Products in the category of 𝕑2n-manifolds

Andrew Bruce; Norbert Poncin

doi:10.1080/14029251.2019.1613051

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Volume 26, Issue 3, May 2019, Pages 420 - 453

Products in the category of 𝕑2n-manifolds

Authors

Andrew Bruce

University of Luxembourg, Mathematics Research Unit, 4364 Esch-sur-Alzette, Luxembourg,andrew.bruce@uni.lu

Norbert Poncin

University of Luxembourg, Mathematics Research Unit, 4364 Esch-sur-Alzette, Luxembourg,norbert.poncin@uni.lu

Received 5 October 2018, Accepted 22 March 2019, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1613051 How to use a DOI?
Keywords: 𝕑2n-geometry; finite categorical product; product morphism; sheafification; locally convex topological algebra; completed tensor product
Abstract: We prove that the category of 𝕑2n-manifolds has all finite products. Further, we show that a 𝕑2n-manifold (resp., a 𝕑2n-morphism) can be reconstructed from its algebra of global 𝕑2n-functions (resp., from its algebra morphism between global 𝕑2n-function algebras). These results are of importance in the study of 𝕑2n Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two 𝕑2n-manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

𝕑2n-Geometry is an emerging framework in mathematics and mathematical physics, which has been introduced in the foundational papers [11] and [14]. This non-trivial extension of standard Supergeometry allows for 𝕑2n-gradings, where

𝕑2n=𝕑2×n=𝕑2×⋯×𝕑2 and n∈ℕ.

The corresponding 𝕑2n-commutation rule for coordinates (u^A)_A with degrees deguA∈𝕑2n does not use the product of the (obviously defined) parities, but the scalar product 〈−, −〉 of 𝕑2n:

uAuB=(−1)〈deguA,deguB〉uBuA.(1.1)

A brief description of the category 𝕑2nMan of 𝕑2n-manifolds ℳ = (M, 𝒪_M) and morphisms Φ = (ϕ, ϕ^*) between them can be found in Section 2. For n = 1, one recovers the category SMan of supermanifolds. A survey on 𝕑2n-Geometry is available in [31]. The differential calculus and the splitting theorem for 𝕑2n-manifolds have been investigated in [13] and [12], respectively. In the introduction of [7], the reader finds motivations for the study of 𝕑2n-Geometry. The present paper uses the main results of [7].

Applications of 𝕑2n-Geometry are based in particular on 𝕑2n Lie groups and their actions on 𝕑2n-manifolds (supergravity), on 𝕑2n vector bundles and their sections (𝕑2n Lie algebroids), on the internal Hom functor in 𝕑2nMan (𝕑ⁿ-gradings and 𝕑2n-parities in field theory), ... All these notions rely themselves on products in the category 𝕑2nMan. On the other hand, a comparison of different approaches to 𝕑2n vector bundles is more challenging than in the supercase [2]. A generalization to 𝕑2n-manifolds of the Schwarz-Voronov-Molotkov embedding is needed. This extension, which embeds 𝕑2nMan into the category of contravariant functors from 𝕑2n-points to a specific category of Fréchet manifolds, uses the reconstructions of 𝕑2n-manifolds and 𝕑2n-morphisms from the 𝕑2n-commutative associative unital ℝ-algebras of global 𝕑2n-functions and the 𝕑2n-graded unital algebra morphisms between them, respectively.

The existence of categorical products and the mentioned reconstruction theorems are the main topics of the present paper. The text is organized as follows. Section 3 contains the proofs of the above-mentioned 𝕑2n reconstruction results. The definition of a product 𝕑2n-manifold and the proof of its meaningfulness are rather obvious, see Definition 4.1. However, the proof of the existence of categorical products in 𝕑2nMan is quite tough. It relies on the generalization of the well-known isomorphism of topological vector spaces

C∞(Ω′)⊗^C∞(Ω″)≃C∞(Ω′×Ω″)

(for open subsets Ω′ ⊂ ℝ^p, Ω″ ⊂ ℝ^r) to an isomorphism of locally convex topological algebras of formal power series

C∞(Ω′)[[ξ]]⊗^C∞(Ω″)[[η]]≃C∞(Ω′×Ω″)[[ξ,η]](1.2)

(for 𝕑2n-domains 𝒰p|q=(Ω′,CΩ′∞[[ξ]])) and 𝒱r|s=(Ω″,CΩ″∞[[η]])), with

ξ=(ξ1,…,ξN(q)) and η=(η1,…,ηN(s))),

see Theorem 4.2. The issue here is the formal power series, which replace the polynomials of standard Supergeometry. Moreover, if ℳ = (M, 𝒪_M) and 𝒩 = (N, 𝒪_N) are two 𝕑2n-manifolds, and ℳ × 𝒩 = (M × N, 𝒪_M×N) is their product 𝕑2n-manifold, one gets from (1.2) that, for an open subset u × v ⊂ M × N of the basis 𝔅 made of products of 𝕑2n-chart domains, we have

ℰ(u×v)≃𝒪M×N(u×v),

where ℰ is the 𝔅-presheaf

ℰ(u×v)=𝒪M(u)⊗^𝒪N(v).

Let now ℱ¯ be the standard extension of the ℬ-presheaf ℱ that assigns to any open subset U × V ⊂ M × N (where U ⊂ M and V ⊂ N are not necessarily chart domains) the algebra

ℱ(U×V)=𝒪M(U)⊗^𝒪N(V).

The presheaf ℱ¯ and the sheaf 𝒪_M×N are thus two extensions of the 𝔅-presheaf ℰ. However, this does not mean that ℱ¯≃𝒪M×N and that ℱ¯ is a sheaf. Indeed, 𝔅-sheaves have unique extensions, but 𝔅-presheaves do not. Also the reconstruction results mentioned above do not allow us to prove that ℱ¯ is a sheaf. Hence, we prove the next best result, i.e., the existence of an isomorphism of sheaves of algebras

𝒪M×N≃ℱ¯+(1.3)

between the structure sheaf of the product 𝕑2n-manifold and the sheafification of the presheaf ℱ¯, see Theorem 4.4. In the case n = 1, we thus recover the definition of a product supermanifold used in [3]. The isomorphism (1.3) allows us to prove the existence of all finite categorical products in 𝕑2nMan, see Theorem 4.8. The proof uses the results on sheafification and presheaves of locally convex topological algebras proven in Subsections A.4 and A.5 of the Appendix. Products of 𝕑2n-morphisms are obtained from the universality property of categorical products. They are explicitly described in Proposition 4.9.

2. 𝕑2n-manifolds and their morphisms

We denote by 𝕑2n the cartesian product of n copies of 𝕑₂. Further, we use systematically the following standard order of the 2ⁿ elements of 𝕑2n: first the even degrees are ordered lexicographically, then the odd ones are also ordered lexicographically. For example,

𝕑23={(0,0,0),(0,1,1),(1,0,1),(1,1,0),(0,0,1),(0,1,0),(1,0,0),(1,1,1)}.

A 𝕑2n-domain has, aside from the usual coordinates x = (x¹,. . ., x^p) of degree degxi=0∈𝕑n2, also formal coordinates or parameters ξ = (ξ¹,. . .,ξ^Q) of non-zero degrees degξa∈𝕑2n. These coordinates u = (x, ξ) commute according to the generalized sign rule

uAuB=(−1)〈deguA,deguB〉uBuA,(2.1)

where 〈−, −〉 denotes the standard scalar product. For instance,

〈(0,1,1),(1,0,1)〉=1.

Observe that, in contrast with ordinary 𝕑₂- or super-domains, even coordinates may anticommute, odd coordinates may commute, and even nonzero degree coordinates are not nilpotent. Of course, for n = 1, we recover the classical situation. We denote by p the number of coordinates xⁱ of degree 0, by q₁ the number of coordinates ξ^a which have the first non-zero degree of 𝕑2n, and so on. We get that way a tuple q = (q₁,. . .,q_N) ∈ ℕ^N with N := 2ⁿ −1. The dimension of the considered 𝕑2n-domain is then given by p|q. Clearly the Q above is the sum |q|=∑i=1Nqi.

We recall the definition of a 𝕑2n-manifold.

Definition 2.1.

A locally 𝕑2n-ringed space is a pair (M, 𝒪_M) made of a topological space M and a sheaf of 𝕑2n-graded 𝕑2n-commutative (in the sense of (2.1)) associative unital ℝ-algebras over it, such that at every point m ∈ M the stalk 𝒪_M,m is a local graded ring.

A smooth 𝕑2n-manifold of dimension p|q is a locally 𝕑2n-ringed space ℳ = (M, 𝒪_M), which is locally isomorphic to the smooth 𝕑2n-domain ℝp|q:=(ℝp,Cℝp∞[[ξ]]), and whose underlying topological space M is second-countable and Hausdorff. Sections of the structure sheaf Cℝp∞[[ξ]] are formal power series in the 𝕑2n-commutative parameters ξ, with coefficients in smooth functions:

Cℝp∞(U)[[ξ]]:={∑α∈ℕ×|q|fα(x)ξα|fα∈C∞(U)} (U open in ℝp).

𝕑2n-morphisms between 𝕑2n-manifolds are just morphisms of 𝕑2n-ringed spaces, i.e., pairs Φ = (ϕ, ϕ^*) : (M, 𝒪_M) → (N, 𝒪_N) made of a continuous map ϕ : M → N and a sheaf morphism ϕ^* : 𝒪_N → ϕ_*𝒪_M, i.e., a family of 𝕑2n-graded unital ℝ-algebra morphisms, which commute with restrictions and are defined, for any open V ⊂ N, by

φV*:𝒪N(V)→𝒪M(φ−1(V)).

We denote the category of 𝕑2n-manifolds and 𝕑2n-morphisms between them by 𝕑2nMan.

Remark 2.2.

Let us stress that the base space M corresponds to the degree zero coordinates (and not to the even degree coordinates), and let us mention that it can be proven that the topological base space M carries a natural smooth manifold structure of dimension p, that the continuous base map ϕ : M → N is in fact smooth, and that the algebra morphisms

φm*:𝒪φ(m)→𝒪m (m∈M)

between stalks, which are induced by the considered 𝕑2n-morphism Φ : ℳ → 𝒩, respect the unique homogeneous maximal ideals of the local graded rings 𝒪_ϕ(m) and 𝒪_m.

3. Reconstructions of 𝕑2n-manifolds and 𝕑2n-morphisms

In this section, we reconstruct a 𝕑2n-manifold (M, 𝒪_M) from the 𝕑2n-commutative unital algebra 𝒪_M (M) of global sections of its function sheaf. We also reconstruct a 𝕑2n-morphism

Φ=(φ,φ*):(M,𝒪M)→(N,𝒪N)

from its pullback 𝕑2n-graded unital algebra morphism

φN*:𝒪N(N)→𝒪M(M)

between global sections.

3.1. Reconstruction of the topological base space

Algebraic characterizations of spaces can be traced back to I. Gel’fand and A. Kolmogoroff [16]. In that paper, compact topological spaces K are characterized by the algebras C⁰(K) of continuous functions on them. In particular, the points m of these spaces are identified with the maximal ideals

Im={f∈C0(K):fm=0}

of these algebras. A similar characterization holds for the points of second countable Hausdorff smooth manifolds.

Let ℳ = (M, 𝒪_M) be a 𝕑2n-manifold. We denote the maximal spectrum of 𝒪(M) (subscript omitted) by Spm(𝒪(M)) (we actually consider here the real maximal spectrum, in the sense that the quotient 𝒪(M)/μ by an ideal μ in the spectrum is isomorphic to the field ℝ of real numbers). Note that any m ∈ M induces a map

ɛm:𝒪(M)∋f↦(ɛMf)(m)∈ℝ,

which is referred to as the evaluation map at m and is a 𝕑2n-graded unital ℝ-algebra morphism

ɛm∈Hom𝕑2nUAlg(𝒪(M),ℝ).

The kernel

μm=kerɛm={f∈𝒪(M):(ɛMf)(m)=0}∈Spm(𝒪(M))

is a maximal ideal. More generally, the kernel of an arbitrary algebra morphism

ψ∈Hom𝕑2nUAlg(𝒪(M),ℝ)

is a maximal ideal, since 𝒪(M)/ker ψ ≃ ℝ. Indeed, to any class [f] in the quotient we can associate the real number ψ(f). This map is well-defined and injective. It is also surjective, since, for any r ∈ ℝ, the image of [r · 1_𝒪(M)] is r. It follows in particular that any class in the quotient is of type [r · 1_𝒪(M)] for a unique r ∈ ℝ. We have the following

Proposition 3.1.

