Construction of $BCK$-neighborhood systems in a $d$-algebra

1.   Introduction

There is growing interest in studying generalized supergeometry, that is, geometry of graded manifolds where the grading group is not $\mathbb Z_2$, but $\mathbb Z^n_2 = \mathbb Z_2\times \ldots \times \mathbb Z_2$. The foundational aspects of the theory of $\mathbb Z^n_2-$manifolds were recently studied in ^[1,2,3,4]. This generalization is used in physics, see ^[5]. Also in Mathematics, there exist many examples of $\mathbb Z^n_2-$graded $\mathbb Z^n_2-$commutative algebras: quaternions and Clifford algebras, the algebra of Deligne differential superforms, etc. Moreover, there exist interesting examples of $\mathbb Z^n_2-$manifolds. In this paper, we study the $\mathbb Z^n_2-$grassmannians as $\mathbb Z^n_2-$manifolds and their constructions.

In the context of manifolds, homogeneous superspaces have been defined and investigated extensively using the functor of points approach in ^[6,7,8]. In this paper, we show that $\mathbb Z^n_2-$grassmannians $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ are homogeneous, c.f. section 3. To this end, we show that the $\mathbb Z^n_2-$Lie group $GL(\overrightarrow{\textbf{m}})$, c.f. section 2, acts transitively on $\mathbb Z^n_2-$grassmannian $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$, c.f. section 3.

In the first section, we recall briefly all necessary basic concepts such as $\mathbb Z^n_2-$ grading spaces, $\mathbb Z^n_2-$manifolds, $\mathbb Z^n_2-$Lie groups and an action of a $\mathbb Z^n_2-$Lie group on a $\mathbb Z^n_2-$manifold. We use these concepts in the case of $\mathbb Z_2-$geometry in ^[6] and ^[8].

In section 2, we study the $\mathbb Z^n_2-$grassmannians extensively. The supergrassmannians are introduced by Manin in ^[9], but here by developing an efficient formalism, we fill in the details of the proof of this statement.

In section 3, by a functor of points approach, an action of the $\mathbb Z^n_2-$Lie group $GL(\overrightarrow{\textbf{m}})$ on the $\mathbb Z^n_2-$grassmannian $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ is defined by gluing local actions. Finally it is shown that this action is transitive.

2.   Preliminaries

Let $\mathbb Z^n_2 = \mathbb Z_2\times \ldots \times \mathbb Z_2$ be the $n-$fold Cartesian product of $\mathbb Z_2.$ From now on, we set $\textbf{q}: = 2^n-1$ and by $\overrightarrow{\textbf{k}}$, we mean $(k_0, k_1, \ldots, k_ \textbf{q})$ such that $k_i\in \mathbb N$. Consider the bi-additive map

The even subgroup $(\mathbb Z^n_2)_0$ consists of elements $\gamma \in \mathbb Z^n_2$ such that $\langle \gamma, \gamma\rangle = 0,$ and the set $(\mathbb Z^n_2)_1$ consists of odd elements $\gamma \in \mathbb Z^n_2$ such that $\langle \gamma, \gamma\rangle = 1.$

One can fix an ordering on $\mathbb Z^n_2$; based on this ordering, each even element is smaller than each odd element. Given two even (odd) elements $(a_1, a_2, \ldots, a_n)$ and $(b_1, b_2, \ldots, b_n)$, the first one is smaller than the second one for the lexicographical order, if $a_i < b_i$, for the first i where $a_i$ and $b_i$ differ. For example, the lexicographical ordering on $\mathbb Z_2^3$ is

Obviously, $\mathbb Z^n_2$ with lexicographical ordering is totally ordered set. Thus it may be diagrammed as an ascending chain as follows

In the supergeometry, the sign rules between generators of the algebra are completely determined by their parity. One can define a grading by (2.1) such that $\epsilon(a, b) = (-1)^{\langle a, b\rangle}$ will be a sign rule what will lead to $\mathbb Z^n_2-$geometry. Also, it has been shown that any other sign rule for finite number of coordinates is obtained from the above sign rule for sufficiently big $n.$ See ^[1] for more details.

2.1.   $\mathbb Z^n_2-$geometry

The $\mathbb Z^n_2-$graded objects like $\mathbb Z^n_2-$algebras, $\mathbb Z^n_2-$ringed spaces, $\mathbb Z^n_2-$domains and $\mathbb Z^n_2-$manifolds have been studied in ^[1,3,4]. In the following, we recall the necessary definitions from these references.

By definition, a $\mathbb Z^n_2-$vector space is a direct sum $V = \bigoplus_{\gamma \in \mathbb Z^n_2}V_{\gamma}$ of vector spaces $V_{\gamma}$ over a field $\mathbb K$ (with characteristic $0$). For each $\gamma\in \mathbb Z^n_2,$ the elements of $V_{\gamma}$ is called homogeneous with degree $\gamma.$ If $x\in V_{\gamma}$ be a homogeneous element of $V,$ then the degree of $x$ is represented by $\tilde x = \gamma.$

A $\mathbb Z^n_2-$ring $\mathcal R = \bigoplus_{\gamma \in \mathbb Z^n_2}\mathcal R_{\gamma}$ is a ring such that its multiplication satisfies

A $\mathbb Z^n_2-$ring $\mathcal R$ is called $\mathbb Z^n_2-$commutative, if for any homogeneous elements $a, b\in \mathcal R$

For any $\mathbb Z^n_2-$algebra $\mathcal R$, let $J$ be an ideal of $\mathcal R$ generated by all homogeneous elements of $R$ having nonzero degree. If $f:R\rightarrow S$ is a morphism of $\mathbb Z^n_2-$algebras, then $f(J_R)\subseteq J_S$. Let $M$ be an $\mathcal R$-module. The collection of sets $\{x+J^kM\}_{k = 0}^{\infty}$ can be considered as a basis for a topology on $M$. This topology is called $J-$adic topology. Note that $x$ runs over all elements of $M$. The $J-$adic topology plays a fundamental role in $\mathbb Z^n_2-$geometry.

A $\mathbb Z^n_2-$ring $\mathcal R$ with respect to $J-$adic topology is Hausdorff complete if the natural ring morphism $\mathcal R\rightarrow \varprojlim_{k\in \mathbb N} \mathcal R/J^k$ is an isomorphism.

