z n 2 -Grassmann algebra Λ, though everything we say generalizes to arbitrary Z n 2 -commutative associative unital R-algebras. A homogeneous matrix X ∈ gl x r|s × p|q, Λ of degree x ∈ Z n 2 is understood to be a block matrix with the entries of each block X ij being elements of the Z n 2 -Grassmann algebra Λ. Here the degree x ij ∈ Z n 2 of X ij is (setting s 0 = r and q 0 = p as usual). Addition of such matrices and multiplication by reals are defined in the obvious way and they endow gl x r|s × p|q, Λ with a vector space structure. We set gl r|s × p|q, Λ := x∈Z n 2 gl x r|s × p|q, Λ ∈ Z n 2 Vec.
Multiplication by an element of Λ requires an extra sign factor given by the row of the matrix, i.e., for any homogeneous λ ∈ Λ γ k , we have that We thus obtain on gl r|s × p|q, Λ a Z n 2 -graded module structure over the Z n 2 -commutative algebra Λ. If r|s = p|q, we write gl p|q, Λ := gl(p|q × p|q, Λ).
Multiplication of matrices in gl(p|q, Λ) is via standard matrix multiplication -now taking care that the entries are from a Z n 2 -commutative algebra. Equipped with this multiplication, the Z n 2graded Λ-module gl p|q, Λ is a Z n 2 -graded associative unital R-algebra. In particular, the degree zero matrices gl 0 p|q, Λ form an associative unital R-algebra. Since multiplication of matrices only uses multiplication and addition in Λ, we can replace Λ not only, as said above, by any Z n 2 -commutative associative unital R-algebra, but also by any Z n 2 -commutative ring R and then get a ring gl 0 (p|q, R). We denote by GL(p|q, R) the group of invertible matrices in gl 0 (p|q, R). For further details the reader may consult [23].

Invertibility of Z n 2 -matrices
Let R be a Z n 2 -commutative ring which is Hausdorff-complete in the J-adic topology, where J is the (proper) homogeneous ideal of R that is generated by the elements of non-zero degree γ j ∈ Z n 2 , j ∈ {1, . . . , N }. The Z n 2 -graded ring morphism ε : R → R/J, where Linear Z n 2 -Manifolds and Linear Actions 39 vanishes in all non-zero degrees, induces a ring morphism ε : gl 0 p|q, R X →ε(X) ∈ Diag p|q, R/J , whereε(X) is the block-diagonal matrix with diagonal blocksε(X ii ) (with commuting entries).
The following proposition appeared as Proposition 5.1 in [22]: Let R be a J-adically Hausdorff-complete Z n 2 -commutative ring and let X ∈ gl 0 p|q, R be a degree zero p|q × p|q matrix with entries in R, written in the standard block format We have: In this work, we are of course mainly interested in the case R := Λ = R ⊕Λ and J =Λ, so that R/J = R.

Z n 2 -Lie groups and their functor of points
Groups, or, better, group objects can easily be defined in any category with finite products, i.e., any category C with terminal object 1 and binary categorical products c × c (c, c ∈ C). If C is a concrete category, the definition of a group object is very simple. For instance, if C is the concrete category AFM of Fréchet manifolds over a Fréchet algebra A, a group object G in C is just an object G ∈ C that is group whose structure maps µ : G× G → G and inv : G → G are C-morphisms, i.e., A-smooth maps. We refer of course to a group object in AFM as a Fréchet A-Lie group.
If C is the category Z n 2 Man of Z n 2 -manifolds, the definition of a group object is similar, but all the (natural) requirements (above) have to be expressed in terms of arrows (since there are no points here). More precisely, a group object G in C is an object G ∈ C that comes equipped with C-morphisms inv : G → G and e : 1 → G (the terminal object 1 is here the Z n 2 -manifold R 0|0 = ({ }, R)), which are called multiplication, inverse and unit, and satisfy the standard group properties (expressed by means of arrows): µ is associative, inv is a two-sided inverse of µ and e is a two-sided unit of µ. To understand the arrow expressions of these properties, we need the following notations. We denote by ∆ : G → G × G the canonical diagonal C-morphism and we denote by e G : G → G the composite of the unique C-morphism 1 G : G → 1 and the unit C-morphism e : 1 → G. The left inverse condition now reads µ • (inv × id G ) • ∆ = e G and the left unit condition reads µ • (e G × id G ) • ∆ = id G (and similarly for the right conditions). The associativity of µ is of course encoded by (3.1) We refer to a group object in Z n 2 Man as a Z n 2 -Lie group. A morphism F : G → G of Fréchet A-Lie groups is of course defined as an A-smooth map that is a group morphism. Analogously, a morphism F : G → G from a Fréchet A-Lie group to a Fréchet A -Lie group is a morphism of AFM that is also a group morphism. We denote the category of Fréchet A-Lie groups by AFLg and we write AFLg for the category of Fréchet Lie groups over any Fréchet algebra.