The maps

𝔟:M∋m↦μm∈Spm(𝒪(M))

and

♭:Hom𝕑2nUAlg(𝒪(M),ℝ)∋ψ↦kerψ∈Spm(𝒪(M))

are 1 : 1 correspondences.

Proof.

To prove that 𝔟 is bijective, consider a maximal ideal μ ∈ Spm(𝒪(M)). The image ε_M(μ) ⊂ C^∞(M) is a maximal ideal. Indeed, it is an ideal, since the map ε_M is surjective (the short sequence of sheaves [7, Equation (3)] is exact for the good reason that it is exact for any open subset of M). To see that it is maximal, assume there is an ideal ν, such that ε_M(μ) ⊂ ν ⊂ C^∞(M), so that μ⊂ɛM−1(ν)⊂𝒪M(M). It follows that ɛM−1(ν)=μ or ɛM−1(ν)=𝒪M(M), and that ν = ε_M(μ) or ν = C^∞(M). Hence,

ɛM(μ)=Im={f∈C∞(M):f(m)=0},

since any maximal ideal of C^∞(M) is known to be of type I_m for a unique m ∈ M. Finally, we get

μ⊂ɛM−1(Im)={f∈𝒪(M):(ɛMf)(m)=0}=μm⊂𝒪(M).

Since μ_m ≠ 𝒪(M), we have μ = μ_m, which proves the bijectivity of 𝔟. Indeed, if μ = μ_n, we obtain ε_M(μ_n) ⊂ I_n ⊂ C^∞(M), so that

In=ɛM(μn)=ɛM(μm)=Im,

and m = n.

Since any μ ∈ Spm(𝒪(M)) reads μ = μ_m = ker ε_m = ♭(ε_m), the map ♭ is surjective. Let ψ, ϕ be unital algebra morphisms, such that ker ψ = ker ϕ = μ. For any f ∈ 𝒪(M), there exists a unique r ∈ ℝ, such that [f] = [r · 1_𝒪(M)]. Thus ψ(f) = r = ϕ(f) and ψ = ϕ, so that ♭ is also injective.

The next proposition relies on the Zariski and the Gel’fand topologies. The Gel’fand topology is possibly less known than the Zariski topology. We recall its precise definition in the proof of the proposition.

Proposition 3.2.

The map

ɛM−1:Spm(C∞(M))∋Im↦ɛM−1(Im)=μm∈Spm(𝒪(M))

is a homeomorphism with inverse ε_M, both, if the maximal spectra are endowed with their Zariski topology and if they are endowed with their Gel’fand topology. Hence, the Zariski and Gel’fand topologies coincide on Spm(𝒪(M)). Further, the bijection

𝔟:M∋m↦μm∈Spm(𝒪(M))

is a homeomorphism.

Proof.

The maps

ɛM−1:Spm(C∞(M))∋Im↦ɛM−1(Im)=μm∈Spm(𝒪(M))

and

ɛM:Spm(𝒪(M))∋μm↦ɛM(μm)=Im∈Spm(C∞(M))

are inverses of each other.

We first equip the spectrum Spm(C^∞(M)) as usual with the Zariski topology, which is defined by its basis of open subsets V_C^∞(f), f ∈ C^∞(M), given by

VC∞(f)={Im∈Spm(C∞(M)):f∉Im},

and we proceed similarly for Spm(𝒪(M)). It is straightforwardly checked that, if f = ε_M(F), we have

ɛM−1(VC∞(f))=VO(F) and ɛM(V𝒪(F))=VC∞(f).

Hence the announced homeomorphism result for the Zariski topologies.

The Gel’fand topology of Spm(C^∞(M)) is defined by the basis of open subsets B_C^∞(m, ∊; f₁,. . ., f_n), indexed by m ∈ M, ∊ > 0, n ∈ ℕ, and f₁,. . ., f_n ∈ C^∞(M), and defined by

BC∞(m,∊;f1,…,fn)={In∈Spm(C∞(M)):|fi(n)−fi(m)|<∊,∀i}.

The Gel’fand topology of Spm(𝒪(M)) is defined analogously by

B𝒪(m,∊;F1,…,Fn)={μn∈Spm(𝒪(M)):|ɛMFi(n)−ɛMFi(m)|<∊,∀i},

where F_i ∈ 𝒪(M). If f_i = ε_M(F_i), we have obviously

ɛM−1(BC∞(m,∊;f1,…,fn))=B𝒪(m,∊;F1,…,Fn),

and similarly for ε_M, so that the homeomorphism result holds also for the Gel’fand topologies.

Since the Zariski and Gel’fand topologies coincide on Spm(C^∞(M)), it follows from the above that there is a homeomorphism from Spm(𝒪(M)) endowed with the Zariski topology to itself endowed with the Gel’fand topology.

It is well-known that the map 𝔟_C^∞ : M ∍ m ↦ I_m ∈ Spm(C^∞(M)) is a homeomorphism (see for example [28]). Hence, the bijection 𝔟=ɛM−1∘𝔟C∞ is a homeomorphism as well.

3.2. Reconstruction of the structure sheaf

Proposition 3.3.

Let (M, 𝒪_M) be a 𝕑2n-manifold and let U ⊂ M be open. A 𝕑2n-function F ∈ 𝒪_M(U) is invertible if and only if its base projection f=ɛU(F)∈CM∞(U) is invertible.

Proof.

It is obvious that f is invertible if F is. Assume now that there exists f⁻¹ ∈ C^∞(U) and consider a cover of U by 𝕑2n-chart domains V_i. For any i, we have f⁻¹|_{V_i} = (f|_{V_i})⁻¹, i.e., the base function ε_{V_i} (F|_{V_i}) = ε_U(F)|_{V_i} is invertible in C^∞(V_i), so that F|_{V_i} ∈ 𝒪(V_i) ≃ C^∞(V_i)[[ξ]] has an inverse G_{V_i} ∈ 𝒪(V_i). It follows that, for any V_i and V_j with intersection V_ij,

GVi|Vij=(F|Vij)−1=GVj|Vij.

Hence, there is a unique 𝕑2n-function G ∈ 𝒪(U), such that G|_{V_i} = G_{V_i}. It is clear that G is the inverse of F.

Reconstructions of a sheaf from its global sections have been thoroughly studied in algebraic and differential geometry. A survey on such results can be found in [5] and [6]. The probably best known example is the construction of the structure sheaf 𝒪_X of an affine scheme X = Spec R from its global sections commutative unital ring 𝒪_X(X) = R. In this case, the ring 𝒪_X(V_f) of functions on a Zariski open subset V_f, f ∈ R, is defined as a localization of 𝒪_X(X). In the case of a 𝕑2n-manifold (M, 𝒪_M), we reconstruct 𝒪_M(U) as the localization of 𝒪_M(M) with respect to the multiplicative subset

SU={F∈𝒪M0(M):(ɛMF)|U is invertible}.

The chosen localization comes with a morphism that sends global sections with invertible projection in CM∞(U) to invertible sections in 𝒪_M(U), see Proposition 3.3.

Since the 𝕑2n-functions in S_U are of degree 0, no sign issues do appear and the definitions of the equivalence of fractions and the operations on their equivalence classes are the standard ones [1].

Proposition 3.4.

The localization 𝒪M(M)⋅SU−1 is a 𝕑2n-commutative associative unital ℝ-algebra structure, whose grading is naturally induced by the grading of 𝒪_M(M) (for a homogeneous r, the degree of rs⁻¹ is the degree of r), and whose zero (resp., unit) is represented by 01⁻¹ (resp., 11⁻¹).

Proof.

Straightforward verification.

We thus get a presheaf

ℒM:Open(M)∋U↦𝒪M(M)⋅SU−1∈𝕑2nUAlg

on M valued in the category of 𝕑2n-commutative associative unital ℝ-algebras. Indeed, if V ⊂ U is open, the obvious inclusion ιVU:SU↪SV provides a natural well-defined restriction

rVU:ℒM(U)∋Fs−1↦F(ιVUs)−1∈ℒM(V),

and these restrictions satisfy the usual cocycle condition.

As indicated above, we will show (in several steps) that the presheaf ℒ_M coincides with the structure sheaf 𝒪_M.

First, since it follows from Proposition 3.3 that, for any s ∈ S_U, the restriction s|_U is invertible in 𝒪_M(U), we have a map

λU:ℒM(U)∋Fs−1↦F|U(s|U)−1∈𝒪M(U).

This map is well-defined. Indeed, if Fs⁻¹ = F′s′⁻¹, there is σ ∈ S_U, such that (F|_Us′|_U − F′|_Us|_U) σ|_U = 0. Since the restrictions s|_U, s′|_U, and σ|_U are invertible in 𝒪_M(U), the claim follows. Further, it can be straightforwardly checked that λ_U is a morphism of 𝕑2n-graded unital ℝ-algebras.

In fact:

Proposition 3.5.

For any open U ⊂ M, the localisation map λ_U : ℒ_M(U) → 𝒪_M(U) is a 𝕑2n-graded unital ℝ-algebra isomorphism.

The proof of this result uses a method that can be found in various works, see for instance [3], [8], and [30]. We give this proof for completeness, as well as to show that it goes through in our 𝕑2n-graded stetting.

Proof.

It suffices to explain why λ_U is bijective.

(1)
Injectivity: Assume that F|_U(s|_U)⁻¹ = 0, i.e., that F|_U = 0, and show that Fs⁻¹ ∼ 01⁻¹, i.e., that there is σ ∈ S_U, such that Fσ = 0. Let (V_i, ψ_i) be a partition of unity of ℳ, such that the V_i are 𝕑2n-chart domains, so that 𝒪M|Vi≃CM∞|Vi[[ξ]]. For any i, we have
F|U∩Vi(x,ξ)=∑αFα|U∩Vi(x)ξα=0, i.e., Fα|U∩Vi=0, ∀α.

Let
σi∈CM∞(Vi)⊂𝒪M0(Vi), such that σi|U∩Vi>0 and σi|Vi\(U∩Vi)=0.

It follows that F|_{V_i} σ_i = 0. The 𝕑2n-function σ=∑iσiψi∈𝒪M0(M) has the required properties. Indeed, the open subsets V_i and Ω_i = M \ suppψ_i cover M and Fσ_iψ_i vanishes on both, V_i and Ω_i, so that Fσ=∑iFσiψi=0. In addition, for any m ∈ U, we have (εψ_i)(m) ≥ 0, for all i, and there is at least one j, such that (εψ_j)(m) > 0. Since (εψ_j)|_{Ω_j} = 0, we get m ∈ U ∩ V_j, so that (σ_jεψ_j)(m) > 0,
(ɛσ)(m)=∑i(σiɛψi)(m)>0,
and σ ∈ S_U.
(2)
Surjectivity: We must express an arbitrary f ∈ 𝒪_M(U) as a product f = F|_U(s|_U)⁻¹, with F ∈ 𝒪_M(M) and s ∈ S_U. To construct the global sections F and s, consider an increasing countable family of seminorms p_n that implements the locally convex topology of the Fréchet space 𝒪_M(M). Take also a countable open cover U_n of U, such that Ū_n ⊂ U, as well as bump functions γn∈𝒪M0(M), which satisfy γ_n|_{U_n} = 1 (which implies that (ε_Mγ_n)|_{U_n} = 1), supp γ_n ⊂ U, and ε_Mγ_n ≥ 0. The following series converge in 𝒪_M(M) and provide us with the required global sections:
F:=∑n=0∞12nγnf1+pn(γn)+pn(γnf) and s:=∑n=0∞12nγn1+pn(γn)+pn(γnf).

Indeed, convergence follows, if we can show that the series are Cauchy, i.e., if they are Cauchy with respect to each p_m. If r, s → ∞, we get r ≥ m, and, since the seminorms are increasing, we have

pm(∑n=rs12nγnf1+pn(γn)+pn(γnf))≤∑n=rs12npm(γnf)1+pn(γn)+pn(γnf)<∑n=rs12n→0,

whether the factor f is present or not. On the other hand, as restrictions are continuous, it is clear that F|_U = fs|_U, so that f = F|_U(s|_U)⁻¹, provided we show that s∈𝒪M0(M) belongs to S_U, i.e., that (ε_Ms)(m) ≠ 0, for all m ∈ U. To see this, note that (id,ɛ):(M,CM∞)→(M,𝒪M) is a morphism of 𝕑2n-manifolds, so that ɛM:𝒪M(M)→CM∞(M) is continuous, see [7, Theorem 19]. For any U_m of the cover of U, we thus get

(ɛMs)|Um:=∑n=0∞12n(ɛMγn)|Um1+pn(γn)+pn(γnf)>0,

in view of the properties of γ_n.

Theorem 3.6.