Example 2.1. Let $R$ be a ring and $\xi_1, \ldots, \xi_{ \textbf{q}}$ be indeterminates with degree $\gamma_1, \ldots, \gamma_{ \textbf{q}}\in \mathbb Z^n_2$ respectively such that

Then $R[[\xi_1, \ldots, \xi_{ \textbf{q}}]]$ is the $\mathbb Z^n_2-$commutative associative unital $R-$algebra of formal power series in the $\xi_a$ with coefficients in $R.$ If $J$ be an ideal generated by all formal power series $\sum_{ \overrightarrow{\textbf{k}}}a_{ \overrightarrow{\textbf{k}}}\xi_1^{k_0} \ldots \xi_q^{k_q}$ whose first term $a_{ \overrightarrow{\textbf{k}}}$ is equal to zero, then One can see $R[[\xi_1, \ldots, \xi_{ \textbf{q}}]]$ is $J-$adically Hausdorff complete.

By a $\mathbb Z^n_2-$ringed space, we mean a pair $(X, \mathcal{O}_X)$ where $X$ is a topological space and $\mathcal O_X$ is a sheaf of $\mathbb Z^n_2-$commutative $\mathbb Z^n_2-$graded rings on $X$. A morphism between two $\mathbb Z^n_2-$ringed spaces $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ is a pair $\psi: = (\overline{\psi}, \psi^*)$ such that $\overline{\psi}:X\rightarrow Y$ is a continuous map and $\psi^*:\mathcal O_Y\rightarrow \overline{\psi}_*\mathcal O_X$ is a homomorphism of weight zero between the sheaves of $\mathbb Z^n_2-$commutative $\mathbb Z^n_2-$graded rings. Let $\textbf{q} = 2^n-1,$ the $\mathbb Z^n_2$-ringed space

is called $\mathbb Z^n_2$-domain such that $C^\infty_{\mathbb R^{m_0}}$ is the sheaf of smooth functions on $\mathbb R^{m_0}$. By evaluation of $f = \sum_{}f_I\xi^I$ at $x \in U$, denoted by $ev_x(f)$, we mean $f_\emptyset(x)$. Also for each open $U\subset \mathbb R^{m_0}$,

is the $\mathbb Z^n_2-$commutative associative unital $\mathbb Z^n_2-$algebra of formal power series in formal variables $\xi_i^j$'s of degrees $\gamma_i$ which commuting as follows:

Let $\mathcal{J}(U)$ be the ideal generated by all homogeneous formal power series of nonzero degree. Then it is easily seen that $\mathcal O_{\mathbb{R}^{ \overrightarrow{\textbf{m}}}}(U)$ is Hausdorff complete with respect to $\mathcal{J}(U)$-adic topology. Equivalently there is a canonical ring isomorphism between $\mathcal{O}(U)$ and $\varprojlim_k\mathcal{O}(U)/\mathcal{J}^k(U)$. Let $V\subset U$. Then the following diagram is commutative

where $r_{UV}:\mathcal{O}(U)\to \mathcal{O}(V)$ is the restriction map and $R_{UV}$ is induced morphism. This shows that the notion of adic topology may be extended for sheaf $\mathcal{O}$ such that each stalk $\mathcal{O}_p$, $p\in \mathbb{R}^{m_0}$, is Hausdorff complete with respect to $\mathcal{J}_p-$adic topology.

A $\mathbb Z^n_2$-manifold of dimension $\overrightarrow{\textbf{m}}$ is a $\mathbb Z^n_2$-ringed space $(\overline {M}, \mathcal{O}_M)$ that is locally isomorphic to $\mathbb R^{ \overrightarrow{\textbf{m}}}$. In addition $\overline {M}$ is a second countable and Hausdorff topological space. A morphism between two $\mathbb Z^n_2$-manifolds $M = (\overline {M}, \mathcal{O}_M)$ and $N = (\overline{N}, \mathcal{O}_N)$ is a local morphism between two local $\mathbb Z^n_2$-ringed spaces.

Let $(M, \mathcal O_M)$ be a $\mathbb Z^n_2-$manifold, and let $\mathcal J$ be the ideal sheaf $\mathcal J$, defined by

The structure sheaf $\mathcal{O}_M$ is $\mathcal J-$adically Hausdorff complete as a sheaf of $\mathbb Z^n_2-$ commutative $\mathbb Z^n_2-$rings. See Proposition 6.9 in ^[3] for more details.

Analogous with supergeometry, one can obtain a $\mathbb Z^n_2-$manifold by gluing $\mathbb Z^n_2-$ domains. We will use this method to construct the $\mathbb Z^n_2-$grassmannian as a $\mathbb Z^n_2-$ manifold in section 3.

In order to introduce the concept of Jacobian, we need a few definitions and a proposition from ^[10].

Definition 2.2. Let $M$ be a $\mathbb Z^n_2$-manifold and $m\in \overline{M}$. The tangent space of $M$ at $m$, denoted by $T_mM$, is the real $\mathbb Z^n_2$-vector space of $\mathbb R-$derivations $\mathcal O_m\rightarrow \mathbb R$.

Definition 2.3. Let $\psi:M\rightarrow N$ be a morphism of $\mathbb Z^n_2$-manifolds. Assume $\overline{\psi}(m) = n\in \overline{N}$ for a point $m\in \overline{M}$. The tangent map of $\psi$ at $m$ is a morphism of $\mathbb Z^n_2$-vector spaces denoted by $(d\psi)_m:T_mM\rightarrow T_nN$ and defined by

Let $\psi:M\rightarrow N$ and $\varphi:N\rightarrow Q$ be two morphisms of $\mathbb Z^n_2$-manifolds, one can use the above definition to show that for any point $m\in \overline{M}$,

Here we state the chain rule for $\mathbb Z^n_2$-manifolds and use it to relate the tangent map to the $\mathbb Z^n_2$-graded Jacobian matrix. See Proposition $2.10$ in ^[10].

Proposition 2.4. Let $U^{ \overrightarrow{\textbf{p}}}$, $U^{ \overrightarrow{\textbf{r}}}$ be $\mathbb Z^n_2$-domains, with coordinates $u^\alpha$, $v^\beta$ respectively. Let $\psi:U^{ \overrightarrow{\textbf{p}}}\rightarrow U^{ \overrightarrow{\textbf{r}}}$ be a morphism of $\mathbb Z^n_2$-manifolds. Then,

Let us study the matrix representation of the equation (2.2). Assume $\psi = \Big(v^{\gamma}(u)\Big)$ and $\varphi = \Big(w^{\alpha}(v)\Big)$ are local representations of $\mathbb Z^n_2$-morphisms $\psi:M\rightarrow N$ and $\varphi:N\rightarrow S$ around $m$ and $\overline{\psi}(m)$ respectively, one has

Definition 2.5. Let $\psi$ be a $\mathbb Z^n_2$-morphism between $\mathbb Z^n_2$-domains $U$ and $V$. If $u\mapsto \Big(v^i(u)\Big)$ is a representation of $\psi$, then the Jacobian of $\psi$ is a $\mathbb Z^n_2$-matrix of degree zero as follow

At the end of this section, it is worth mentioning that some of the required concepts of category theory are given in the Appendix. At there, among other things, we talked about Yoneda lemma (Lemma A.1). According to this lemma two objects $X, Y\in Obj(\mathcal{C})$ are isomorphic if and only if the functor of points associated to them are isomorphic. Indeed, the Yoneda embedding is an equivalence between $\mathcal{C}$ and a subcategory of representable functors in $[\mathcal{C}, \textbf{SET}]$ since not all functors are representable. See (Appendix A.1) for more details.