Further, a morphism Φ : G → G of Z n 2 -Lie groups is a Z n 2 -morphism that respects the multiplications, the inverses and the units (obvious arrow definitions). The category of Z n 2 -Lie groups we denote by Z n 2 Lg. The functor of points of Z n 2 -manifolds induces a fully faithful functor of points of Z n 2 -Lie groups: Therefore, in order to prove that the functor (3.2) restricts to a functor (3.3), it suffices to show that S sends objects G and morphisms Φ of Z n 2 Lg to objects and morphisms of the functor category with target AFLg. Observe first that, for any M, N ∈ Z n 2 Man, we have the functor equality  4). Now, if G ∈ Z n 2 Lg with structure Z n 2 -morphisms µ, inv (and e), then the AFM-valued functor S(G) = G(−) is actually AFLg-valued. This means that it sends any Z n 2 -Grassmann algebra Λ and any Z n 2 Alg-morphism ϕ * : Λ → Λ to an object G(Λ) and a morphism G(ϕ * ) of AFLg. For G(Λ) ∈ Λ 0 FM, notice that the natural transformations S(µ) = µ − , S(inv) = inv − (and S(e) = e − ) have Λ 0 -smooth Λ-components (the Fréchet Λ 0 -manifold 1(Λ) is the singleton that consists of the Z n 2 Alg-morphism ι Λ that sends any real number to itself viewed as an element of Λ) that define a group structure on G(Λ) (with unit 1 Λ := e Λ (ι Λ )), which is therefore a Fréchet Λ 0 -Lie group. The group properties of these structure maps are consequences of the group properties of the structure maps of G. For instance, when we apply S to the associativity equation (3.1) and then take the Λ-component of the resulting natural transformation, we get As for G(ϕ * ) : G(Λ) → G(Λ ), we know that it is an AFM-morphism and have to show that it respects the multiplications µ Λ and µ Λ , i.e., that However, this equality is nothing other than the naturalness property of µ − . Finally, let Φ : G → G be a Z n 2 Lg-morphism and denote the multiplications of the source and target by µ and µ , respectively. In order to prove that the natural transformation S(Φ) = Φ − : G(−) → G (−) of the functor category with target AFM is a natural transformation of the functor category with target AFLg, it suffices to show that Φ Λ is a morphism of AFLg, which results from the application of the functor S to the commutative diagram The next task is to show that the functor (3.3) is fully faithful, i.e., that the map is a 1 : 1 correspondence, for any Z n 2 -Lie groups G, G . Since the functor (3.2) is fully faithful, any natural transformation in the target set of (3.7) is implemented by a unique Z n 2 -morphism φ : G → G and it suffices to show that φ respects the group operations, for instance, that is satisfies equation (3.6). However, equation (3.6) is satisfied if and only if for all Λ. The latter condition holds, since φ Λ is, by assumption, a group morphism.
define visibly a natural transformation with Λ 0 -smooth Λ-components. Hence, it is implemented by a unique Z n 2 -morphism e : 1 → M . We leave it to the reader to check that µ, inv and e satisfy (3.1) and the other group properties.

The general linear Z n 2 -group
We want to define the general linear Z n 2 -group of order p|q so that it is a Z n 2 -Lie group GL p|q . In view of Theorem 3.2, it suffices to define a functor GL p|q (−) ∈ Fun 0 Z n 2 Pts op , AFLg that is represented by a Z n 2 -manifold GL p|q .