The 𝕑2n-commutative associative unital ℝ-algebra 𝒪_M(M) of global sections of the structure sheaf of a 𝕑2n-manifold (M, 𝒪_M) fully determines this sheaf. More precisely, there is a presheaf isomorphism λ : ℒ_M → 𝒪_M, so that the presheaf ℒ_M, which is obtained from 𝒪_M(M), is actually a sheaf, which is isomorphic to the structure sheaf 𝒪_M.

Proof.

It suffices to check that the family λ_U : ℒ_M(U) → 𝒪_M(U), U ∈ Open(M), of 𝕑2n-graded unital ℝ-algebra isomorphisms, commutes with the restrictions rVU in ℒ_M and ρVU in 𝒪_M (V ⊂ U, V ∈ Open(M)). This is actually obvious:

λV(rVU(Fs−1))=F|V(s|V)−1=ρVU(λU(Fs−1)).

3.3. Reconstruction of a 𝕑2n-morphism

In algebraic geometry, any commutative unital ring morphism ψ : S → R defines a morphism of affine schemes Φ = (ϕ, ϕ^*) : (Spec R, 𝒪_SpecR) → (Spec S,𝒪_SpecS), whose continuous base map ϕ associates to each prime ideal 𝔭 the prime ideal ψ⁻¹(𝔭). A similar result exists in the category of 𝕑2n-manifolds and 𝕑2n-morphisms, with the same definition of the continuous base map.

Theorem 3.7.

Let ℳ = (M, 𝒪_M) and 𝒩 = (N, 𝒪_N) be 𝕑2n-manifolds. The map

β:Hom𝕑2nMan(ℳ,𝒩)∋Φ=(φ,φ*)↦φN*∈Hom𝕑2nUAlg(𝒪N(N),𝒪M(M))

is a bijection.

Proof.

To show that β is surjective, we consider ψ∈Hom𝕑2nUAlg(𝒪N(N),𝒪M(M)) and construct Φ∈Hom𝕑2nMan(ℳ,𝒩), such that φN*=ψ.

Since M (resp., N) endowed with its base space topology is homeomorphic to Spm(𝒪_M(M)) (resp., Spm(𝒪_N (N))) endowed with the Zariski topology, we define ϕ by

φ:Spm(𝒪M(M))∋kerɛm↦ker(ɛm∘ψ)=kerɛn∈Spm(𝒪N(N)),

see Propositions 3.2 and 3.1. This map is continuous. Indeed, for any F ∈ 𝒪_N (N), the preimage by ϕ of the open subset

V(F)={kerɛn∈Spm(𝒪N(N)):ɛn(F)≠0}

is the subset

φ−1(V(F))={kerɛm∈Spm(𝒪M(M)):ɛm(ψ(F))≠0}=V(ψ(F)).

To define, for any open V ⊂ N, a 𝕑2n-graded unital ℝ-algebra morphism

φV*:𝒪N(V)→(φ*𝒪M)(V),

we rely on the isomorphism of 𝕑2n-graded unital ℝ-algebras 𝒪_N (V) ≃ ℒ_N (V) and the similar isomorphism in M. Hence, we define φV* by

φV*:ℒN(U)∋Fs−1↦ψ(F)ψ(s)−1∈ℒM(φ−1(V)).

This map is actually well-defined. Since s ∉ ker ε_n, for all n ∈ V, we have s ∉ ker(ε_m ○ ψ), for all m ∈ ϕ⁻¹(V), what means that ψ(s) ∈ S_ϕ⁻¹(V). In view of this, it is easy to see that the image is independent of the representative. The map φV* is a 𝕑2n-graded unital R-algebra morphism, because is.

As the family φV*, V ∈ Open(N), commutes obviously with restrictions, the continuous base map ϕ and the family of algebra morphisms φV*, V ∈ Open(N), define a 𝕑2n-morphism Φ : ℳ → 𝒩. To see that β(Φ)=φN*=ψ, it suffices to note that λ_N : ℒ_N (N) → 𝒪_N (N) sends the fraction Fs⁻¹ to the section Fs⁻¹ (and similarly for M), so that φN* and ψ coincide.

It remains to prove that β is injective. Let thus Φ = (ϕ, ϕ^*) and Ψ = (ψ, ψ^*) be two 𝕑2n-morphisms from ℳ to 𝒩, such that φN*=ψN* . Since the pullbacks by a 𝕑2n-morphism commute with the base projections, we get, for any m ∈ M,

φ(m)≃kerɛφ(m)={F∈𝒪(N):((ɛNF)∘φ)(m)=0}={F∈𝒪(N):(ɛM(φN*F))(m)=0}.

Hence, the continuous base maps ϕ and ψ coincide. Similarly, for any open V ⊂ N, each F_V ∈ 𝒪^N (V) reads uniquely F_V = λ_V (Fs⁻¹) = F|_V (s|_V)⁻¹, with F ∈ 𝒪_N (N) and s ∈ S_V ⊂ 𝒪_N (N), see

Proposition 3.5. As the family of pullbacks ϕ^* commutes with restrictions, we obtain

φV*(FV)=(φV*F|V)(φV*s|V)−1=(φN*F)|φ−1(V)(φN*s)|φ−1(V)−1.

Hence φV*=ψV*.

The preceding theorem, which allows us to characterize ℳ-points Hom𝕑2nMan(ℳ,𝒩) of a 𝕑2n-manifold 𝒩 by algebra morphisms, has some noteworthy corollaries.

Corollary 3.8.

The covariant functor

ℱ:𝕑2nMan→𝕑2nUAlgop,

which is defined on objects by ℱ(ℳ) = 𝒪_M(M) and on morphisms by ℱ(Φ)=φN*, is fully faithful, so that 𝕑2nMan can be viewed as full subcategory of 𝕑2nUAlgop.

The statement regarding the full subcategory is based on the well-known fact that any fully faithful functor is injective up to isomorphism on objects. This means that the existence of an isomorphism 𝒪_M(M) ≃ 𝒪_N (N) of 𝕑2n-graded unital ℝ-algebras implies the existence of an isomorphism ℳ ≃ 𝒩 of 𝕑2n-manifolds.

Corollary 3.9.

Let ℳ = (M, 𝒪_M) and 𝒩 = (N, 𝒪_N) be 𝕑2n-manifolds. The 𝕑2n-manifolds ℳ and 𝒩 are diffeomorphic if and only if their 𝕑2n-commutative associative unital ℝ-algebras 𝒪_M(M) and O_N(N) of global 𝕑2n-functions are isomorphic.

Such Pursell-Shanks type results have been studied extensively by one of the authors of this paper. Algebraic characterizations similar to Corollary 3.9 exist for instance for the Lie algebras of first order differential operators, of differential operators, and symbols of differential operators on a smooth manifold, for the super Lie algebras of vector fields and first order differential operators on a smooth supermanifold, as well as for the Lie algebra of sections of an Atiyah algebroid, see [18], [19], [20], [21].

Corollary 3.10.

The 𝕑2n-manifold ℰ = (∅, 0) (resp., ℝ^0|0 = ({pt}, ℝ)) is the initial (resp., terminal) object of the category of 𝕑2n-manifolds.

Proof.

For any 𝕑2n-manifold ℳ = (M, 𝒪_M), we have bijections

Hom𝕑2nMan(ℰ,ℳ)≃Hom𝕑2nUAlg(𝒪M(M),0)≃{F↦0}

and

Hom𝕑2nMan(ℳ,ℝ0|0)≃Hom𝕑2nUAlg(ℝ,𝒪M(M))≃{r↦r⋅1}.

4. Finite products in the category of 𝕑2n-manifolds

4.1. Cartesian product of 𝕑2n-manifolds

Let ℳ = (M, 𝒪_M) and 𝒩 = (N, 𝒪_N) be two 𝕑2n-manifolds of dimension p|q and r|s, respectively. The products U × V, U ⊂ M and V ⊂ N open, form a basis ℬ of the (second-countable, Hausdorff) product topology of M × N. Better, since the 𝕑2n-chart domains U_i in M (resp., V_j in N) are a basis of the topology of M (resp., of N), the products U_i × V_j form a basis 𝔅 of the product topology of M × N. As 𝕑2n-chart domains are diffeomorphic to open subsets of some coordinate space ℝⁿ, we identify the U_i and the V_j with the diffeomorphic U_i ⊂ ℝ^p and V_j ⊂ ℝ^r. Further, we denote the coordinates of the charts with domains U_i (resp., V_j) by (x_i, ξ_i) (resp., (y_j, η_j)), or, in case we use only two domains U_i (resp., V_j), we write also (x, ξ) and (x′, ξ′) (resp., (y, η) and (y′, η′)).

Definition 4.1.

Let ℳ = (M, 𝒪_M) and 𝒩 = (N, 𝒪_N) be two 𝕑2n-manifolds of dimension p|q and r|s, respectively. The product 𝕑2n-manifold ℳ × 𝒩, of dimension p + r|q + s, is the locally 𝕑2n-ringed space (M × N, 𝒪_M×N), where M × N is the product topological space and where the sheaf 𝒪_M×N is glued from the sheaves CUi×Vj∞(xi,yj)[[ξi,ηj]] associated to the basis 𝔅:

𝒪M×N|Ui×Vj≃CUi×Vj∞(xi,yj)[[ξi,ηj]].(4.1)

Recall that sheaves can be glued. More precisely, if (U_i)_i is an open cover of a topological space M, if ℱ_i is a sheaf on U_i, and if φ_ji : ℱ_i|_{U_i∩U_j} → ℱ_j|_{U_i∩U_j} is a sheaf isomorphism such that the usual cocycle condition φ_kjφ_ji = φ_ki holds, then there is a unique sheaf ℱ on M such that ℱ|_{U_i} ≃ ℱ_i. In the following, we set U_ij = U_i ∩ U_j.

Let now CUi×Vj∞[[ξ,η]] be the standard sheaf of 𝕑2n-graded 𝕑2n-commutative associative unital ℝ-algebras of formal power series in (ξ, η) with coefficients in sections of the sheaf CUi×Vj∞. The isomorphisms φ_𝔦𝔧,ij between the appropriate restrictions of the sheaves of algebras CUi×Vj∞[[ξ,η]] on the open cover (U_i × V_j)_i,j of M × N are induced as follows. Since ℳ is a 𝕑2n-manifold, there are 𝕑2n-isomorphisms

Φi=(φi,φi*):(Ui,𝒪M|Ui)→(Ui,CUi∞[[ξ]]),

which induce 𝕑2n-isomorphisms or coordinate transformations

Ψ𝔦i=Φ𝔦Φi−1:(Ui𝔦,CUi∞|Ui𝔦[[ξ]])→(Ui𝔦,CUi∞|Ui𝔦[[ξ′]]).

As we view U_i as both, an open subset of M and an open subset of ℝ^p, we implicitly identify U_i with its diffeomorphic image ϕ_i(U_i), so that ϕ_i = id_{U_i}. Hence, the coordinate transformations reduce to the isomorphisms

ψ𝔦i*=(φi*)−1φ𝔦*(4.2)

of sheaves of 𝕑2n-commutative ℝ-algebras:

ψ𝔦i*:CUi∞|Ui𝔦[[ξ′]]→CUi∞|Ui𝔦[[ξ]].(4.3)

Similar coordinate transformations exist for 𝒩:

ψ𝔦j*:CV𝔧∞|Vj𝔦[[η′]]→CVj∞|V𝔧𝔦[[η]].(4.4)

We denote the base coordinates in U_i (resp., U_𝔦) by x (resp., x′) and those in V_j (resp., V_𝔧) by y (resp., y′). The coordinate transformations (4.3), x = x(x′, ξ′), ξ = ξ(x′, ξ′), and (4.4), y = y(y′, η′), η = η(y′, η′), implement coordinate transformations or isomorphisms of sheaves of 𝕑2n-commutative ℝ-algebras

ϕ𝔦𝔦,ij=ψ𝔦i*×ψ𝔧j*:CU𝔦×V𝔦∞|Ui𝔦×Vj𝔧[[ξ′,η′]]→CUi×Vj∞|Ui𝔦×Vj𝔧[[ξ,η]].

In view of (4.2), the φ_𝔦𝔧,ij satisfy the cocycle condition. We thus get a unique glued sheaf 𝒪_M×N of 𝕑2n-commutative ℝ-algebras over M × N which restricts on U_i ×V_j to

𝒪M×N|Ui×Vj≃CUi×Vj∞[[ξ,η]],

i.e., we obtain a 𝕑2n-manifold, which we refer to as the product ℳ × 𝒩 of ℳ and 𝒩.

4.2. Fundamental isomorphisms

Theorem 4.2.