2.2.   $\mathbb Z^n_2-$Lie groups

Let $\textbf{ZSM}$ be the category of $\mathbb Z^n_2-$manifolds. This is a category whose objects are $\mathbb Z^n_2-$manifolds whose morphisms are morphisms between two $\mathbb Z^n_2-$manifolds. $\textbf{ZSM}$ is a locally small category and has finite product property, see ^[11] for more details. In addition it has a terminal object $\mathbb{R}^{\overrightarrow{\textbf{0}}}$, that is the constant sheaf $\mathbb{R}$ on a singleton $\{0\}$.

Let $M = (\overline {M}, \mathcal{O}_M)$ be a $\mathbb Z^n_2-$manifold and $p\in \overline {M}$. There is a map $j_p = (\overline{j}_p, {j_p}^*)$ where:

So, for each $\mathbb Z^n_2-$manifold $T$, one can define the morphism

as a composition of $j_p$ and the unique morphism $T\rightarrow \mathbb{R}^{\overrightarrow{\textbf{0}}}$.

By $\mathbb Z^n_2-$Lie group, we mean a group-object in the category $\textbf{ZSM}$. This group is a $\mathbb Z^n_2-$graded group. Graded Lie groups are extensively studied in ^[12]. The category $\textbf{ZSM}$ has group-object because of existence categorical products and terminal object. Also, one can show that any $\mathbb Z^n_2-$Lie group $G$ induced a group structure over its $T$-points for any arbitrary $\mathbb Z^n_2-$manifold $T$. This means that the functor $T\rightarrow G(T)$ takes values in category of groups. Moreover, for any other $\mathbb Z^n_2-$manifold $S$ and morphism $T\rightarrow S$, the corresponding map $G(S)\rightarrow G(T)$ is a homomorphism of groups. See Appendix for more details. As another form, one can also define a $\mathbb Z^n_2-$Lie group as a representable functor $T\rightarrow G(T)$ from category $\textbf{ZSM}$ to category of groups.

Example 2.6. Consider the $\mathbb Z^n_2-$domain $\mathbb R^{ \overrightarrow{\textbf{m}}}$ and an arbitrary $\mathbb Z^n_2-$manifold $T$. Let $f^j_i\in \mathcal{O}(T)_{\gamma_i}$, $0\leq i \leq \textbf{q}$, $1\leq j \leq m_j$, be $\gamma_i-$degree elements. By Theorem 6.8 in ^[1] (Fundamental theorem of $\mathbb Z^n_2$-morphisms), One may define a unique morphism $\psi:T\rightarrow \mathbb R^{ \overrightarrow{\textbf{m}}}$, by setting $\xi_i^j\mapsto f_i^j$ where $(\xi_i^j)$ is a global coordinates system on $\mathbb R^{ \overrightarrow{\textbf{m}}}$. Thus $\psi$ may be represented by $(f_i^j)$.

One can see a degree zero square matrix with entries in the standard block format

is invertible if and only if $B_{ii}$ is an invertible matrix for all $0\leq i\leq \textbf{q}$, see Proposition 1.5 and Proposition 6.1 in ^[10].

Let $V$ be a finite dimensional $\mathbb Z^n_2$-vector space of dimension $\overrightarrow{\textbf{m}} = m_0|m_1|\ldots|m_ \textbf{q}$. One can define the $\mathbb Z^n_2-$Lie group $GL(V)$ which is denoted by $GL(\overrightarrow{\textbf{m}})$ if $V = \mathbb{R}^{ \overrightarrow{\textbf{m}}}$. For more details, see (Appendix A.2). It can be shown that $T-$points of $GL(\overrightarrow{\textbf{m}})$ are the $\overrightarrow{\textbf{m}}\times \overrightarrow{\textbf{m}}$ invertible $\mathbb Z^n_2-$matrices of weight zero where the elements of the $m_k\times m_u$ block $B_{ku}$ have degree $\gamma_k+\gamma_u$ and the multiplication is the matrix product.

Let $x\in\overline{G}$, one can define the left and right translation by $x$ as

where $\Delta_G$ is the diagonal map on $G$ and $\hat{x}_G$ is as above. One can show that pullbacks of above morphisms are as following

One may also use the language of functor of points to describe two morphisms (2.4) and (2.5).

One can see the definition of vector fields on a graded Lie group $G$ in ^[12]. According to the standard superspace, we have the following definition:

Definition 2.7. Let $G$ be a $\mathbb Z^n_2-$Lie group. A Vector field $X$ on $G$ is called right invariant vector field, if we have

Similarly, one can use $(1\otimes X)\circ \mu^* = \mu^* \circ X$, for a left invariant vector field $X.$

The bracket of two right invariant vector fields is right invariant. So we have

Definition 2.8. Let $G$ be a $\mathbb Z^n_2-$ Lie group. The set of all right invariant vector fields is denoted by $\mathfrak g$ and is called the $\mathbb Z^n_2-$Lie algebra associated with the $\mathbb Z^n_2-$Lie group $G$, and we write $\mathfrak g = Lie(G)$

Similar to standard supergeometry, one can show that $\mathfrak g = Lie(G)$ is a finite dimensional $\mathbb Z^n_2-$vector space canonically identified with the tangent space at the identity of the $\mathbb Z^n_2-$Lie group $G$.

Definition 2.9. Let $M$ be a $\mathbb Z^n_2-$manifold and let $G$ be a $\mathbb Z^n_2-$Lie group with $\mu, i$ and $e$ as its multiplication, inverse and unit morphisms respectively. A morphism $a:M\times G\rightarrow M$ is called a (right) action of $G$ on $M$, if the following diagrams commute

where $\hat{e}_{_M}, \Delta_M$ are as above. In this case, we say G acts from right on $M$. One can define left action analogously.

According to the above diagrams, one has:

By Yoneda lemma (Lemma A.1), one may consider, equivalently, the action of G as a natural transformation:

As in ^[13], for each $p\in M$ and $g\in G$, one can define the morphisms $a_p$ and $a^g$ and use the functor point language to show that the maps $a_p$ and $a^g$ satisfy the relations

where $r_g$ is the right translation on Lie $\mathbb Z^n_2$-group $G.$

For proof of the following proposition, see (Appendix A.3).