Theorem 3.4. The maps GL p|q (−) of Definition 3.3 define a representable functor. We refer to the representing object GL p|q ∈ Z n 2 Lg as the general linear Z n 2 -group of dimension p|q.
Proof . Recall that: 1. It follows from equation (2.64) that where u n is given by (2.61) (t = u 0 ). In particular, a matrix

It follows from
is invertible if and only if X ii ∈ GL(q i , R), for all i. It follows that as well as its functor of points (see [13]).
Linear Z n 2 -Manifolds and Linear Actions 43 On the other hand, we get so that U t|u (−) and GL p|q (−) "coincide" on objects Λ: if we denote the coordinates of R t|u as usually by (u a ) = x a , ξ A , this "equality" reads Moreover, U t|u (−) and GL p|q (−) coincide on morphisms ϕ * : Λ → Λ . Indeed, the map GL p|q (ϕ * ) acts on a matrix by acting on all its entries x * (u a ) by ϕ * , whereas the map U t|u (ϕ * ) acts on a Z n 2 Alg-morphism x * ∈ U t|u (Λ) by left composition ϕ * •x * ; if we identify x * with the tuple (x * (u a )) a , then U t|u (ϕ * ) acts by acting on each x * (u a ) by ϕ * , which proves the claim.
It follows that GL p|q (−) is a functor GL p|q (−) ∈ Fun 0 Z n 2 Pts op , AFM that is represented by so that it now suffices to prove that this functor is valued in AFLg, i.e., it suffices to show that GL p|q (Λ) ∈ Λ 0 FLg and that GL p|q (ϕ * ) is an AFLg-morphism.
Recall that gl 0 p|q, Λ is an associative unital R-algebra for the standard matrix multiplication · (standard matrix addition, standard matrix multiplication by reals and standard unit matrix I) (see Section 3.1). It is clear that the subset GL p|q (Λ) ⊂ gl 0 p|q, Λ is closed under ·: is an associative unital multiplication on GL p|q (Λ). Therefore, µ Λ and inv Λ : GL p|q (Λ) X → X −1 ∈ GL p|q (Λ) (3.12) endow GL p|q (Λ) with a group structure (with unit I). Finally, the Fréchet Λ 0 -manifold GL p|q (Λ) together with its group structure µ Λ , inv Λ (and I) is a Fréchet Λ 0 -Lie group, if its structure maps µ Λ and inv Λ are Λ 0 -smooth. This condition is actually satisfied (see below). As for GL p|q (ϕ * ), we know that it is an AFM-morphism and need to show that it respects the multiplications µ Λ , µ Λ . This condition is clearly met because GL p|q (ϕ * ) acts entry-wise by the Z n 2 Alg-morphism ϕ * . It remains to explain why µ Λ and inv Λ are Λ 0 -smooth. Notice first that the source of the multiplication (3.11) is the open subset Ω(Λ) := U t|u (Λ) × U t|u (Λ) of the Fréchet space F (Λ) := R t|u (Λ) × R t|u (Λ) (see [13]) and that we can choose the Fréchet vector space (and Fréchet Λ 0 -module) R t|u (Λ) as its target. Since Λ is the (Z n 2commutative nuclear) Fréchet R-algebra of global Z n 2 -functions of some Z n 2 -point R 0|m , its addition and internal multiplication (its multiplication by reals and subtraction) are continuous maps. It follows that each component function of the standard matrix multiplication µ Λ is continuous, so that µ Λ is itself continuous. We must now explain why all directional derivatives of µ Λ exist everywhere and are continuous, and why the first derivative is Λ 0 -linear. Let (X, Y ) ∈ Ω(Λ) and (V, W ) ∈ F (Λ). We get Hence, the first derivative exists everywhere, is continuous and Λ 0 -linear. Indeed, for any a ∈ Λ 0 , we have It is easily checked that we start computing the directional derivative of (∆ Λ is the diagonal map), assuming continuity of inv Λ , for the time being. For any V ∈ R t|u (Λ), we have It follows that so that the first derivative is defined everywhere, is continuous, as well as Λ 0 -linear. Also the higher order derivatives exist everywhere and are continuous. For instance, the second order derivative is given by Finally, the inverse map inv Λ is Λ 0 -smooth, provided we prove its still pending continuity. We will show that the continuity of (3.12) boils down to the continuity of the inverse map ι Λ : Λ × λ → λ −1 ∈ Λ × in Λ. Here Λ × ⊂ Λ is the group of invertible elements of Λ. Since Λ is a (unital) Fréchet R-algebra, its inverse map ι Λ is continuous if and only if Λ × is a G δ -set, i.e., if and only if it is a countable intersection of open subsets of Λ [47]. We will show that Λ × is actually open in the specific Fréchet R-algebra Λ considered. In