Let ℝ^p|q (resp., ℝ^r|s) be the usual 𝕑2n-domain (ℝ^p, Cℝp∞[[ξ]]) (resp., (ℝ^r, Cℝr∞[[η]])), and let Ω′ ⊂ ℝ^p and Ω″ ⊂ ℝ^r be open. There is an isomorphism of topological algebras

Cℝp∞(Ω′)[[ξ]]⊗^Cℝr∞(Ω″)[[η]]≃Cℝp×ℝr∞(Ω′×Ω″)[[ξ,η]],(4.5)

where the completion is taken with respect to any locally convex topology on the algebraic tensor product Cℝp∞(Ω′)[[ξ]]⊗Cℝr∞(Ω″)[[η]].

Proof.

Let R be a commutative von Neumann regular ring. For any families (M_α)_α and (N_β)_β of free R-modules, the natural R-linear map

(∏αMα)⊗R(∏βNβ)→∏αβ(Mα⊗RNβ)

is injective, if and only if R is injective as a module over itself [17]. Since any field is von Neumann regular, the regularity and injective module conditions are satisfied for R = ℝ. Hence, the linear map

(∏αC∞(Ω′))⊗(∏βC∞(Ω″))→∏αβ(C∞(Ω′)⊗C∞(Ω″))

is injective. Further, in view of [7, Corollary 17], the map

C∞(Ω′)[[ξ]]∋∑α∈𝒜fα(x)ξα↦(fα)α∈𝒜∈∏α∈𝒜C∞(Ω′)(4.6)

is a TVS-isomorphism between the source and the target equipped with the standard topology and the product topology of the standard topologies, respectively. In the sequence of canonical maps

C∞(Ω′)[[ξ]]⊗C∞(Ω″)[[η]]≃(∏α∈𝒜C∞(Ω′))×(∏β∈ℬC∞(Ω″))→∏α∈𝒜,β∈ℬ(C∞(Ω′)⊗C∞(Ω″))→∏α∈𝒜,β∈ℬC∞(Ω′×Ω″)≃C∞(Ω′×Ω″)[[ξ,η]],(4.7)

the first ≃ is a linear bijection, the first → is a linear injection, and the second ≃ is a TVS-isomorphism for the topologies used in (4.6). In

C∞(Ω′)⊗C∞(Ω″)→C∞(Ω′)⊗^C∞(Ω″)≃C∞(Ω′×Ω″), (4.8)

the isomorphism ≃ is the well-known TVS-isomorphism [22] (the target is endowed with its standard topology and the source with the topology of the completion with respect to any (C^∞(Ω′) is nuclear) locally convex topology on C^∞(Ω′) ⊗ C^∞(C) – we will not specify the latter topology), and the arrow → is the continuous linear inclusion (any TVS is a topological vector subspace (TVSS) of its completion, see Proposition A.6). This → induces the second → in (4.7), which is the inclusion of the source vector subspace into the target vector space. The source becomes a TVSS of the target when endowed with the induced topology (the induced topology is coarser than the product topology of the induced topologies). Finally, we equip the first space in (4.7) with the initial topology with respect to the first →, so that the first space gets promoted to a TVSS of the second, see Proposition A.4, and the first → becomes the continuous linear inclusion. The composite

ι:C∞(Ω′)[[ξ]]⊗C∞(Ω″)[[η]]∋∑αfαξα⊗∑βgβηβ↦∑αβfα⊗gβξαηβ∈C∞(Ω′×Ω″)[[ξ,η]]

of the maps of (4.7) is now the inclusion of a the source TVSS into the target TVS.

Note that, since the target is a LCTVS, see [7, Lemma 16], the source TVSS is also a LCTVS, see Proposition A.9 (since C^∞(Ω′)[[ξ]] is nuclear, see [7, Lemma 16], the completion of the source is independent of the chosen locally convex topology). In view of Proposition A.8, the completion of the source is a TVSS of the completion of the target, which, as the target is complete, see [7, Lemma 16], can be identified with the target due to Remark A.7. In other words, the continuous extension

ι^:C∞(Ω′)[[ξ]]⊗^C∞(Ω″)[[η]]→C∞(Ω′×Ω″)[[ξ,η]](4.9)

of the inclusion ι is an injective continuous linear map, see text above Proposition A.8. We will now prove that this map is surjective.

Let

S=∑αβFαβξαηβ∈C∞(Ω′×Ω″)[[ξ,η]]

be a formal series in the target space. In view of (4.8), we have [22], for any (α, β) ∈ 𝒜 × ℬ,

Fαβ=limN→+∞∑j=0Nfαβj⊗gαβj,

where fαβj∈C∞(Ω′) and gαβj∈C∞(Ω″), and where the limit is taken in C^∞(Ω′ × Ω″). Recall that 𝒜=ℕ×|q′|×𝕑2×|q″|, and similarly for ℬ. The product 𝒜 × ℬ is countable, since it is a finite product of countable sets. Let I : 𝒜 × ℬ → ℕ be an injective map valued in ℕ. The map J : 𝒜 × ℬ → ℐ, with ℐ = I(𝒜 × ℬ), is thus a 1:1 correspondence. We identify 𝒜 × ℬ with ℐ via J. For any j ∈ ℕ, we set,

for any α ∈ 𝒜 and any i ∈ ℐ,
C∞(Ω′)∋φαij={0,if i≃(γ,δ)≠(α,δ),fαδj,if i≃(γ,δ)=(α,δ), and,
for any β ∈ ℬ and any i ∈ ℐ,
C∞(Ω″)∋ψiβj={0,if i≃(γ,δ)≠(γ,β),gγβj,if i≃(γ,δ)=(γ,β).

Note that ℐ is a finite set {0,1,. . ., L}, L ∈ ℕ, (resp., is ℕ), if 𝒜 × ℬ is finite (resp., if 𝒜 × ℬ is countably infinite). For all j ∈ ℕ and all (α, β) ∈ 𝒜 × ℬ, we get

∑i=0Mφαij⊗ψiβj=fαβj⊗gαβj,

when M ∈ ℐ ∩ [J(α, β),+∞[. Indeed, if i ≃ (γ, δ) ≠ (α, β), then, either γ ≠ α and φαij=0, or δ ≠ β and ψiβj=0. However, if i ≃ (γ, δ) = (α, β), then φαij=fαβj and ψiβj=gαβj, so that the announced result follows. Hence, for any j ∈ ℕ and any (α, β) ∈ 𝒜 × ℬ, we have

limM→+∞∑i=0Mφαij⊗ψiβj=fαβj⊗gαβj,

where the sequence is constant for M ≥ J(α, β) and where the limit is computed in the topology of C^∞(Ω′ × Ω″). If a finite number of sequences of a TVS do converge, then their sum converges to the sum of the limits. It follows that, for any (α, β) ∈ 𝒜 × ℬ and any N ∈ ℕ,

limM→+∞∑j=0N∑i=0Mφαij⊗ψiβj=∑j=0Nfαβj⊗gαβj,

so that, for all (α, β) ∈ 𝒜 × ℬ,

limN→+∞limM→+∞∑j=0N∑i=0Mφαij⊗ψiβj=limN→+∞∑j=0Nfαβj⊗gαβj=Fαβ

in C^∞(Ω′ × Ω ″), and

limN→+∞limM→+∞(∑j=0N∑i=0Mφαij⊗ψiβj)(α,β)∈𝒜×ℬ=(Fαβ)(α,β)∈𝒜×ℬ(4.10)

in the product topology of Π_αβC^∞(Ω ′ × Ω ″), i.e., in the topology of the TVS C^∞(Ω ′ × Ω ″)[[ξ, η]].

Therefore, the sequence

(∑j=0N∑i=0Mφαij⊗ψiβj)(α,β)=∑j=0N∑i=0M(∑αφαijξα⊗∑βψiβjηβ)∈C∞(Ω′)[[ξ]]⊗C∞(Ω″)[[η]]

is a Cauchy sequence in C^∞(Ω′ × Ω″)[[ξ, η]], so a Cauchy sequence in the TVSS C^∞(Ω′)[[ξ]] ⊗ C^∞(Ω″)[[η]], and also in the topological vector supspace C∞(Ω′)[[ξ]]⊗^C∞(Ω″)[[η]]. Since this completion is sequentially complete, the Cauchy sequence considered converges in this space:

limN→+∞limM→+∞∑j=0N∑i=0M(∑αφαijξα⊗∑βψiβjηβ)∈C∞(Ω′)[[ξ]]⊗^C∞(Ω″)[[η]],

where the limit is taken in the topology of C∞(Ω′)[[ξ]]⊗^C∞(Ω″)[[η]]. Since the inclusion î, see Equation (4.9), is sequentially continuous, we get

ι^limN→+∞limM→+∞∑j=0N∑i=0M(∑αφαijξα⊗∑βψiβjηβ)=limN→+∞limM→+∞ι^∑j=0N∑i=0M(∑αφαijξα⊗∑βψiβjηβ)=limN→+∞limM→+∞∑αβ∑j=0N∑i=0Mφαij⊗ψiβjξαηβ=∑αβFαβξαηβ=S,

in view of (4.10). This shows that the continuous linear inclusion

ι^:C∞(Ω′)[[ξ]]⊗^C∞(Ω″)[[η]]→C∞(Ω′×Ω″)[[ξ,η]]

is bijective, so that the source TVSS of the target coincides with the target as TVS.

Since the completed tensor product of two nuclear Fréchet algebras is again a nuclear Fréchet algebra [15, Lemma 1.2.13], the source and target are actually topological algebras. We leave it to the reader to check that the preceding identification respects the multiplications.

Remark 4.3.

If p|q = p|0 and r|s = 0|s, it follows from Theorem 4.2 that

C∞(Ω)⊗^ℝℝ[[ξ]]≃C∞(Ω)[[ξ]],

and, if p|q = 0|q and r|s = 0|s, we get

ℝ[[ξ]]⊗^ℝℝ[[η]]≃ℝ[[ξ,η]].

Conversely, the general isomorphism of Theorem 4.2 is a consequence of the preceding particular cases and the fact that the category of complete nuclear spaces is a symmetric monoidal category with respect to the completed tensor product [9].

Theorem 4.4.

There is an isomorphism of sheaves of 𝕑2n-commutative ℝ-algebras

𝒪M×N≃(𝒪M⊗^𝒪N)−+

between the structure sheaf of a product 𝕑2n-manifold and the sheafification of the standard extension of the ℬ-presheaf

𝒪M⊗^𝒪N:U×V↦𝒪M(U)⊗^𝒪N(V).

Proof.

Recall that ℬ (resp., 𝔅) is the basis of the product topology of M × N made of the rectangular subsets U × V, where U ⊂ M and V ⊂ N are open (resp., of the rectangular subsets U_i × V_j, where U_i ⊂ M and V_j ⊂ N are 𝕑2n-chart domains). Let

ℱ(U×V):=𝒪M(U)⊗^𝒪N(V)(4.11)

be the completed tensor product of the nuclear 𝕑2n-graded Fréchet algebras 𝒪_M(U) and 𝒪_N(V) (with respect to any (reasonable) locally convex topology, e.g., the projective one). If U′ ×V′ ⊂ U ×V, the restrictions ρU′U:𝒪M(U)→𝒪M(U′) and ρV′V:𝒪N(V)→𝒪N(V′) of the Fréchet sheaves 𝒪_M and 𝒪_N are continuous linear maps. The continuous extension of the continuous linear map ρU′U⊗ρV′V is a continuous linear map [22]

ρU′U⊗^ρV′V:𝒪M(U)⊗^𝒪N(V)→𝒪M(U′)⊗^𝒪N(V′),

which we denote by

ρU′×V′U×V:ℱ(U×V)→ℱ(U′×V′).

Since the ρU′U and the ρV′V satisfy the standard presheaf conditions and the linear maps ρU′×V′U×V are continuous, it is clear that the latter satisfy these conditions as well. Hence, the pair (ℱ, ρ) is a Set-valued ℬ-presheaf.

This ℬ-presheaf can be extended to a Set-valued presheaf (ℱ¯,ρ¯). Indeed, set, for any open Ω ⊂ M × N,

ℱ¯(Ω):={(fab)ab:fab∈ℱ(Ua×Vb),Ua×Vb⊂Ω, such that ρUa𝔞×Vb𝔟Ua×Vb(fab)=ρUa𝔞×Vb𝔟Ua×Vb(f𝔞𝔟)},(4.12)

and consider, for any Ω′ ⊂ Ω, the map

ρ¯Ω′Ω:ℱ¯(Ω)→ℱ¯(Ω′)

which sends any element of ℱ¯(Ω) to the element of ℱ¯(Ω′) that we obtain by suppressing the f_ab for which U_a ×V_b is not a subset of Ω′. The ρ¯Ω′Ω satisfy of course the standard presheaf conditions. Further, the presheaf (ℱ¯,ρ¯) extends the ℬ-presheaf (ℱ, ρ). Indeed, for Ω = U × V, any f ∈ ℱ(Ω) provides a unique family fab=ρUa×VbU×V(f) in ℱ¯(Ω), thus defining a map ♭Ω:ℱ(Ω)→ℱ¯(Ω). If ♭_Ω(f) = ♭_Ω(g), then, in particular,

f=ρU×VU×V(f)==ρU×VU×V(g)=g.