Proposition 2.10. Let $a:M\times G \rightarrow M$ be an action of a Lie $\mathbb Z^n_2$-group $G$ on a $\mathbb Z^n_2$-manifold $M$. Then

(1) $a^g$ is a $\mathbb Z^n_2$-diffeomorphism for all $g\in \overline{G}.$

$(2)$ $a_p$ has constant rank for all $p\in \overline{M}.$

Before next definition, we recall that a morphism between $\mathbb Z^n_2-$manifolds, say $\psi:M\rightarrow N$ is a submersion at $x\in\overline {M}$, if $(d\psi)_x$ is surjective and $\psi$ is called submersion, if it is surjective at each point. (For more details, refer to ^[10]). Also $\psi$ is a surjective submersion, if in addition $\psi_0$ is surjective.

Definition 2.11. Let $G$ acts on $M$ with action $a:M\times G \rightarrow M$. The action $a$ is called transitive, if there exist $p\in \overline {M}$ such that $a_p$ is a surjective submersion.

It is shown that, if $a_p$ is a submersion for one $p\in \overline {M}$, then it is a submersion for all point in $\overline {M}$. Also one can show that $a$ is transitive if and only if $\overline{a}$ is a transitive action in classical geometry and $(da_p)_e$ is a surjective because of Proposition 2.10. The following proposition will be required in the last section.

Proposition 2.12. Let $a:M\times G \rightarrow M$ be an action of a Lie $\mathbb Z^n_2$-group $G$ on a $\mathbb Z^n_2$-manifold $M$. Let $p\in\overline{M}$ and $\dim G = \overrightarrow{\textbf{r}} = (r_0, r_1, \ldots, r_ \textbf{q})$ and $\overrightarrow{\textbf{r}}^\prime: = (0, r_1, \ldots, r_ \textbf{q}).$ If the map

is surjective, then $a$ is a transitive action.

Proof. According to above argument, it is enough to show that $\overline{a}$ is a transitive action in classical geometry and $(da_p)_e$ is surjective.

Let $(a_p)_{\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime}}$ be surjective. Looking at the reduced part of each morphism in

we have that

is surjective. So $\overline{a}$ is a classical transitive action. Let now $\{t, \xi_1, \ldots, \xi_ \textbf{q}\}$ be coordinates in a neighbourhood $U$ of $m\in\overline{M}$. Consider the element $\Phi\in M(\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime})$ defined by

By surjectivity of $(a_p)_{\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime}}$, there exists $\psi\in G(\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime})$ such that $(a_p)_{\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime}}(\psi) = \Phi$ and we have

This implies that $(T_mM)_{\gamma\neq 0}$ is in the image of $(da_p)_{\overline{\psi}}$. Since, by our previous considerations, $\overline{a_p}$ is a submersion, $(T_mM)_{0}$ is in the image $\overline{a_p}$. Hence, due to Proposition 8.1.5 we are done.

In the following, after introducing the concept of stabilizer, we state some related results without any proofs, since the proofs are the same as the proofs of the similar results in supergeometry with appropriate modifications without any extra difficulties regarding $\mathbb Z^n_2-$geometry. For more details see ^[6] and ^[8].

Definition 2.13. Let $G$ be a $\mathbb Z^n_2-$Lie group and let $a$ be an action of $G$ on $\mathbb Z^n_2-$manifold $M$. By stabilizer of $p\in \overline {M},$ we mean a $\mathbb Z^n_2-$manifold $G_p$ equalizing the diagram

Proposition 2.14. Let $a:M\times G \rightarrow M$ be an action, then

1. The following diagram admits an equalizer $G_p$

2. $G_p$ is a $\mathbb Z^n_2-$sub Lie group of $G$.

3. The functor $T\rightarrow \Big(G(T)\Big)_{\hat{p}_{_T}}$ is represented by $G_p$, where $\Big(G(T)\Big)_{\hat{p}_{_T}}$ is the stabilizer in $\hat{p}_{_T}$ of the action of $G(T)$ on $M(T)$.

Proposition 2.15. Suppose $G$ acts transitively on $M$. There exists a $G$-equivariant isomorphism

3.   $\mathbb Z^n_2$-grassmannian

Supergrassmannians $G_{k|l}(m|n)$ are introduced and studied by Manin in ^[9] and ^[14]. Also the authors have studied them in more details in ^[13] and ^[15]. In this section, we introduce the $\mathbb Z_2^n$-grassmannian which is denoted by $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ shortly, or $G_{k_0|k_1|\ldots|k_ \textbf{q}}(m_0|m_1|\ldots|m_ \textbf{q})$. For convenience from now, we set

and also decompose any $\mathbb Z^n_2-$matrix into $2^n\times 2^n$ blocks

such that the elements of block $B_{ku}$ have degree $\gamma_k+\gamma_u$. By a $\mathbb Z^n_2$-grassmannian, $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$, we mean a $\mathbb Z^n_2$-manifold which is constructed by gluing the following $\mathbb Z^n_2$- domains

For $i = 0, 1, \ldots, \textbf{q},$ let $I_i \subset \{1, \ldots, m_i\}$ be a sorted subset in ascending order with $k_i$ elements. The elements of $I_i$ are called $\gamma_i$-degree indices. The multi-index $\overrightarrow{\textbf{I}} = (I_1, ..., I_ \textbf{q})$ is called $\overrightarrow{\textbf{k}}$-index. Set $\mathcal U_{ \overrightarrow{\textbf{I}}}: = (\overline{U}_{ \overrightarrow{\textbf{I}}}, \mathcal{O}_{ \overrightarrow{\textbf{I}}})$, where

Let each $\mathbb Z^n_2$-domain $\mathcal U_{ \overrightarrow{\textbf{I}}}$ be labeled by a $\mathbb Z^n_2-$matrix $\overrightarrow{\textbf{k}}\times \overrightarrow{\textbf{m}}$ of weight zero, say $A_{ \overrightarrow{\textbf{I}}}$, with $2^n\times 2^n$ blocks $B_{ij}$ each of which is a $k_i\times m_j$ matrix. In addition, except for columns with indices in $I_0 \cup I_1\cup \ldots \cup I_ \textbf{q}$, which together form a $\mathbb Z^n_2-$submatrix denoted by $M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{I}}}$, the matrix is filled from up to down and left to right by $x_a^{ \overrightarrow{\textbf{I}}}, \xi_b^{ \overrightarrow{\textbf{I}}}$, the free generators of $\mathcal{O}_{ \overrightarrow{\textbf{I}}}(\mathbb R^{\beta_0})$ each of them sits in a block with same degree. This process impose an ordering on the set of generators. In addition $M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{I}}}$ is supposed to be the identity matrix.