view of Equation (16) in [14], the topology of Λ = R[[θ]] (Λ R 0|m ) is induced by the countable family of seminorms where ε is the projection ε : Λ → R. This means that the topology is made of the unions of finite intersections of the open semiballs is the open ball in R with center ν β and radius ε . Since and which implies that Λ × is open and that ι Λ is continuous, as announced. Before we are able to deduce from this that inv Λ is continuous, we need an inversion formula for X ∈ GL p|q (Λ). Notice first that, in view of [23,Proposition 4.7], an invertible 2 × 2 block matrix with square diagonal blocks A and D and entries (of all blocks) in a ring, has a block UDL decomposition if and only if D is invertible. In this case, the UDL decomposition is As upper and lower unitriangular matrices are obviously invertible, it follows that the diagonal matrix is invertible, hence that A − BD −1 C is invertible. Similarly, the invertible matrix X has a block LDU decomposition if and only if A is invertible and in this case D − CA −1 B is invertible. Moreover, in view of Proposition 3.1, a matrix X ∈ gl 0 p|q, Λ is invertible if and only if all its diagonal blocks X ii are invertible. Let now be a 2 × 2 block decomposition of X ∈ gl 0 p|q, Λ that respects the (N + 1) × (N + 1) block decomposition Since A (resp., D) is invertible if and only if A = A 0 0 I resp.,D = I 0 0 D is invertible, hence, if and only if the X kk on the diagonal of A (resp., D) are invertible, we get that X is invertible if and only if A and D are invertible. If we combine everything we have said so far in this paragraph, we find that if X ∈ GL p|q (Λ), then A, D, A − BD −1 C, D − CA −1 B are all invertible. Therefore, we can use the formula for any X ∈ GL p|q (Λ). In order to simplify proper understanding, we consider for instance the case n = 2, p|q = p|q 1 , q 2 , q 3 = 1|2, 1, 1 and and so on. We focus for instance on the first of the four block matrices in X −1 , i.e., on A − BD −1 C −1 . The matrix D is a 2 × 2 invertible matrix with square diagonal blocks and entries in Λ. Since the four diagonal block matrices in X are invertible, it follows from what we have said above that the inverse D −1 is given by equation (3.13) with A = t ∈ Λ, B = u ∈ Λ, C = y ∈ Λ and D = z ∈ Λ. Hence all entries of D −1 are composites of the addition, the subtraction, the multiplication and the inverse in Λ, and so are all entries in the invertible 2 × 2 block matrix with square diagonal blocks (which are invertible) and with entries in Λ (the square diagonal blocks have entries in Λ 0 ). Hence, the inverse (A − BD −1 C) −1 can again be computed by (3.13). We focus on its entry Notice that here we cannot conclude that ε and ξ are invertible and apply (3.13) to compute the internal inverse. However, this inverse is the inverse of a square matrix with entries in the commutative ring Λ 0 , for which the standard inversion formula holds (recall that a square matrix with entries in a commutative ring is invertible if and only if its determinant is invertible): Since all the entries of (3.14) are composites of the addition, subtraction, multiplication and inverse in Λ, it follows from (3.15) and (3.16) that the same is true for the entry κ of X −1 . More precisely the entry κ corresponds to a mapκ that is a composite of the inclusion of GL p|q (Λ) into its topological supspace Λ ×(t+|u|) (continuous), the projection of Λ ×(t+|u|) onto Λ ×v (v ≤ t+|u|) (continuous) and of products of the identity map id of Λ (continuous), the diagonal map ∆ of Λ (continuous), the switching map σ of Λ × Λ (continuous), the addition a of Λ (continuous), the scalar multiplication e of Λ (continuous), its subtraction s (continuous), multiplication m (continuous) and its inverse ι (continuous). Indeed, it is for instance easily seen that the map Λ ×4 (t, u, y, z) → −z −1 y t − uz −1 y −1 ∈ Λ is a (continuous) composite of products of these continuous maps. We thus understand that the entry κ of X −1 corresponds to a continuous mapκ : GL p|q (Λ) → Λ. The same holds of course also for all the other entries of X −1 . Finally, the inverse map is continuous and it remains continuous when view as valued in the subspace GL p|q (Λ).