In fact ♭_Ω is a 1:1 correspondence. Indeed, any family f_ab in ℱ¯(Ω) contains f ∈ ℱ(Ω), and fab=ρUa×VbU×V(f), so that ♭_Ω(f) = (f_ab)_ab. Hence,

♭U×V:ℱ(U×V)→∼ℱ¯(U×V).(4.13)

Moreover, if Ω′ = U′ ×V′ ⊂ Ω = U × V, and if f ∈ ℱ(U × V), then

ρ¯U′×V′U×V(♭U×V(f))=♭U′×V′(ρU′×V′U×V(f)),

since both sides are made of the family of restrictions ρUa×VbU×V(f), for all U_a × V_b ⊂ U′ ×V′.

Let now U_i × V_j ∈ 𝔅 be a Cartesian product of 𝕑2n-chart domains, and let Ω ⊂ U_i × V_j be any open subset. Recall Definition (4.12). Since U_a ⊂ U_i and V_b ⊂ V_j are 𝕑2n-chart domains, we get in view of (4.5) and of (4.1),

ℱ(Ua×Ub)=𝒪M(Ua)⊗^𝒪N(Vb)=C∞(Ua)[[ξi]]⊗^C∞(Vb)[[ηj]]≃C∞(Ua×Vb)[[ξi,ηj]]=𝒪M×N(Ua×Vb).(4.14)

Due to the continuity and linearity of the restrictions ρU′×V′U×V of ℱ and the restrictions PΩ′Ω of 𝒪_M×N, the restrictions in (4.12) coincide with the corresponding restrictions P of the structure sheaf of the product 𝕑2n-manifold, as both reduce to the same restrictions of classical functions. It follows from (4.12) and (4.14) that any family fab∈ℱ¯(Ω) is made of 𝕑2n-functions f_ab ∈ 𝒪_M×N (U_a ×V_b), which are defined on the cover of Ω by all the U_a × V_b ⊂ Ω, and whose P-restrictions coincide on all intersections. Hence, any family fab∈ℱ¯(Ω) can be glued in the sheaf 𝒪_M×N and thus provides a unique 𝕑2n-function f ∈ 𝒪_M×N(Ω) such that PUa×VbΩ(f)=fab. The resulting map 𝔟Ω:ℱ¯(Ω)→𝒪M×N(Ω) is clearly injective. It is also surjective, since the restrictions fab:=PUa×VbΩ(f) of any f ∈ 𝒪_M×N(Ω) define a family fab∈ℱ¯(Ω) whose image by 𝔟_Ω is f. Therefore, if Ω ⊂ U_i × V_j, we have a 1:1 correspondence

𝔟Ω:ℱ¯(Ω)→∼𝒪M×N(Ω).(4.15)

Moreover,

PΩ′Ω∘𝔟Ω=𝔟Ω′∘ρ¯Ω′Ω.(4.16)

Indeed, if (fab)ab∈ℱ¯(Ω), the LHS map sends this family f_ab, U_a × V_b ⊂ Ω, first to the unique f ∈ 𝒪_M×N(Ω) such that PUa×VbΩ(f)=fab, and then to the restriction PΩ′Ω(f). The RHS map sends this family first to the subfamily f_αβ, U_α × V_β ⊂ Ω′, then to the unique g ∈ 𝒪_M×N (Ω′) such that PUα×VβΩ′(g)=fαβ. It is clear that g=PΩ′Ω(f). Hence,

𝔟:ℱ¯|Ui×Vj→∼𝒪M×N|Ui×Vj

is a presheaf isomorphism, so that ℱ¯|Ui×Vj is a sheaf.

Remark 4.5.

Let us recall and emphasize that the superscript + refers to sheafification.

We denote the sheafification of the presheaf ℱ¯ by ϕ:ℱ¯→ℱ¯+ (ρ¯+ refers to the restrictions of ℱ¯+). Recall that any presheaf and its sheafification have the same stalks, i.e., that the maps

ϕm,n:ℱ¯m,n→∼ℱ¯m,n+,

(m, n) ∈ M × N, induced on stalks by the presheaf morphism φ are isomorphisms. Therefore, the sheaf morphism ϕ|Ui×Vj:ℱ¯|Ui×Vj→ℱ¯+|Ui×Vj is a sheaf isomorphism. This means that

ϕΩ:ℱ¯(Ω)→∼ℱ¯+(Ω)

is an isomorphism, or, here, a 1:1 correspondence, for any open Ω ⊂ U_i ×V_j, so that

ιΩ:=𝔟Ω∘ϕΩ−1:ℱ¯+(Ω)→∼𝒪M×N(Ω)

is also 1:1. In particular, for any U_i ×V_j ∈ 𝔅, we have

ιUi×Vj:ℱ¯+(Ui×Vj)→∼𝒪M×N(Ui×Vj).(4.17)

Since φ commutes with restrictions, we get, for any U_𝔦 ×V_𝔧 ⊂ U_i × U_j,

ρ¯U𝔦×V𝔧Ui×Vj∘ϕUi×Vj−1=ϕU𝔦×V𝔧−1∘ρ¯+Ui×VjU𝔦×V𝔧,

so that, when taking also (4.16) into account, we obtain

PU𝔦×V𝔧Ui×Vj∘ιUi×Vj=ιU𝔦×V𝔧∘ρ¯+U𝔦×V𝔧Ui×Vj.(4.18)

Due to (4.17) and (4.18), the map ι is a 𝔅-sheaf isomorphism between ℱ¯+ and 𝒪_M×N viewed as 𝔅-sheaves. Since a 𝔅-sheaf morphism extends to a unique sheaf morphism, there exists a sheaf isomorphism

I:ℱ¯+→∼𝒪M×N.

The morphism I is actually an isomorphism of sheaves of 𝕑2n-commutative ℝ-algebras. It suffices to show that ι is an isomorphism of 𝔅-sheaves of such algebras, i.e., that ι_{U_i}×V_j is a morphism of 𝕑2n-graded unital ℝ-algebras. We will prove that ι_Ω, Ω ⊂ U_i × V_j, is an algebra morphism, leaving the remaining checks to the reader. The space ℱ(U × V) is a nuclear Fréchet algebra, because it is the completed tensor product of nuclear Fréchet algebras [15, Lemma 1.2.13]. Its multiplication •_ℱ is continuous. It is given by

∑i=0∞fi⊗gi•ℱ∑j=0∞hj⊗kj=∑i=0∞∑j=0∞(−1)〈gi,hj〉(fihj)⊗(gikj).

The multiplication •_ℱ induces a multiplication •ℱ¯ on ℱ¯(Ω), Ω ⊂ M × N, which is defined by

(fab)ab•ℱ¯(gab)ab=(fab•ℱgab)ab.

Addition and scalar multiplication on ℱ¯(Ω) are defined similarly. As ℱ¯ is thus a presheaf of algebras, its sheafification ℱ¯+ is a sheaf of algebras and the ℱ¯(Ω)→ℱ¯+(Ω) are algebra morphisms, see Subsection A.4 of the Appendix. For Ω ⊂ U_i × V_j, this morphism φ_Ω is an algebra isomorphism and so is ϕΩ−1. The map ℱ¯(Ω)→𝒪M×N(Ω) associates to each (f_ab)_ab the f such that PUa×VbΩ(f)=fab. Hence, the image by 𝔟_Ω of a product (fab)ab•ℱ¯(gab)ab is the function h such that PUa×VbΩ(h)=fab•ℱgab. On the other hand, the product f · g in 𝒪_M×N (Ω) of the images by 𝔟_Ω satisfies

PUa×VbΩ(f⋅g)=PUa×VbΩ(f)⋅PUa×VbΩ(g)=fab⋅gab=fab•ℱgab,

see Theorem 4.2. It follows that h = f · g. The map 𝔟_Ω is in fact an algebra morphism. Finally ιΩ=𝔟Ω∘ϕΩ−1 is a morphism of algebras, as needed.

Remark 4.6.

In view of (4.15), (4.13), and (4.11) (as well as in view of (4.1) and (4.5)), it is clear that, for any U_i ×V_j ∈ 𝔅, we have

𝒪M×N(Ui×Vj)=𝒪M(Ui)⊗^𝒪N(Vj),

but we were unable to convince ourselves that the same holds true for any U ×V ∈ ℬ.

Indeed, it is well-known that tensor products of sheaves (and in particular completed tensor products of function sheaves) require a sheafification (see [32, Section 3]). However, section spaces of the sheafification of a presheaf do not agree with the corresponding section spaces of the presheaf.

On the other hand, attempts to get rid of the problem in Remark 4.6 using the reconstruction results from Section 3 below, are not really promising.

Further, although 𝒪_M×N and ℱ¯ are two presheaves that extend the 𝔅-presheaf 𝒪M⊗^𝒪N, they do not necessarily coincide: 𝔅-sheaves have unique extensions, but 𝔅-presheaves do not. Indeed, to show that 𝒪M×N≃ℱ¯, we would have to decompose sections of 𝒪_M×N into sections of the 𝔅-presheaf and then reglue them in ℱ¯, which is impossible, since ℱ¯ is only a presheaf.

There is actually a condition for the presheaf ℱ¯ that extends the ℬ-presheaf ℱ=𝒪M⊗^𝒪N to be a sheaf.

The explanation of this result needs some preparation.

For any open U ×V ⊂ M × N, we set 𝒪M×NM(U×V):=𝒪M(U). Similarly, for any open U′ ×V′ ⊂ U ×V, we define

rU′×V′U×V:𝒪M×NM(U×V)→𝒪M×NM(U′×V′)

to be ρU′U:𝒪M(U)→𝒪M(U′). It is straightforwardly checked that 𝒪M×NM is a nuclear Fréchet sheaf of algebras, hence, in particular a nuclear locally convex topological sheaf of algebras. The assignment

ℱ:U×V↦𝒪M×NM(U×V)⊗^𝒪M×NM(U×V)=𝒪M(U)⊗^𝒪N(V)

defines a presheaf ℱ¯ on M × N. Applying [25, Equation (2.2)], we would get

𝒪M×N(Ui×Vj)≃ℱ¯+(U×V)≃𝒪M(U)⊗^𝒪N(V),(4.19)

for any open U × V ⊂ M × N, if 𝒪M×NM or 𝒪M×NN determined a topologically dual weakly flabby precosheaf.

Just as a presheaf 𝒢 on a topological space T with values in a concrete category C is a contravariant functor 𝒢 : Open(T)^op → C, a precosheaf ℋ on T with values in C is a covariant functor ℋ : Open(T) → C. The point is that, for open subsets V ⊂ U, there is a C-morphism eUV:ℋ(V)→ℋ(U), which we refer to as extension morphism. The relevant example for our purpose is the topologically dual precosheaf 𝒱′ of vector spaces of a topological sheaf 𝒱 of vector spaces on a Hausdorff space T. This precosheaf is defined, for any open U ⊂ T, by

𝒱′(U)=HomTVS(𝒱(U),ℝ),

and, for any open subsets V ⊂ U, by

eUV=RtVU:𝒱′(V)∋ℓ↦[(RtVU)ℓ:𝒱(U)∋v↦ℓ(RVUv)∈ℝ]∈𝒱′(U),

where RVU denotes the restriction in 𝒱. The precosheaf 𝒱′ is weakly flabby if, for any open U ⊂ T, the morphism eTU:𝒱′(U)→𝒱′(T) is surjective.

We should prove that the topologically dual precosheaf of 𝒱=𝒪M×NM is weakly flabby, i.e., that, for any open U ⊂ M and for any L ∈ Hom_TVS(𝒪_M(M), ℝ), there exists ℓ ∈ Hom_TVS(𝒪_M(U), ℝ), such that ρtUMℓ=L. It turns out that this condition is not satisfied, so that we cannot conclude that (4.19) holds. Indeed, assume that the condition is satisfied, so that it is in particular valid for ℳ=(ℝ,Cℝ∞). Choose now any x ∈ ℝ and any open interval I ⊂ ℝ that does not contain x. The evaluation map

ɛx:Cℝ∞(ℝ)∋f↦f(x)∈ℝ

is linear. It is also continuous, since there exists a compact C ⊂ ℝ that contains x and since, for any f∈Cℝ∞(ℝ), we have |f(x)| ≤ sup_C |f|. In view of our assumption, there exists ℓ∈HomTVS(Cℝ∞(I),ℝ), such that, for any f∈Cℝ∞(ℝ), we have ℓ(f|_I) = f(x). If we take now two functions f, g∈Cℝ∞(ℝ) that coincide in I and have different values at x, we get the contradiction

f(x)=ℓ(f|I)=ℓ(g|I)=g(x).