For example, consider $G_{1|2|1|1|}{2|2|2|2}$. Then let $I_0 = \{1\}, I_1 = \{1, 2\}, I_2 = \{1\}, I_3 = \{2\}$, so $\overrightarrow{\textbf{I}}$ is a $1|2|1|1$-index. In this case the set of generators of $\mathcal{O}_{ \overrightarrow{\textbf{I}}}(\mathbb{R}^{\beta_0})$ is

and $A_{ \overrightarrow{\textbf{I}}}$ is:

Note that, in this example,

is corresponding total ordered set of generators.

By $\widetilde{U}_{{ \overrightarrow{\textbf{I}}}, { \overrightarrow{\textbf{J}}}},$ we mean the set of all points of $\overline{U}_{ \overrightarrow{\textbf{I}}}$, on which $M_{ \overrightarrow{\textbf{J}}}A_{ \overrightarrow{\textbf{I}}}$ is invertible. Obviously $\widetilde{U}_{{ \overrightarrow{\textbf{I}}}, { \overrightarrow{\textbf{J}}}}$ is an open set. The transition map between the two $\mathbb Z^n_2$-domains $U_{ \overrightarrow{\textbf{I}}}$ and $U_{ \overrightarrow{\textbf{J}}}$ is denoted by

Note that $g_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}} = (\overline{g}_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}, g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}),$ where $g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ is an isomorphism between sheaves determined by defining on each entry of $D_{ \overrightarrow{\textbf{I}}}(A_{ \overrightarrow{\textbf{I}}})$ as a rational expression which appears as the corresponding entry provided by the pasting equation

where $D_{ \overrightarrow{\textbf{I}}}(A_{ \overrightarrow{\textbf{I}}})$ is a matrix which is remained after omitting $M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{I}}}$. Clearly, the left hand side of (3.2) is defined whenever $M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{J}}}$ is invertible. The morphism $g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ induces the continuous map $\overline{g}_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ (in the case $n = 1$, see ^[16], lemma 3.1).

For example in $G_{1|2|1|1}(2|2|2|2)$ suppose

so $\overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}$ are $1|2|1|1$-indices. We have:

The maps $g_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ are gluing morphisms. In fact, a straightforward computation shows the following proposition holds.

Proposition 3.1. Let $g_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}} = \big(\overline{g}_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}, g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}\big)$ be as above, then

1. $g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{I}}}} = id.$

2. $g^*_{_{ \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{I}}}} \circ g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}} = id.$

3. $g^*_{_{ \overrightarrow{\textbf{S}}, \overrightarrow{\textbf{I}}}} \circ g^*_{_{ \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{S}}}}\circ g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}} = id.$

Proof. For first equality, note that the map $g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{I}}}}$ is obtained from the following equality:

where the matrix $M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{I}}}$ is identity. So $g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{I}}}}$ is defined by the following equality:

This shows the first equality. For second equality, let $\overrightarrow{\textbf{J}}$ be an another $\overrightarrow{\textbf{k}}$-index, so $g^*_{_{ \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{I}}}}$ is obtained by the following equality:

One may see that $g^*_{_{ \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{I}}}} \circ g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ is obtained by following equality:

For left side, we have

Accordingly the map $g^*_{_{ \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{I}}}} \circ g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ is obtained by $D_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{I}}} = D_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{I}}}$ and it shows that this map is identity. For third equality, it is sufficient to show that the map $g^*_{_{ \overrightarrow{\textbf{S}}, \overrightarrow{\textbf{I}}}} \circ g^*_{_{ \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{S}}}}\circ g^*_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}$ is obtained from

This case obtains from case $2$ analogously.

So the sheaves $(\overline{U}_{ \overrightarrow{\textbf{I}}}, \mathcal O_{ \overrightarrow{\textbf{I}}})$ may be glued through the $g_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}$ to construct the $\mathbb Z^n_2-$grassmannian $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$. Indeed, according to Lemma 3.1 in ^[10], the conditions of the above proposition are necessary and sufficient for gluing.

4.   $\mathbb Z^n_2-$grassmannian as homogeneous $\mathbb Z^n_2-$space

In this section, we study construction of the quotients of $\mathbb Z^n_2-$Lie groups. Although this study is parallel its analogue in supergeometry, see ^[18] for $\mathbb Z_2$ graded case, but generalizing related results in $\mathbb Z^n_2$ setting necessitates working with $\mathcal{J}-$adic topology.

Let $G = (\overline{G}, \mathcal{O}_G)$ be a $\mathbb Z^n_2-$Lie group and $H = (\overline{H}, \mathcal{O}_H)$ be a closed $\mathbb Z^n_2-$Lie supgroup of $G$. One can define a $\mathbb Z^n_2-$manifold structure on the topological space $\overline{X} = \overline{G}/\overline{H}$ as follows:

Let $\mathfrak{g} = Lie(G)$ and $\mathfrak{h} = Lie(H)$ be the $\mathbb Z^n_2-$Lie algebras corresponding with $G$ and $H$. For each $Z\in \mathfrak{g}$, let $D_{Z}$ be the left invariant vector field on $G$ associated with $Z$. For $\mathfrak{h},$ a $\mathbb Z^n_2-$subalgebra of $\mathfrak{g},$ set:

On the other hand, for any open subset $U\subset \overline{G}$ set:

where $r_{x_0}$ is the right translation by $x_0$ in (2.5). If $\overline{H}$ is connected, then $\mathcal{O}_{inv}(U) = \mathcal{O}_{\mathfrak{h}}(U)$. Let $\overline \pi:\overline G\rightarrow \overline X$ be the natural projection. For each open subset $\overline W\subset \overline{X} = \overline{G}/\overline{H}$, the structure sheaf $\mathcal{O}_X$ is defined as following

where $W = \overline{\pi}^{-1}(\overline W).$ One can show that $\mathcal{O}_{X}$ is a sheaf on $\overline{X}$ and the ringed space $X = (\overline{X}, \mathcal{O}_{X})$ is a $\mathbb Z^n_2-$domain locally. Indeed, according to the definition of distribution in ^[19], we have a distribution spanned by the vector fields in $\mathfrak h$ which is denoted by $\mathfrak D_{\mathfrak h}$. This distribution is involutive, because $\mathfrak h$ is a $\mathbb Z^n_2-$Lie algebra. By using the local Frobenius theorem, $\mathfrak D_{\mathfrak h}$ is integrable. For more details about Frobenius theorem in the $\mathbb Z^n_2-$graded category, see ^[19]. So there is an open neighborhood $U$ of identity element $1$ and coordinates $(x, \xi_1, \ldots, \xi_ \textbf{q})$ such that $\mathfrak D_{\mathfrak h}$ is spanned on $U$ by

Let $\mathfrak D^\prime_{\mathfrak h}$ be spanned by

on $U.$ One can use the local Frobenius theorem again, so $\mathfrak D^\prime_{\mathfrak h}$ is involutive distribution. One can show that $\mathbb Z^n_2-$manifold $X$ is obtained by gluing the integral $\mathbb Z^n_2-$manifolds of the local distributions $\mathfrak D^\prime_{\mathfrak h}$. So $X$ is a $\mathbb Z^n_2-$manifold and is called homogeneous $\mathbb Z^n_2-$space. The details of proof are similar to standard supergeometry in ^[6].