Smooth linear actions
In this section we define linear actions of Z n 2 -Lie groups G on finite dimensional Z n 2 -vector spaces V V (we identify the isomorphic categories Z n 2 FinVec and Z n 2 LinMan). The definition can be given in the category of Z n 2 -manifolds, but it is slightly more straightforward if we use the functor of points. Notice that the functors of points of G ∈ Z n 2 Lg ⊂ Z n 2 Man and V ∈ Z n 2 LinMan ⊂ Z Definition 3.6. Let G ∈ Z n 2 Lg and V ∈ Z n 2 LinMan. A smooth linear action of G on V is a natural transformation in Fun 0 Z n 2 Pts op , AFM (natural transformation with Λ 0 -smooth Λ-components) that satisfies the following conditions: where 1 Λ is the unit of G(Λ).
where · is the action of Λ 0 on V (Λ).
is fully faithful (for more details, see [13,14,15]), there is a 1 : 1 correspondence between natural transformations σ − as above and Z n 2 -morphisms This correspondence implies in particular that condition (ii) is equivalent to the equality The same holds for condition (i) and the equality

Canonical action of the general linear group
We will now define the canonical action of the general linear Z n 2 -group GL p|q = U t|u ∈ Z n 2 Lg on the Cartesian Z n 2 -manifold R p|q ∈ Z n 2 LinMan. To do this, we use both, the fully faithful functor (3.17)  We start defining a natural transformation σ − of Fun Z n 2 Man op , Set from U t|u (−) × R p|q (−) to R p|q (−). We will denote the coordinates of U t|u (resp., R p|q ) here by X a b (resp., x c ), where a, b ∈ {1, . . . , p + |q|} (resp., where c ∈ {1, . . . , p + |q|}). For this, we must associate to any S ∈ Z n 2 Man, a set-theoretical map σ S that assigns to any (X, φ) ∈ U t|u (S) × R p|q (S) = Hom Z n 2 Man S, U t|u × Hom Z n 2 Man S, R p|q , i.e., to any (appropriate) coordinate pullbacks a unique element σ S (X, φ) ∈ R p|q (S), i.e., unique (appropriate) coordinate pullbacks Since x c S c is viewed as a tuple (horizontal row), the natural definition of this image (horizontal row) is where the sum and products are taken in the global Z n 2 -function algebra O(S) of S. It is clear that the elements of this target-tuple have the required degrees, as the same holds for the elements of the source-tuple. The transformation σ − we just defined is clearly natural. Indeed, for any Z n 2morphism ψ : S → S, the induced set-theoretical mapping between the Hom-sets with source S and the corresponding ones with source S is − • ψ, so that the induced set-theoretical mapping between the tuples of global Z n 2 -functions of S and S is ψ * . The naturalness of σ − follows now from the fact that ψ * is a Z n 2 Alg-morphism. Since (3.20) is fully faithful, the natural transformation σ − is implemented by a unique Z n 2morphism which in turn implements, via (3.17), a unique natural transformation in Fun 0 Z n 2 Pts op , AFM between the same functors, but restricted to Z n 2 Pts op . Since this transformation is the restriction of σ − to Z n 2 Pts op , we use this symbol for both transformations (provided that any confusion can be excluded). It is easily seen that with sum and products in Λ, has the properties (i), (ii) and (iii) of Definition 3.6, so that we defined a smooth linear action of GL p|q on R p|q . The interesting aspect here is that we are able to compute the Z n 2 -morphism (3.23). Indeed, in view of the proof of the full faithfulness of the standard Yoneda embedding c → Hom C (−, c) of an arbitrary locally small category C into the functor category Fun C op , Set , the morphism σ ∈ Hom C (c, c ) that implements a natural transformation σ − : Hom C (−, c) → Hom C (−, c ) is σ = σ c (id c ) ∈ Hom C (c, c ).