4.3. Categorical products of 𝕑2n-manifolds

We recommend to first read Subsections A.4 and A.5 of the Appendix.

Lemma 4.7

Let ℳ₁, ℳ2∈𝕑2nMan. The presheaf ℱ¯ considered in Theorem 4.4 is an object of the category PSh(M₁ × M₂, LCTAlg) of presheaves of locally convex topological algebras over M₁ × M₂.

Proof.

In the proof of Theorem 4.4 we showed that ℱ¯∈Psh(M1×M2,Alg) for the obvious restrictions and algebra operations.

Recall that, for any open Ω ⊂ M₁ × M₂, the algebra ℱ¯(Ω) is given by

ℱ¯(Ω):={(fab)ab:fab∈ℱ(Ua×Vb),Ua×Vb⊂Ω, such that ρUa𝔞×Vb𝔟Ua×Vb(fab)=ρUa𝔞×Vb𝔟Ua×Vb(f𝔞𝔟)}

and is thus a subalgebra of

∏Ua×Vb⊂Ωℱ(Ua×Vb)=∏Ua×Vb⊂Ω𝒪M1(Ua)⊗^𝒪M2(Vb).

We equip ℱ¯(Ω) with the topology induced by the product of the topologies of the locally convex topological algebras (LCTA-s) 𝒪M1(Ua)⊗^𝒪M2(Vb). Since a product of LCTVS-s and a subspace of a LCTVS are themselves LCTVS-s, the algebra ℱ¯(Ω) is a LCTVS. Its multiplication

ab)ab•ℱ¯(gab)ab=(fab•ℱgab)ab

is continuous, since the multiplication •_ℱ is, see Proof of Theorem 4.4 and Lemma A.12. Hence, the space ℱ¯(Ω) is a LCTA.

A restriction ρ¯Ω′Ω(Ω′⊂Ω) (Ω′ ⊂ Ω) – it sends any family (f_ab)_ab of ℱ¯(Ω) indexed by the U_a × V_b ⊂ Ω to the family (f_ab)_ab of ℱ¯(Ω′) indexed by the U_a ×V_b ⊂ Ω′ – is known to be an algebra morphism. It is continuous, since it is continuous as a map

ρ¯Ω′Ω:∏Ua×Vb⊂Ωℱ(Ua×Vb)→∏Ua×Vb⊂Ω′ℱ(Ua×Vb),

in view of the definition of the product topology.

Theorem 4.8.

The category 𝕑2nMan has all finite products.

Proof.

Since 𝕑2nMan has a terminal object (see Corollary 3.10), it suffices to prove that it has binary products. Let ℳ₁, ℳ2∈𝕑2nMan. We will show that the product 𝕑2nMan-manifold ℳ₁ × ℳ₂ (see Definition 4.1) is the categorical binary product of ℳ₁ and ℳ₂.

We first define 𝕑2n-morphisms

Πi=(πi,πi*):ℳ1×ℳ2→ℳi, i∈{1,2}.

The base maps π_i : M₁ × M₂ → M_i are the canonical smooth projections. In the following, we consider the case i = 1 and use the notation introduced above.

The maps

TU:𝒪M1(U)∋f↦f⊗1∈(π1,*ℱ¯)(U),

U ∈ Open(M₁), define a morphism T:𝒪M1→π1,*ℱ¯ in PSh(M₁, LCTAlg). Indeed, we have 𝒪_M₁,π_1,*ℱ¯∈Psh(M1,LCTAlg), see [7, Theorem 14] and Proof of Proposition A.19. It is easy to see that the maps T_U commute with restrictions and are algebra morphisms. To show that T_U is continuous, it suffices to check that the linear map

TU:𝒪M1(U)∋f↦f⊗1∈𝒪M1(U)⊗𝒪M2(M2)(4.20)

is continuous for the projective tensor topology on the target. We apply Theorem A.10 and Proposition A.11. Let p_C,D ⊗ p_K,Δ be any of the seminorms that induce the projective tensor topology. Recall that C ⊂ U and K ⊂ M₂ are compact subsets, and that D and Δ are differential operators acting on 𝒪_M₁(U) and 𝒪_M₂(M₂), respectively. We must prove that there is a finite number of seminorms p_{C_k},D_k on the source and a constant C > 0, such that

pC,D(f)⋅pK,Δ(1)≤CmaxkpCk,Dk(f),

for any f ∈ 𝒪_M₁(U). It suffices to use a single source seminorm p_{C_k},D_k, namely p_C,D. Indeed, if ℭ = p_K,Δ(1) = 0, the previous condition is satisfied with C = 1, and if ℭ > 0, it is satisfied with C = ℭ.

It follows from Proposition A.15 that T+:𝒪M1+→(π1,*ℱ¯)+ is a morphism in Sh(M₁, LCTAlg). Moreover, in view of Proposition A.19, there is an Sh(M₁, LCTAlg)-morphism ι:(π1,*ℱ¯)+→π1,*ℱ¯+. The composite ι∘T+:𝒪M1+→π1,*ℱ¯+ is a morphism in Sh(M₁, LCTAlg). Proposition A.15 and Theorem 4.4 allow us to interpret it as a morphism

π1*=ι∘T+:𝒪M1→π1,*𝒪M1×M2(4.21)

in Sh(M₁, Alg). The map π1* is actually a morphism of sheaves of 𝕑2n-commutative associative unital algebras, so that the map Π1=(π1,π1*):ℳ1×ℳ2→ℳ1 is a morphism in 𝕑2nMan (the results of the appendix we use in this proof extend obviously to the graded unital setting: when speaking in the rest of the proof about algebras, we actually mean 𝕑2n-commutative associative unital algebras).

It remains to check the universality of our construction. Let 𝒩∈𝕑2nMan and let Φi=(φi,φi*):𝒩→ℳi be morphisms in 𝕑2nMan. We will prove that there exists a unique 𝕑2nMan-morphism Ψ = (ψ, ψ^*) : 𝒩 → ℳ₁ × ℳ₂, such that Π_i ○ Ψ = Φ_i, for both i.

We set ψ = (ϕ₁, ϕ₂) : N → M₁ × M₂. As for ψ^*, observe that, for any open U_i ⊂ M_i, the map φi,Ui*:𝒪Mi(Ui)→𝒪N(φi−1(Ui)) is a continuous algebra (𝕑2n-graded unital algebra) morphism, see [7, Theorem 19]. Denote now by V = V₁ ∩V₂ the open subset

V=V1∩V2=φ1−1(U1)∩φ2−1(U2)=ψ−1(U1×U2)⊂N,

and denote by m_V = − · − the multiplication of 𝒪_N(V). In view of [7, Theorem 14] and Propositions A.17 and A.18, the map

𝔭U1×U2=m^V∘(ρVV1⊗^ρVV2)∘(φ1,U1*⊗^φ2,U2*):𝒪M1(U1)⊗^𝒪M2(U2)→𝒪N(V),

where ρVV1 and ρVV2 are restrictions in 𝒪_N, is a continuous algebra morphism between nuclear Fréchet algebras. The maps

𝔭U1×U2=ℱ(U1×U2)∋∑j=0∞fj⊗gj↦∑j=0∞ρVV1φ1,U1*fj⋅ρVV2φ2,U2*gj∈(ψ*𝒪N)(U1×U2)(4.22)

define a morphism of ℬ-presheaves of locally convex topological algebras. The latter extends to a morphism 𝔭¯:ℱ¯→ψ*𝒪N of presheaves of locally convex topological algebras over M₁ × M₂.

In view of Propositions A.15 and A.19, there are morphisms

𝔭¯+:ℱ¯→(ψ*𝒪N)+ and ι:(ψ*𝒪N)+→ψ*𝒪N+

in Sh(M₁ × M₂, LCTAlg). As above, we can view their composite as a morphism

ψ*=ι∘𝔭¯+:𝒪M1×M2→ψ*𝒪N(4.23)

of sheaves of algebras, so that Ψ = (ψ, ψ^*) : 𝒩 → ℳ₁ × ℳ₂ is a morphism in 𝕑2nMan.

To prove that Π_i ○ Ψ = Φ_i, it suffices to show that

ψM1×M2*∘πi,Mi*=β(Πi∘Ψ)=β(Φi)=φi,Mi*,

see Theorem 3.7. The latter is a straightforward consequence of Equations (4.20), (4.21), (4.22), (4.23), (A.3), and (A.5).

Let now X = (χ, χ^*) : 𝒩 → ℳ₁ × ℳ₂ be another 𝕑2nMan-morphism that satisfies Π_i ○ X = Φ_i. As the category of smooth manifolds has finite products, we get χ = ψ. We will check that β(X) = β(Ψ), i.e., that, for all σ∈ℱ¯(M1×M2), we have

χM1×M2*σ=ψM1×M2*σ∈𝒪N(N).

It suffices to show that these sections coincide in a neighborhood of an arbitrary point n₀ ∈ N. We use the compact notation m ∈ M instead of (m₁, m₂) ∈ M₁ × M₂. Recall that

σ=([s]m)m∈M,

where s ∈ ℱ (U) (U = U₁ × U₂ ∍ m) reads s=∑j=0∞fj⊗gj (f_j ∈ 𝒪_M₁(U₁), g_j ∈ 𝒪_M₂(U₂)).

In view of (4.22), (4.23), (A.3), and (A.5),

ψM1×M2*σ=([∑j=0∞ρVV1φ1,U1*fj⋅ρVV2φ2,U2*gj]n)n∈N,(4.24)

where we take the germ at n of the section induced by the representative s of the germ at ψ(n) ∈ M. Let m₀ = ψ(n₀) ∈ M. Since s is constant in a neighborhood U₀ = U_1,0 ×U_2,0 of m₀, the representative in the RHS of the preceding equation is constant in the neighborhood

V0=V1,0∩V2,0=φ1−1(U1,0)∩φ2−1(U2,0)=ψ−1(U0)

of n₀. Hence,

ρV0NψM1×M2*σ=∑j=0∞ρV0V1,0φ1,U1,0*fj⋅ρV0V2,0φ2,U2,0*gj∈𝒪N(V0).

On the other hand, since Π_i ○ X = Φ_i, we have, for any open U₁ ⊂ M₁ and any f ∈ 𝒪_M₁(U₁),

χU1×M2*([f⊗1]m)m∈U1×M2=φ1,U1*f,

due to (4.20), (4.21), (A.3), and (A.5). For any open U₂ ⊂ M₂, we get

χU*([f⊗1]m)m∈U=ρψ−1(U)ψ−1(U1×M2)χU1×M2*([f⊗1]m)m∈U1×M2=ρVV1φ1,U1*f.(4.25)

An analogous result holds for i = 2. Observe now that

σ=([∑j=0∞fj⊗gj]m)m∈M=(πUmlimN∑j=0Nfj⊗gj)m∈M=(limN∑j=0N[fj⊗gj]m)m∈M=∑j=0∞([fj⊗gj]m)m∈M,(4.26)

see Proof of Proposition A.15. As the pullbacks χ^* and the restriction ρV0N are continuous algebra morphisms, we get

ρV0NχM1×M2*σ=∑j=0∞ρψ−1(U0)ψ−1(M)χM*([fj⊗gj]m)m∈M=∑j=0∞χU0*([fj⊗1]m)m∈U0⋅χU0*([1⊗gj]m)m∈U0=∑j=0∞ρV0V1,0φ1,U1,0*fj⋅ρV0V2,0φ2,U2,0*gj,

due to (4.25).

4.4. Products of 𝕑2n-morphisms

We use again abbreviations of the type n = (n₁, n₂) ∈ N = N₁ × N₂ (which we introduced in the proof of Theorem 4.8).

Proposition 4.9.

Let Ψ_i : ℳ_i → 𝒩_i, i ∈ {1, 2}, be a 𝕑2n-morphism with base map ψ_i and pullback sheaf morphism ψ1*. Due to the universality of the product of 𝕑2n-manifolds, there is a canonical 𝕑2n-morphism

Ψ=Ψ1×Ψ2:ℳ1×ℳ2→𝒩1×𝒩2.

Its base map is ψ = ψ₁ × ψ₂ and its pullback sheaf morphism ψ^* = (ψ₁ × ψ₂)^* is given, for each open subset Ω ⊂ N₁ × N₂, by

ψΩ*=(∏m∈ψ−1(Ω)(ψ1*⊗^ψ2*)ψ(m))∘(pψ(m))m∈ψ−1(Ω).