In this section, we want to show that the $\mathbb Z^n_2-$grassmannian $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ is a homogeneous $\mathbb Z^n_2-$space. According to the section 1, it is enough to find a $\mathbb Z^n_2-$Lie group which acts on $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ transitively. For this, we need the following remark and the next lemma

Remark 4.1. Let $\mathcal X$ be an element of $U_{ \overrightarrow{\textbf{I}}}(T)$ where $\overrightarrow{\textbf{I}}$ is an arbitrary index. One can correspond to $\mathcal X$ a $\mathbb Z^n_2-$matrix $\overrightarrow{\textbf{k}}\times \overrightarrow{\textbf{m}}$ called $[\mathcal X]_{ \overrightarrow{\textbf{I}}}$ as follows: Except for columns with indices in $I_0 \cup I_1 \cup \ldots \cup I_ \textbf{q}$, the blocks are filled from up to down and left to right by $f_i, g_j$'s where

according to the ordering (3.1), where $(x_i; \xi_j)$ is the global coordinates of the $\mathbb Z^n_2-$domain $U_{ \overrightarrow{\textbf{I}}}$. The columns with indices in $I_0 \cup I_1 \cup \ldots \cup I_ \textbf{q}$ form an identity matrix.

Lemma 4.2. Let $\psi:T\rightarrow \mathbb{R}^{ \overrightarrow{\textbf{r}}}$ be a $T$-point of $\mathbb{R}^{ \overrightarrow{\textbf{r}}}$ and $(z_{tu})$ be a global coordinates of $\mathbb{R}^{ \overrightarrow{\textbf{r}}}$ with ordering as the one introduced in ($3.1$). If $B = (\psi^*(z_{tu}))$ is the $\mathbb Z^n_2-$matrix corresponding to $\psi$, then the $\mathbb Z^n_2-$matrix corresponding to $\big(g_{_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}}\big)_{_T}(\psi)$ is as follows:

where $[B]_{ \overrightarrow{\textbf{J}}}$ is introduced in Remark 4.1.

Proof. Note that $g^*_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}$ may be represented by a $\mathbb Z^n_2-$matrix as follows:

where $A_{ \overrightarrow{\textbf{J}}}$ is the label of $U_{ \overrightarrow{\textbf{J}}}.$ Let $M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{J}}} = (m_{tu})$ and $(M_{ \overrightarrow{\textbf{I}}}A_{ \overrightarrow{\textbf{J}}})^{-1} = (m^{tu})$. If $z = (z_{ij})$ be a coordinates system on $U_{ \overrightarrow{\textbf{I}}}$, then one has

Then

For second equality one may note that $\psi^*$ is a homomorphism of $\mathbb Z^n_2-$algebras and $m^{tk}(z)$ is a rational function of $z$. Obviously, the last expression is the $(t, u)$-entry of the matrix $D_{ \overrightarrow{\textbf{I}}}\Big((M_{ \overrightarrow{\textbf{I}}}[B]_{ \overrightarrow{\textbf{J}}})^{-1}[B]_{ \overrightarrow{\textbf{J}}}\Big)$. This completes the proof.

Theorem 4.3. The $\mathbb Z^n_2-$Lie group $GL(\overrightarrow{\textbf{m}})$ acts on $\mathbb Z^n_2-$grassmannian $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$.

Proof. First, we have to define a morphism $a:G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})\times GL(\overrightarrow{\textbf{m}}) \rightarrow G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$. For this, by Yoneda lemma, it is sufficient to define $a_{_T}$:

for each $\mathbb Z^n_2-$manifold $T$ or equivalently define

where $\mathcal P$ is a fixed arbitrary element in $GL(\overrightarrow{\textbf{m}})(T)$. For brevity, we denote $(a_{_T})^ \mathcal P$ by $\textbf{A}$. One may consider $GL(\overrightarrow{\textbf{m}})(T)$, as the set of $\overrightarrow{\textbf{m}}\times \overrightarrow{\textbf{m}}$ invertible $\mathbb Z^n_2-$matrices with entries in $\mathcal{O}(T)$, but there is not such a description for $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$, because it is not a $\mathbb Z^n_2-$domain. We know each $\mathbb Z^n_2-$grassmanian is constructed by gluing $\mathbb Z^n_2-$domains (c.f. section 2), so one may define the actions of $GL(\overrightarrow{\textbf{m}})$ on $\mathbb Z^n_2-$domains $(\overline{U}_{ \overrightarrow{\textbf{I}}}, \mathcal{O}_{ \overrightarrow{\textbf{I}}})$ and then shows that these actions glued to construct $a_{_T}$.

For defining $\textbf{A}$, it is needed to refine the covering $\{U_{ \overrightarrow{\textbf{I}}}(T)\}_{ \overrightarrow{\textbf{I}}}$. Set

where $[\mathcal P]$ is the matrix form of the fixed arbitrary element $\mathcal P$ in $GL(\overrightarrow{\textbf{m}})(T)$, see ^[8] and ^[21]. One can show that $\{U_{ \overrightarrow{\textbf{I}}}^{ \overrightarrow{\textbf{J}}}(T)\}_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}$ is a covering for $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$ and $\textbf{A} \Big(U_{ \overrightarrow{\textbf{I}}}^{ \overrightarrow{\textbf{J}}}(T)\Big)\subseteq U_{ \overrightarrow{\textbf{J}}}(T).$ Now consider all maps

where, $[\psi]_{ \overrightarrow{\textbf{I}}}$ is as above. We have to show that these maps may be glued to construct a global map on $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$. For this, it is sufficient to show that the following diagram commutes:

where $\big(g_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}\big)_{_T}$ is the induced map from $g_{ \overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}}$ on $T$-points. The following proposition is used to show commutativity of the above diagram.

Proposition 4.4. The last diagram commutes.