In our case of interest C = Z n 2 Man, the previous Yoneda embedding is the functor (3.20) and the morphism σ ∈ Hom Z n 2 Man GL p|q × R p|q , R p|q is σ = σ c (id c ), with c = GL p|q × R p|q .
Since the pullback of the identity Z n 2 -morphism id c is identity and the coordinate pullbacks (3.21) are

Equation (3.22) yields
with sum and products in O(c). In other words: Proposition 3.7. The canonical action σ of the general linear Z n 2 -group GL p|q on the linear Z n 2 -manifold or Z n 2 -graded vector space R p|q , is the Z n 2 -morphism that is defined by the coordinate pullbacks

24)
where we denoted the coordinates of GL p|q (resp., R p|q ) by X a b (resp., x c ).
An obvious identification leads now to Hom Z n 2 LinMan R p|q , R p|q (Λ) When comparing (3.26) and (3.25), we see that the internal Hom of linear Z n 2 -manifolds consists of the pullbacks of the internal Hom of arbitrary Z n 2 -manifolds which are defined by the canonical action of gl 0 p|q (Λ) on R p|q , in the sense of (3.24).

Equivalent definitions of a smooth linear action
Section 3.5 already implicitly contained the idea that a smooth linear action of a Z n 2 -Lie group G on a linear Z n 2 -manifold V in the sense of Definition 3.6, is equivalent to a Z n 2 -morphism σ : G × V → V that satisfies the conditions (3.18) and (3.19) and additionally has a certain linearity property with respect to V . A natural idea is that σ * should send linear Z n 2 -functions of V to Z n 2 -functions of G × V that are linear along the fibers. The meaning of this concept becomes clear when we think of the classical differential geometric case in which the functions of a trivial vector bundle E = M × R r are C ∞ (E) = Γ(∨E * ) = C ∞ (M ) ⊗ ∨ (R r ) * (∨ is the symmetric tensor product), i.e., are the functions that are smooth in the base and polynomial along the fiber. Hence, linear functions of E are the functions that are smooth in the base and linear along the fiber, i.e., We can choose the same definition in the case of the trivial Z This definition is of course in particular valid for G = GL p|q ∈ Z n 2 Lg. However, let us mention that the linear functions ("linear along the fibers") of the trivial Z n 2 -vector bundle E = GL p|q × V that are defined on | GL p|q | × |V | do not coincide with the linear functions ("globally linear") of the linear Z n 2 -manifold M = R t|u × V (see (2.57)) that are defined on the open subset | GL p|q | × |V | of its base R t × |V |: Given what we have just said, we expect the following proposition to hold: Proposition 3.9. A smooth linear action σ − of the Z n 2 -Lie group G = GL p|q on a linear Z n 2 -manifold V in the sense of Definition 3.6, is equivalent to a Z n 2 -morphism σ : G × V → V that satisfies the conditions (3.18) and (3.19) and has the linearity property Since G and V have global coordinates, it follows from [15] that the target of σ * is given by which shows that it contains and that the requirement (3.27) actually makes sense. Another fact is also worth noting. We know from standard supergeometry that the classical Berezinian defines a super-Lie group morphism Ber : GL(p|q) → GL(1|0), so that we get a linear action of GL(p|q) on R 1|0 . The point here is that linear actions of GL(p|q) are not limited to actions on R p|q .

A The category of modules over a variable algebra
We define the category AMod (resp., FAMod) of modules (resp., Fréchet modules) over any (unital) algebra (resp., any (unital) Fréchet algebra) A. The algebra A can vary from object to object. The objects are the modules over some A (resp., the Fréchet vector spaces that come equipped with a (compatible) continuous A-action). We denote such modules by M A . Morphisms consist of pairs (ϕ, Φ), where is an algebra morphism (resp., a continuous algebra morphism), and is a map (resp., a continuous map) that satisfies Φ(am + a m ) = ϕ(a)Φ(m) + ϕ(a )Φ(m ), for all a, a ∈ A and m, m ∈ M A . It is evident that we do indeed obtain a category in this way.