The first map in the RHS is the product of the morphisms between stalks induced by the morphism

ψ1*⊗^ψ2*:ℱ¯N→ψ*ℱ¯M

(of presheaves of locally convex topological algebras), where ℱ¯N is the presheaf defined by 𝒪N1⊗^𝒪N2, and similarly for ℱ¯M. The second map in the RHS is the tuple of morphisms

pψ(m):∏n∈Ωℱ¯N,n→ℱ¯N,ψ(m).

To understand this claim, recall that

ψΩ*:𝒪N(Ω)≃ℱ¯N+(Ω)⊂∏n∈Ωℱ¯N,n→𝒪M(ψ−1(Ω))≃ℱ¯M+(ψ−1(Ω))⊂∏m∈ψ−1(Ω)ℱ¯M,m.

Note now that

(pψ(m))m∈ψ−1(Ω):𝒪N(Ω)→∏m∈ψ−1(Ω)ℱ¯N,ψ(m)

and that

∏m∈ψ−1(Ω)(ψ1*⊗^ψ2*)ψ(m):∏m∈ψ−1(Ω)ℱ¯N,ψ(m)→∏m∈ψ−1(Ω)ℱ¯M,m,

so that the composite of these maps may coincide with ψΩ*.

Proof.

For i ∈ {1, 2}, we denote by ΠiS (resp., ΠiT) the 𝕑2n-morphism ΠiS:ℳ1×ℳ2→ℳi (resp., ΠiT:𝒩1×𝒩2→𝒩i). The composite Φi=Ψi∘ΠiS is a 𝕑2n-morphism Φ_i : ℳ₁ × ℳ₂ → 𝒩_i. In view of the universality of the product 𝒩₁ × 𝒩₂, there is a unique 𝕑2n-morphism Ψ : ℳ₁ × ℳ₂ → 𝒩₁ × 𝒩₂, such that ΠiT∘Ψ=Φi. We denote this morphism Ψ by Ψ₁ × Ψ₂ and refer to it as the product of the 𝕑2n-morphisms Ψ_i. We showed in the proof of Theorem 4.8 that the base map of Ψ is

ψ=(ψ1∘π1S,ψ2∘π2S)=ψ1×ψ2.

We investigate now the pullback morphisms ψ^*. Let Ω ⊂ N be open, so that ψΩ*:𝒪N(Ω)→𝒪M(ψ−1(Ω)), and let σ∈𝒪N(Ω)≃ℱ¯N+(Ω). Recall once again that σ = ([s]_n)_n∈Ω, where s ∈ ℱ_N(U) (U = U₁ × U₂ ∍ n) reads

s=∑j=0∞fj⊗gj (fj∈𝒪N1(U1),gj∈𝒪N2(U2)).

It follows from Equation (4.24) that

ψΩ*σ=([∑j=0∞ρVV1(π1S)W1*ψ1,U1*fj⋅ρVV2(π2S)W2*ψ2,U2*gj]m)m∈ψ−1(Ω),(4.27)

where

V=V1∩V2, Vi=(πiS)−1(Wi), and Wi=ψi−1(Ui) (obviously V=W1×W2).

We interpret ψ1,U1*fj∈𝒪M1(W1) as

([ψ1,U1*fj]m1)m1∈W1∈𝒪M1+(W1),

so that

ρVV1(π1S)W1*ψ1,U1*fj=([ψ1,U1*fj⊗1]μ)μ∈V,

due to Equations (4.20) and (4.21), as well as to the observation that m₁ ∈ W₁ is equivalent to m1=π1S(m)(m∈V1). A similar result holds for the second factor in the RHS of (4.27), so that, in view of (4.26), we obtain

ψΩ*σ=([∑j=0∞([ψ1,U1*fj⊗ψ2,U2*gj]μ)μ∈V]m)m∈ψ−1(Ω)=([([ψ1,U1*⊗^ψ2,U2*∑j=0∞fj⊗gj]μ)μ∈V]m)m∈ψ−1(Ω).(4.28)

This image is a family (indexed by m) of elements of ℱ¯M,m+≃ℱ¯M,m. The isomorphism between a stalk of a presheaf and the corresponding stalk of its sheafification is described in the proof of Lemma 6.17.2 of the Stacks Project. The description shows that

ψΩ*σ=([ψ1,U1*⊗^ψ2,U2*∑j=0∞fj⊗gj]m)m∈ψ−1(Ω)=(∏m∈ψ−1(Ω)(ψ1*⊗^ψ2*)ψ(m))((pψ(m))m∈ψ−1(Ω)σ).

A. Appendices

We prove and recall results on topological vector spaces and on topological algebras.

A.1. Topological vector subspaces

Definition A.1

A topological vector subspace S of a TVS V (TVSS for short) is a subset S ⊂ V which is a TVS for the linear operations and the topology of V.

Proposition A.2

A subset S ⊂ V of a TVS V is a TVSS of V if and only if S is a linear subspace of V and is endowed with the topology induced by the topology of V.

Proof.

The restrictions to S of the continuous addition and scalar multiplication in V are continuous in the topology induced on S by V.

Note also that, if V is a TVS, and if S ⊂ V is a TVSS, then S is a TVS and the inclusion ι : S ∍ s ↦ s ∈ V is an injective continuous linear map. Conversely, if S ⊂ V is a TVS and the inclusion ι is an injective continuous linear map, then the linear structure on S ⊂ V is the same as in V, but S can have a topology that is finer than the induced one, so that S is not necessarily a TVSS.

Proposition A.3

Let ι : V → W be an injective linear map between TVS-s. When equipped with the induced topology, the linear subspace ι(V) is a TVSS of W. The bijective linear map ĩ: V → ι(V) is a TVS-isomorphism, i.e., a linear homeomorphism, if and only if the topology of V is the initial topology of ι : V → W. In this case, the space V can be viewed as a TVSS V ≃ ι(V) of W.

Proof.

It suffices to prove the second claim. If ĩ is an isomorphism, then the topology 𝒯(V) of V is given by

𝒯(V)={ι˜−1(ι(V)∩UW)=ι−1(ι(V)∩UW)=ι−1(ι(V))∩ι−1(UW)=ι−1(UW):UW∈𝒯(W)},

hence, it is the initial topology of ι : V → W. Conversely, if 𝒯(V) is the initial topology of ι, the maps ĩ and ĩ⁻¹ are continuous. Indeed, as just explained, we have ĩ⁻¹(ι(V) ∩ U_W) = ι⁻¹(U_W) ∈ 𝒯(V), so that ĩ is continuous (but it would still be continuous for a finer topology on V). For ĩ⁻¹ we have ĩ(ι⁻¹(U_W)) = ι(V) ∩ U_W ∈ 𝒯(ι(V)).

Proposition A.4

Let ι : V → W be an injective linear map from a vector space V to a TVS W. When equipped with the initial topology of ι, the linear space V is a TVS and can be viewed as a TVSS of W.

Proof.

The linear operations on V are continuous when V has the initial topology. Indeed, denote by +_V (resp., +_W) the addition in V (resp., W), and let ι⁻¹(U_W), U_W ∈ 𝒯(W), be an arbitrary open subset in 𝒯 (V). Then,

(+V)−1(ι−1(UW))={(v,v′)∈V×V:ι(v)+Wι(v′)∈UW}={(v,v′)∈V×V:(ι(v),ι(v′))∈(+W)−1(UW)}=(ι×ι)−1((+W)−1(UW))

is open in V ×V, since ι and +_W are continuous. The case of the scalar multiplication is similar. The second claim follows now from Proposition A.3.

Remark A.5

When passing above from topological vector subspace structures on included subsets to topological vector subspace structures on injected subsets, we replaced the induced topology by the initial topology. Of course, if the injection is the inclusion, the initial topology with respect to it coincides with the induced topology.

A.2. Completions of topological vector spaces

We recall now well-known properties of the completion of a TVS [29].

For any TVS V, there is a complete TVS V^, and an injective linear map ι:V→V^, such that the linear subspace ι:V⊂V^ is dense in V^. Moreover, when endowed with the induced topology, the image ι:V⊂V^ becomes a TVSS of V^, and the map ĩ : V → ι(V) is promoted to a linear homeomorphism, or, equivalently, to a TVS-isomorphism. It follows from Proposition A.3 that the topology of V is the initial topology of ι:V→V^, and that V ≃ ι(V) is a TVSS of V^. In short, the complete TVS V^, which we refer to as the completion of the TVS V, contains V as a dense TVSS:

Proposition A.6

The completion V^ of V contains V as a dense subset, and it induces on V the original topology and original linear structure.

Remark A.7

If V is already a complete TVS, then ι:V→V^ is a TVS-isomorphism.

Further, let S ⊂ V be a TVSS of V and let ι : S → V be the injective continuous linear inclusion. The continuous extension of ι, which we denote by ι^:S^→V^, is an injective continuous linear map. If î(Ŝ) carries the induced topology, the map ι^˜:S^→ι^(S^) is a TVS-isomorphism. In view of Proposition A.3 this means that the topology of Ŝ is the initial topology of ι^:S^→V^, and that Ŝ ≃ î(Ŝ) is a TVSS of V^. In short:

Proposition A.8

The completion of a TVSS S ⊂ V is a TVSS S^⊂V^ of the completion.

A.3. Locally convex spaces

The initial topology of a linear map ℓ : V → W from a vector space V to a LCTVS W endows V with a LCTVS structure. In particular, the induced topology on a vector subspace S ⊂ W of a LCTVS endows S with a LCTVS structure.

Proof.

The topology of W has a convex basis 𝔅. As the preimage of a convex set by a linear map is convex, the family ℓ⁻¹(𝔅) is a convex basis of the initial topology. In view of the proof of Proposition A.4, the initial topology endows V with a LCTVS structure. The second claim is a special case of the first one.

We close this subsection recalling two results.

Theorem A.10

Let V and W be two LCTVS-s and let (p_i)_i∈I and (q_j)_j∈J be two families of semi-norms that induce the topologies of V and W, respectively. The projective tensor topology π on V ⊗ W is induced by (p_i ⊗ q_j)_ij. Moreover, for any t ∈ V ⊗ W, we have

(pi⊗qj)(t)=inf{∑k=1Npi(vk)qj(wk):t=∑k=1Nvk⊗wk,vk∈V,wk∈W,N∈ℕ},

and, for any v ∈ V and w ∈ W, we have

(pi⊗qj)(v⊗w)=pi(v)qj(w).

Proposition A.11

Let V and W be two LCTVS-s and let (p_i)_i∈I and (q_j)_j∈J be two families of semi-norms that induce the topologies of V and W, respectively. A linear map ℓ : V → W is continuous if and only if, for any j ∈ J, there exist i₁,. . .,i_N ∈ I and C > 0, such that, for all v ∈ V, one has

qj(ℓ(v))≤Cmaxkpik(v).

A.4. Presheaves of topological algebras and sheafification

In the following, we need two lemmas.

Unless otherwise stated, all cartesian products Π_αT_α of topological spaces T_α are endowed with the product topology, i.e., the weakest topology for which all projections pβ : Π_αT_α →T_β are continuous.

Lemma A.12

Let (Tαi)α (i ∈ {1,2,3}) be families of topological spaces, and let (m_α)_α be a family mα:Tα1×Tα2→Tα3 of continuous maps. Then the map

m:∏αTα1×∏αTα2∋((τα)α,(𝔱α)α)↦(mα(τα,𝔱α))α∈∏αTα3

is continuous.

Proof.

Since the cartesian product Π_αT_α of topological spaces T_α equipped with the product topology, is the product in the category of topological spaces, it follows from the universal property that a map f : → Π_αT_α from a topological space to a product space is continuous if and only if all the p_β ○ f : T → T_β are continuous. However, the map pβ3∘m:((τα)α,(t)α)↦mβ(τβ,tβ) is the composite mβ∘(pβ1×pβ2), which is of course continuous.

Lemma A.13.

Let V be a vector space, let (V_α)_α be a family of LCTVS, let ℓ_α : V_α → V be a family of linear maps, and let V be equipped with the finest locally convex vector space topology, for which all the ℓ_α are continuous (V is then a LCTVS). If W is a LCTVS, a linear map ℓ : V → W is continuous if and only if all the maps ℓ ○ ℓ_α : V_α → W are continuous.

Proof.

See [4, Proposition 2.3.5]

In the present text:

Definition A.14.

A topological algebra (TA for short) (resp., a locally convex topological algebra (LCTA for short)) is a (real) topological vector space (resp., a locally convex (real) topological vector space), with an associative, bilinear, and (jointly) continuous multiplication. A morphism of topological algebras (resp., a morphism of locally convex topological algebras) is a continuous algebra morphism. We denote the category of topological algebras (resp., of locally convex topological algebras) and morphisms between them by TAlg (resp., by LCTAlg).