Proof. We have to show that

for arbitrary $\overrightarrow{\textbf{k}}$-indices $\overrightarrow{\textbf{I}}, \overrightarrow{\textbf{J}}, \overrightarrow{\textbf{Q}}, \overrightarrow{\textbf{L}}.$ Let $\psi\in U_{ \overrightarrow{\textbf{I}}}^{ \overrightarrow{\textbf{J}}}(T)\cap U_{ \overrightarrow{\textbf{Q}}}^{ \overrightarrow{\textbf{L}}}(T)$ be an arbitrary element. One has $\psi\in U_{ \overrightarrow{\textbf{I}}}^{ \overrightarrow{\textbf{J}}}(T)$, so

From left side of (4.1), we have:

For right side of equation (4.1), we have

This shows that the above diagram commutes.

Therefore $GL(\overrightarrow{\textbf{m}})$ acts on $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ with action $a$. Now it is needed to show that this action is transitive.

Theorem 4.5. $GL(\overrightarrow{\textbf{m}})$ acts on $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ transitively.

Proof. By proposition 2.12, it is sufficient to show that the map

is surjective, where $\overrightarrow{\textbf{r}} = (r_0, r_1, \ldots, r_{_ \textbf{q}})$ is dimension of $GL(\overrightarrow{\textbf{m}})$ and one can consider $\overrightarrow{\textbf{r}}^\prime = (0, r_1, \ldots, r_{_ \textbf{q}}).$ Let

be an element and $\bar{p}_0, \bar{p}_1, \ldots, \bar{p}_{_ \textbf{q}}$ be the matrices corresponding to $p_0, p_1, \ldots, p_ \textbf{q}$ respectively as subspaces. As an element of $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$, one may represent $\hat{p}_{_T}$, as follows

where $T$ is an arbitrary $\mathbb Z^n_2-$manifold. For surjectivity, let

be an arbitrary element. We have to show that there exists an element $V\in GL(\overrightarrow{\textbf{m}})(\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime})$ such that $\hat{p}_{_T}V = W$. Since the Lie group $GL(m_i)$ acts on manifold $G_{k_i}(m_i)$ transitively, then there exists an invertible matrix $H_{ii}\in GL(m_i)$ such that $\bar{p}_iH_{ii} = W_{ii}$. In addition, the equations $\bar{p}_iZ = W_{ij}$ have solutions since $rank(\bar{p}_i) = k_i$. Let $H_{ij}$ be solutions of these equations respectively. Clearly, One can see

satisfy in the equation $\hat{p}_{_T}V = W$. So $(a_p)_{\mathbb{R}^{ \overrightarrow{\textbf{r}}^\prime}}$ is surjective. By Proposition 2.12, $GL(\overrightarrow{\textbf{m}})$ acts on $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ transitively. Thus according to Proposition 2.15, $G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ is a homogeneous $\mathbb Z^n_2-$space.

A.   Basic Concepts

A.1.   Category theory

By a locally small category, we mean a category such that the collection of all morphisms between any two of its objects is a set. Let $X$, $Y$ are objects in a category and $\alpha, \beta:X\rightarrow Y$ are morphisms between these objects. An universal pair $(E, \epsilon)$ is called equalizer if the following diagram commutes:

i.e., $\alpha \circ \epsilon = \beta \circ \epsilon$ and also for each object $T$ and any morphism $\tau:T\rightarrow X$ which satisfy $\alpha \circ \tau = \beta \circ \tau$, there exists unique morphism $\sigma:T\rightarrow E$ such that $\epsilon \circ \sigma = \tau$. If equalizer existed then it is unique up to isomorphism. For example, in the category of sets, which is denoted by $\textbf{SET}$, the equalizer of two morphisms $\alpha, \beta:X\rightarrow Y$ is the set $E = \{x\in X | \alpha(x) = \beta(x)\}$ together with the inclusion map $\epsilon:E\hookrightarrow X.$

Let $\mathcal{C}$ be a locally small category, and $X$ be an object in $\mathcal{C}$. By $T$-points of $X$, we mean $X(T): = Hom_\mathcal{C}(T, X)$ for any $T\in Obj(\mathcal{C})$. The functor of points of $X$ is a functor which is denoted by $X(.)$ and is defined as follows:

where $X(\varphi):f\mapsto f\circ \varphi.$ A functor $F:\mathcal{C} \to \textbf{SET}$ is called representable if there exists an object $X$ in $\mathcal{C}$ such that $F$ and $X(.)$ are isomorphic. Then one may say that $F$ is represented by $X$. The category of functors from $\mathcal{C}$ to $\textbf{SET}$ is denoted by $[\mathcal{C}, \textbf{SET}]$. It is shown that the category of all representable functors from $\mathcal{C}$ to $\textbf{SET}$ is a subcategory of $[\mathcal{C}, \textbf{SET}]$.

Corresponding to each morphism $\psi:X\rightarrow Y$, there exists a natural transformation $\psi(.)$ from $X(.)$ to $Y(.)$. This transformation corresponds the mapping $\psi(T):X(T)\rightarrow Y(T)$ with $\xi\mapsto \psi\circ \xi$ for each $T\in Obj(\mathcal{C})$. Now set:

Obviously, $\mathcal{Y}$ is a covariant functor and it is called Yoneda embedding. The following lemma, is brought from ^[8] or the section $III.2$ in ^[20].

Lemma A.1. The Yoneda embedding is full and faithful functor, i.e. the map

is a bijection for each $X, Y\in Obj(\mathcal{C}).$

Thus according to this lemma, $X, Y\in Obj(\mathcal{C})$ are isomorphic if and only if their functor of points are isomorphic. The Yoneda embedding is an equivalence between $\mathcal{C}$ and a subcategory of representable functors in $[\mathcal{C}, \textbf{SET}]$ since not all functors are representable.

A.2.   The $\mathbb Z^n_2-$Lie group $GL(V)$

Let $V$ be a finite dimensional $\mathbb Z^n_2$-vector space of dimension $\overrightarrow{\textbf{m}} = m_0|m_1|\ldots|m_ \textbf{q}$ and let

be a basis of $V$ for which the elements $R_{m_0+...+m_{i-1}+ k},$ $1\leq k\leq m_i,$ are of weight $\gamma_i$ for $0\leq i\leq \textbf{q}$.