The preceding categories AMod and FAMod are similar to the category AFMan that we used in [13]. They naturally appear when considering the zero degree rules functor or the functor of points. See for instance equations (2.1) and (2.22).
B Basics of Z n 2 -geometry B.1 Z n 2 -manifolds and their morphisms The locally ringed space approach to Z n 2 -manifolds was pioneered in [18]. We work over the field R of real numbers and set Z n 2 := Z 2 × Z 2 × · · · × Z 2 (n-times). A Z n 2 -graded algebra is an Ralgebra A with a decomposition into vector spaces A := ⊕ γ∈Z n 2 A γ , such that the multiplication, say ·, respects the Z n 2 -grading, i.e., A α · A β ⊂ A α+β . We will always assume the algebras to be associative and unital. If for any pair of homogeneous elements a ∈ A α and b ∈ A β we have that a · b = (−1) α,β b · a, (B.1) where −, − is the standard scalar product on Z n 2 , then A is a Z n 2 -commutative algebra. Essentially, Z n 2 -manifolds are "manifolds" equipped with both, standard commuting coordinates and formal coordinates of non-zero Z n 2 -degree that Z n 2 -commute according to the general sign rule (B.1). Note that in general we need to deal with formal coordinates that are not nilpotent.
In order to keep track of the various formal coordinates, we need to introduce a convention on how we fix the order of elements in Z n 2 and we choose the lexicographical order. For example, with this choice of ordering It is clear that Z n 2 -manifolds, as they are locally ringed spaces, are not fully determined by their topological points. To "claw back" a fully useful notion of a point, one can employ Grothendieck's functor of points. This is, of course, an application of the Yoneda embedding (see [35, Chapter III, Section 2]). For the case of supermanifolds, it is well-known, via the seminal works of Schwarz and Voronov [39,40,46], that superpoints are sufficient to act as "probes" for the functor of points. That is, we only need to consider supermanifolds that have a single point as their underlying topological space. Dual to this, we may consider finite dimensional Grassmann algebras Λ = Λ 0 ⊕ Λ 1 as parameterizing the "points" of a supermanifold. One can thus view supermanifolds as functors from the category of finite dimensional Grassmann algebras to sets. However, it turns out that the target category is not just sets, but (finite dimensional) Λ 0 -smooth manifolds. That is, the target category consists of smooth manifolds that have a Λ 0 -module structure on their tangent spaces. Morphisms in this category respect the module structure and are said to be Λ 0 -smooth (we will explain this further later on). In [13], it was shown how the above considerations generalize to the setting of Z n 2 -manifolds. We will use the notations and results of [13] rather freely. We encourage the reader to consult this reference for the subtleties compared to the standard case of supermanifolds.
A Z n 2 -Grassmann algebra we define to be a formal power series algebra R[[θ]] in Z n 2 -graded, Z n 2 -commutative parameters θ j . All the information about the number of generators is specified by the tuple q as before. We will denote a Z n 2 -Grassmann algebra by Λ, as usually we do not need to specify the number of generators. A Z n 2 -point is a Z n 2 -manifold (that is isomorphic to) R 0|q . It is clear, from Definition B.1, that the algebra of global sections of a Z n 2 -point is precisely a Z n 2 -Grassmann algebra. There is an equivalence between Z n 2 -Grassmann algebras and Z n 2 -points: Z n 2 GrAlg ∼ = Z n 2 Pts op .
The Yoneda functor of points of the category Z n 2 Man of Z n 2 -manifolds is the fully faithful embedding Y: Z n 2 Man M → Hom Z n 2 Man (−, M ) ∈ Fun Z n 2 Man op , Set .
In [13], we showed that Y remains fully faithful for appropriate restrictions of the source and target of the functor category, as well as of the resulting functor category. More precisely, we proved that the functor S: Z n 2 Man M → Hom Z n 2 Man (−, M ) ∈ Fun 0 Z n 2 Pts op , A(N)FM is fully faithful. The category A(N)FM is the category of (nuclear) Fréchet manifolds over a (nuclear) Fréchet algebra, and the functor category is the category of those functors that send a Z n 2 -Grassmann algebra Λ to a (nuclear) Fréchet Λ 0 -manifold, and of those natural transformations that have Λ 0 -smooth Λ-components.