The category of Fréchet algebras is a full subcategory of the category of locally convex topological algebras, itself a full subcategory of the category of topological algebras.

We denote by PSh(T, TAlg) (resp., Sh(T, TAlg)) the category of presheaves (resp., sheaves) of TA-s over a topological space T. Similarly, the category PSh(T, LCTAlg) (resp., Sh(T, LCTAlg)) is the category of presheaves (resp., sheaves) of LCTA-s over T, and the category PSh(T, Alg) (resp., Sh(T, Alg)) is the category of presheaves (resp., sheaves) of algebras over T.

Proposition A.15.

Denote by + the sheafification functor

+:PSh(T,Alg)→Sh(T,Alg):For,

i.e., the left adjoint of the forgetful functor For. If ℱ ∈ PSh(T, TAlg), we have ℱ⁺ ∈ Sh(T, TAlg), and the morphism i : ℱ → ℱ⁺ of presheaves of algebras is a morphism of presheaves of TA-s. Moreover, if φ : ℱ → ℱ′ is a morphism in PSh(T, TAlg), then φ⁺ : ℱ⁺ → ℱ′⁺ is a morphism in Sh(T, TAlg). The same results hold, if we replace TA-s by LCTA-s.

Proof.

We study the case ℱ ∈ PSh(T, LCTAlg) (in particular ℱ ∈ PSh(T, Alg)).

Since sheafification is based on stalks ℱ_x (x ∈ T), i.e., on inductive limits, and since the inductive limit of a directed system of sets endowed with a same algebraic structure has as underlying set the inductive limit of the directed system of underlying sets, it is natural that the same result holds for sheafification functors. Let + be the sheafification functor

+:PSh(T,Alg)∋ℱ↦ℱ+∈Sh(T,Alg):For.

Recall that ℱ⁺(U) (U ∈ Open(T)) is defined as the subset of (Πℱ)(U):=∏x∈Uℱx, which is made of those elements σ = (σ_x)_x_∈_U = ([s]_x)_x_∈_U, for whom the section s is constant in a neighborhood of any point of U. The algebra operations on ℱ⁺(U) are naturally induced by those of the stalks. Restrictions are the obvious algebra morphisms. The morphism i : ℱ → ℱ⁺ of presheaves of algebras is defined by

iU:ℱ(U)∋s↦([s]x)x∈U∈ℱ+(U),(A.1)

and i_U is injective, if ℱ is separated. If 𝒢 ∈ Sh(T, Alg), the morphism i : 𝒢 → 𝒢⁺ is an isomorphism of sheaves of algebras. Further, if j : ℱ → 𝒢 is a morphism of presheaves of algebras, the unique morphism 𝔧 : ℱ⁺ → 𝒢 of sheaves of algebras, such that 𝔧 ○ i = j, is given by 𝔦_U = Π_x∈U j_x 𝔧_U = ∏_x∈U j_x, where j_x : ℱ_x → 𝒢_x is the morphism of algebras induced by j :

𝔦U:ℱ+(U)∋([s]x)x∈U↦(jx[s]x)x∈U∈𝒢+(U)≃𝒢(U).(A.2)

Similarly, if ℱ, ℱ′ ∈ PSh(T, Alg) and φ : ℱ → ℱ′ is a presheaf morphism, the components of the sheaf morphism φ⁺ : ℱ⁺ → ℱ′⁺ are

ϕU+:ℱ+(U)∋([s]x)x∈U↦(ϕx[s]x)x∈U∈ℱ′+(U).

The stalk ℱ_x (x ∈ T) is the inductive limit algebra ℱx=lim→U∋xℱ(U) of the directed system of algebras (ℱ(U),ρVU), where U is an open neighborhood of x and ρVU the restriction from U to V ⊂ U. This system is actually a directed system of LCTA-s, in the sense that the ℱ(U) are LCTA-s and the ρVU are continuous algebra morphisms. If we endow the inductive limit algebra ℱ_x with the final locally convex vector space topology with respect to the canonical algebra morphisms πUx:ℱ(U)→ℱx (i.e., with the finest locally convex vector space topology, for which the πUx are all continuous), the limit ℱ_x is a LCTVS, whose multiplication is (jointly) continuous [26, Lemma 2.2], i.e., the stalk ℱ_x is a LCTA.

In the following, the algebra ℱ⁺(U) ⊂ (Πℱ)(U) carries the induced topology of the product topology. Since any product of LCTVS-s and any subspace of a LCTVS are LCTVS-s, the algebra ℱ⁺(U) is a LCTVS. The multiplication on ℱ⁺(U) is continuous in view of Lemma A.12, so that ℱ⁺(U) is a LCTA.

To show that ℱ⁺ ∈ Sh(T, LCTAlg), it suffices to prove that any restriction

rVU:(Πℱ)(U)∋(σx)x∈U↦(σx)x∈V∈(Πℱ)(V)

is continuous, i.e., that, for all y ∈ V, the map

pyV∘rVU=pyU:(Πℱ)(U)∋(σx)x∈U↦σy∈ℱy

is continuous – which is a consequence of the definition of the product topology.

The next step consists in proving that the morphism i : ℱ → ℱ⁺ of presheaves of algebras defined by (A.1), is a morphism of presheaves of LCTA-s, i.e., in proving that, for any y ∈ U, the map

pyU∘iU=πUy:ℱ(U)∋s↦[s]y∈ℱy

is continuous. This holds true by definition of the final topology on ℱ_y.

Let φ : ℱ → ℱ′ be a morphism in PSh(T, LCTAlg). To show that φ⁺ : ℱ⁺ → ℱ′⁺ is a morphism in Sh(T, LCTAlg), it suffices to show that

∏x∈Uϕx:(Πℱ)(U)∋([s]x)x∈U↦(ϕx[s]x)x∈U∈(Πℱ′)(U)(A.3)

is continuous. This is the case if and only if φ_x : ℱ_x → ℱ′_x, x ∈ U, is continuous. In view of Lemma A.13, the algebra morphism φ_x is continuous if and only if

ϕx∘πVx=πV′x∘ϕV:ℱ(V)∋s↦ϕx[s]x=[ϕVs]x∈ℱx′,

V ∍ x, is continuous. This condition is obviously fulfilled.

Corollary A.16.

When equipped with the final locally convex vector space topology with respect to the canonical algebra morphisms πUx:ℱ(U)→ℱx, a stalk ℱ_x, x ∈ T, of a presheaf ℱ ∈ PSh (T, TAlg) is the inductive limit in TAlg of the directed system (ℱ(U),ρVU). The same statement holds in LCTAlg.

Proof.

Clearly, the πUx:ℱ(U)→ℱx are morphisms in (LC)TAlg. Let (F, p_U) be made of F ∈ (LC)TAlg and (LC)TAlg-morphisms p_U : ℱ(U) → F, such that pV∘ρVU=pU. Due to Lemma A.13, the unique Alg-morphism u : ℱ_x → F, such that u∘πUx=pU, is continuous, since the u∘πUx=pU are all continuous. Hence, the claim.

Proposition A.17.

Let α : A → C and β : B → D be two continuous algebra morphisms between nuclear Fréchet algebras. Then α⊗^β:A⊗^B→C⊗^D is a continuous algebra morphism between nuclear Fréchet algebras.

Proof.

Since α and β are continuous linear maps between locally convex spaces, the map α ⊗ β : A ⊗ B → C ⊗ D is continuous linear, and its continuous extension α⊗^β:A⊗^B→C⊗^D is continuous linear as well.

The source and the target of α⊗^β are nuclear Fréchet algebras [15, Lemma 1.2.13]. It remains to show that α⊗^β respects their (continuous) multiplications. Let 𝔞=∑i=0∞ai⊗bi and 𝔟=∑j=0∞aj′⊗bj′ be two elements of A⊗^B. Using continuity, we get

(α⊗^β)(𝔞⋅𝔟)=limnlimm∑i=0n∑j=0m(α(ai)α(aj′))⊗(β(bi)β(bj′)).

It is straightforwardly seen that (α⊗^β)𝔞⋅(α⊗^β)𝔟 is given by the same limit of sums of tensor products.

Proposition A.18.

The multiplication m : A × A → A of a nuclear Fréchet algebra A extends to a continuous algebra morphism m^:A⊗^A→A between nuclear Fréchet algebras.

Proof.

We equip A ⊗ A with the projective tensor topology. The continuous bilinear maps from A× A to A correspond exactly to the continuous linear maps from A ⊗ A to A. Hence, the multiplication m can be viewed as a continuous linear map m : A ⊗ A → A. The latter extends to a continuous linear map m^:A⊗^A→A. This extension respects the (continuous) multiplications. The proof is similar to the one of Proposition A.17.

A.5. Direct image and sheafification

Proposition A.19.

Let ℱ ∈ PSh(T, LCTAlg) and let f ∈ C⁰(T, T′). There is a morphism

ι:(f*ℱ)+→f*ℱ+

in Sh(T′, LCTAlg).

Proof.

The assignment

f*ℱ:Open(T′)∋V↦ℱ(f−1(V))∈LCTAlg

together with the restrictions

ρf−1(V′)f−1(V):(f*ℱ)(V)→(f*ℱ)(V′),

where ρ are the restrictions of ℱ in LCTAlg, is a presheaf f_*ℱ ∈ PSh(T′, LCTAlg). In view of Proposition A.15, we get (f_*ℱ)⁺ ∈ Sh(T′, LCTAlg). Similarly, we have f_*ℱ⁺ ∈ Sh(T′, LCTAlg).

Further, for x ∈ T and U ⊃ U′ ∍ x, the LCTAlg-morphisms πUx:ℱ(U)→ℱx and πU′x:ℱ(U′)→ℱx satisfy πU′x∘ρU′U=πUx. As, due to Corollary A.16, the stalk (f_*ℱ)_f(x) ∈ LCTAlg is the inductive limit in LCTAlg of the directed system

(ℱ(f−1(V)),ρf−1(V′)f−1(V)) (V⊃V′∋f(x)),

there exists a unique LCTAlg-morphism

ux:(f*ℱ)f(x)→ℱx,(A.4)

such that ux∘πVf(x)=πf−1(V)x, i.e., such that u_x [s]_f(x) = [s]_x, for all s ∈ ℱ(f⁻¹(V)) and all V ∍ f(x).

For any V ∈ Open(T′), the map

ιV:(Π(f*ℱ))(V)∋([s]y)y∈V↦(ux[s]f(x))x∈f−1(V)=([s]x)x∈f−1(V)∈(Πℱ)(f−1(V))(A.5)

is a continuous algebra morphism. Indeed, the algebraic operations are defined component-wise. For instance, the image of a product ([s]_y · [s′]_y)_y∈V is the product (u_x[s]_f _(x) · u_x [s′]_f_(x))_{x∈f⁻¹(V)} of the images. Moreover, the map ι_V is continuous if and only if the maps

pxf−1(V)∘ιV:(Π(f*ℱ))(V)∋([s]y)y∈V↦ux[s]f(x)∈ℱx (x∈f−1(V))

are. The preimage by pxf−1(V)∘ιV of any open ω ⊂ ℱ_x is the product over y ∈ V, whose factors indexed by y ≠ f(x) are (f_*ℱ)_y and whose factor indexed by f(x) is the open subset ux−1(ω) of (f_*ℱ)_f(x). The preimage by pxf−1(V)∘ιV is thus open in (Π(f_*ℱ))(V).

The restriction of ι_V to (f_*ℱ)⁺(V), still denoted by ι_V, arrives in (f_*ℱ⁺)(V). Assume that ([s]_y)_y∈V is implemented in a neighborhood V_y₀ of an arbitrary point y₀ ∈ V by a same section t ∈ (f_*ℱ)(V_y₀), let x₀ ∈ f⁻¹(V), and set U_x₀ = f⁻¹(V_f(x₀)). For any x ∈ U_x₀, we have

ux[s]f(x)=ux[t]f(x)=[t]x,

with t ∈ ℱ (U_x₀).

The restriction ι_V : (f_*ℱ)⁺(V) → (f_*ℱ⁺)(V) is obviously a morphism in LCTAlg. Since these ι_V commute with restrictions, they define the sheaf morphism ι announced in Proposition A.19.

References

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 3
Pages: 420 - 453
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2019.1613051 How to use a DOI?
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ris enw bib

TY  - JOUR
AU  - Andrew Bruce
AU  - Norbert Poncin
PY  - 2021
DA  - 2021/01/06
TI  - Products in the category of 𝕑2n-manifolds
JO  - Journal of Nonlinear Mathematical Physics
SP  - 420
EP  - 453
VL  - 26
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1613051
DO  - 10.1080/14029251.2019.1613051
ID  - Bruce2021
ER  -

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