Consider the functor $F$ from the category $\textbf{ZSM}$ to $\textbf{GRP}$ the category of groups which maps each $\mathbb Z^n_2-$manifold $T$ to $Aut_{\mathcal O(T)}(\mathcal O(T)\otimes V)$ the group of zero weight automorphisms of $\mathcal O(T)\otimes V$. Consider the $\mathbb Z^n_2-$manifold $\textbf{End}(V) = \Big(\prod_{i} End(V_i), \mathcal{A}\Big)$ where $\mathcal{A}$ is the following sheaf

where $t_k = \sum_{\gamma_i+\gamma_j = \gamma_k}m_im_j.$ Let $F_{ij}$ be a linear transformation on $V$ defined by $R_k\mapsto \delta_{ik}R_j$, then $\{F_{ij}\}$ is a basis for $\textbf{End}(V)$. If $\{f_{ij}\}$ is the corresponding dual basis, then it may be considered as a global coordinates on $\textbf{End}(V)$. Let $X$ be the open submanifold of $\textbf{End}(V)$ corresponding to the open set:

Thus, we have

It can be shown that the functor $F$ may be represented by $X$. For this, one may show that $Hom(T, X)\cong Aut_{\mathcal O(T)}(\mathcal O(T)\otimes V)$. To this end, first, one can notice Theorem 9 in ^[17] to see

It is known that each $\psi\in Hom\Big(\mathcal A(X), \mathcal O(T)\Big)$ may be uniquely determined by $\{g_{ij}\}$ where $g_{ij} = \psi(f_{ij})$, see Theorem 6.8 in ^[1]. Now set $\Psi(R_j): = \Sigma g_{ij}R_i$. One may consider $\Psi$ as an element of $Aut_{\mathcal{O}(T)}(\mathcal{O}(T)\otimes V)$. Obviously $\psi\mapsto\Psi$ is a bijection from $Hom(T, X)$ to $Aut_{\mathcal{O}(T)}(\mathcal{O}(T)\otimes V)$. Thus the $\mathbb Z^n_2-$manifold $X$ is a $\mathbb Z^n_2-$Lie group and denoted it by $GL(V)$ or $GL(\overrightarrow{\textbf{m}})$ if $V = \mathbb{R}^{ \overrightarrow{\textbf{m}}}$. Therefore $T$- points of $GL(\overrightarrow{\textbf{m}})$ are the $\overrightarrow{\textbf{m}}\times \overrightarrow{\textbf{m}}$ invertible $\mathbb Z^n_2-$matrices of weight zero

where the elements of the $m_k\times m_u$ block $B_{ku}$ have degree $\gamma_k+\gamma_u$ and the multiplication is the matrix product.

A.3.   Proof of (Proposition 2.10)

Proof. The proof of (1) is easy because of (2.10). For (2), let $M$ be a $\mathbb Z^n_2$-manifold of dimension $\overrightarrow{\textbf{m}}$ and $G$ be a Lie $\mathbb Z^n_2$-group of dimension $\overrightarrow{\textbf{k}}$. Let $\mathfrak g$ be the Lie $\mathbb Z^n_2$-algebra of $G$ and let $J_{a_p}$ be the Jacobian $\mathbb Z^n_2$-matrix of $a_p$ in a neighborhood of a point $g\in \overline{G}$. Since

and $a^g$ and $r_{g^{-1}}$ are diffeomorphisms, $\overline{J_{a_p}}(g)$ has rank equal to $\dim \mathfrak g - \dim \ker (da_p)_e$ for each $g\in \overline{G}$. Recall that if $X\in \mathfrak g$ we denote by $D_X^R: = (X \otimes 1_{_{\mathcal O_G(U)}})\mu^*$ the right-invariant vector field associated with $X$. Using equation (2.8) we have, for each $X\in \ker (da_p)_e$,

Because

If $(\xi_{_0} = x, \xi_1, \ldots, \xi_ \textbf{q})$ and $(\eta_{_0} = y, \eta_1, \ldots, \eta_ \textbf{q})$ are coordinates in a neighborhood $U$ of $e\in \overline{G}$, and in a neighborhood $V\supseteq\overline{a_p}(U)$ of $p\in \overline{M}$, respectively, then $J_{a_p}$ may be represented as the following block matrix:

where by $\dfrac{\partial a_p^*(\eta_t)}{\partial \xi_l}$, we mean a matrix as follows

and its sign in (5.3) is equal to

Now, we find a matrix $A\in GL_{ \overrightarrow{\textbf{k}}}(\mathcal O_G(U))$ such that $J_{a_p}A$ has a certain set of columns equal to zero. We are going to use equation (5.2). Let

and $\{X_{u_0}\}$, $\ldots$ and $\{X_{u_ \textbf{q}}\}$ be the bases of $\mathfrak g_0$, $\ldots$ and $\mathfrak g_ \textbf{q}$ such that for all $n_r$ if $u_r\leq n_r$ then $X_{u_r}\in\ker (da_p)_e.$ Consider a block matrix as follows:

where, for each $u_j$, $0\leq j\leq \textbf{q}$, $\alpha_{{u_j}.{i_s}}$'s are the coefficients appear in the following equality:

where $\gamma_j+\gamma_s$ is the degree of $\alpha^s_{u_j, i_s}$ and the sign of $(t, l)$-th block is

Since $X_{u_i}$ and $X_{u_j}$ are linearly independent at each point, the reduced matrix $\overline{A}$ is invertible and $A\in GL_{ \overrightarrow{\textbf{k}}}(\mathcal O_G(U)).$ Now by equation (5.2), one may see that the matrix

has $| \overrightarrow{\textbf{n}}| = n_0+\ldots+n_ \textbf{q}$ zero columns. Thus $\overline{J_{a_{p}}}$ has rank equal to $\overrightarrow{\textbf{k}} - \overrightarrow{\textbf{n}}$ with entries in $\mathcal O_G(U)$. In addition the first $| \overrightarrow{\textbf{k}}| -| \overrightarrow{\textbf{n}}|$ columns are non-zero. Thus one may describe $J_{a_p}$ by $\begin{pmatrix} z & 0\\ w & 0 \end{pmatrix}.$ Then it is not restrictive to assume that $z$ invertible. Thus one has

where

For more explanation, assume $J_{a_p}$ is a matrix of the form as below

Since $\overline{J_{a_{p}}}$ has rank $\overrightarrow{\textbf{k}} - \overrightarrow{\textbf{n}}$, we can suppose that $\alpha_{ii}$ and $\beta_{jj}$ are invertible. By section 4 in ^[10], we can rearrange the matrix so that it takes the form $\begin{pmatrix} z & 0\\ w & 0 \end{pmatrix}$, with $z$ invertible and the matrix $G = \begin{pmatrix} z^{-1} & 0\\ -wz^{-1} & I \end{pmatrix}$ such that $GJ = \begin{pmatrix} I & 0\\ 0 & 0 \end{pmatrix}.$ We can then conclude that $J_{a_{p}}$ has constant rank in $U$ and, by translation, in all of $\overline{G}$.

Conflict of interest

The author declares there is no conflict of interest.