Ind-Sheaves, distributions, and microlocalization

Masaki Kashiwara & Pierre Schapira

04/1999^{∗}

1 Introduction

If C is an abelian category, the category Ind(C) of ind-objects of C has many remarkable properties: it is much bigger than C, it contains C, and furthermore it is dual (in a certain sense) to C.

We introduce here the category of ind-sheaves on a locally compact space X as the category of ind-objects of the category of sheaves with compact supports. This construction has some analogy with that of distributions: the space of distributions is bigger than that of functions, and is dual to that of functions with compact support. This last condition implies the local nature of distributions, and similarly, one proves that the category of ind-sheaves defines a stack.

In our opinion, what makes the theory of ind-sheaves on manifolds really interesting is twofold.

(i) Ind-sheaves allows us to treat in the formalism of sheaves (the “six operations”) functions with growth conditions. For example, on a complex manifold X, one can define the ind-sheaf of “tempered holomorphic functions” O_{X}^{t} or the indsheaf of “Whitney holomorphic functions” O_{X}^{w}, and obtain for example the sheaf of Schwartz’s distributions using Sato’s construction of hyperfunctions, simply replacing O_{X} with O_{X}^{t}.

(ii) On a real manifold, one can construct a microlocalization functor μ_{X} which sends sheaves (i.e. objects of the derived category of sheaves) on X to ind-sheaves on T^{∗}X, and the Sato functor of microlocalization along a submanifold M ⊂ X (see [9]) becomes the usual functor Hom (π^{−1}(k_{M}), ·) (where π : T^{∗}X ⟶ X is the projection) composed with μ_{X}(·).

When combining (i) and (ii), one can treat in a unified way various objects of classical analysis.

The results presented here are extracted from [7]. We refer to [5] for an exposition on derived categories and sheaves, and to [3] for the theory of ind-objects.

∗Talk given at Ecole Polytechnique, May 1999

2 Ind-objects

Let U be a universe (see [8]). If C is a U-category, one denotes by C^{∨} the category of contravariant functors from C to Set, the category of (U-small) sets. One sends C into C^{∨} by the fully faithful functor h^{∨} : X ↦ Hom_{C}(·, X).

Let I be a small filtrant category and let i ↦ X_{i} be an inductive system in C indexed by I. One denotes by “\underrightarrow{lim}_{i}”X_{i} the object of C^{∨} defined by

C ∋ Y ↦ \underrightarrow{lim}_{i∈I} Hom_{C}(Y, X_{i}).

An ind-object in C is an object of C^{∨} which is isomorphic to such a “\underrightarrow{lim}_{i}”X_{i}. One

denotes by Ind(C) the full subcategory of C^{∨} consisting of ind-objects and identifies C with a full subcategory of Ind(C) by the functor h^{∨}.

From now on we shall assume that C is abelian. The subcategory C^{∨,add} of C^{∨} consisting of additive functors is clearly abelian and the functor h^{∨} : C ⟶ C^{∨,add} is left exact.

Since the functor Hom is left exact and filtrant inductive limits are exact, any ind-object defines a left exact functor on C. The converse holds if C is small.

Example 2.1. Let k be a field. Define the inductive system in Mod(k) indexed by ℕ by setting X_{n} = k^{n}, with the natural injections X_{n} ↪ X_{n+1}. Then “\underrightarrow{lim}_{n}”X_{n} is

the functor Y ↦ \underrightarrow{lim}_{n} Hom_{k}(Y, X_{n}). This object does not belong to Mod(k) since

there are no vector space Z such that \underrightarrow{lim}_{n} Hom_{k}(Y, X_{n}) ≃ Hom_{k}(Y, Z) for all vector

spaces Y.

If f : X ⟶ Y is a morphism in Ind(C), one can construct a small filtrant category I such that X = “\underrightarrow{lim}_{i}”X_{i}, Y = “\underrightarrow{lim}_{i}”Y_{i} and f = “\underrightarrow{lim}_{i}”f_{i}, with f_{i} : X_{i} ⟶ Y_{i}. Then

“\underrightarrow{lim}_{i}” ker f_{i} and “\underrightarrow{lim}_{i}” coker f_{i} are the kernel and the cokernel of f in Ind(C).

Theorem 2.2. (i) The category Ind(C) is abelian.

(ii) The natural functor C ⟶ Ind(C) is fully faithful and exact, and C is thick in Ind(C).

(iii) The natural functor Ind(C) ⟶ C^{∨,add} is fully faithful and left exact.

(iv) The category Ind(C) admits right exact small inductive limits. Moreover, inductive limits over small filtrant categories are exact.

(v) If small products of objects of C exist in Ind(C), then Ind(C) admits small projective limits and such limits are left exact.

We shall denote by “\underrightarrow{lim}” the usual inductive limit in Ind(C). With this conven-

tion, one can identify C with its image in Ind(C) by h^{∨} without risk of confusion.

Consider an additive functor F : C ⟶ C′. It defines an additive functor IF : Ind(C) ⟶ Ind(C′). If there is no risk of confusion, we shall write F instead of IF. If F is right exact, or left exact, or fully faithhful, so is IF.

We shall have to consider the derived category of Ind(C). One proves that the natural functor D^{b}(C) ⟶ D_{C}^{b}(Ind(C)) is an equivalence.

Even if C has enough injectives, we cannot prove that Ind(C) has the same property. However, quasi-injective objects, i.e. ind-objects which are exact functors on C, are sufficient for many applications.

3 Ind-sheaves

Let X be a locally compact topological space which is countable at infinity. Let k be a field and let A be a sheaf of commutative k-unitary algebras on X.

One denotes by Mod(A) the abelian category of sheaves of A-modules, and by Mod^{c}(A) the full subcategory consisting of sheaves with compact support.

We call an object of Ind(Mod^{c}(A)) an ind-sheaf on X and we set for short:

I(A) := Ind(Mod^{c}(A)).

If U is an open subset of X, one defines the restriction of F to U as follows. If F = “\underrightarrow{lim}_{i}”F_{i}, one sets

F |_{U} = “ \underrightarrow{lim}_{i,V ⊂⊂U}”(F_{iV} |_{U}).

Then one proves that for F and G in I(A), the presheaf U ↦ Hom_{I(A|_{U})}(F |_{U}, G|_{U}) is a sheaf. We shall denote it by Hom (F, G).

In fact, there is a better result: U ↦ I(A|_{U}) is a stack. Roughly speaking, this means a sheaf of categories.

Note that Ind(Mod(A)) is not a stack.

Example 3.1. Let X = ℝ, and let A = k_{X}. Let F = k_{X}, G_{n} = k_{[n,+∞[}, G = “\underrightarrow{lim}_{n}”G_{n}. Then G|_{U} = 0 for any relatively compact open subset U of X. On the

other hand, Hom_{Ind(Mod(k_{X}))}(k_{X}, G) ≃ \underrightarrow{lim}_{n} Hom_{k_{X}}(k_{X}, G_{n}) ≃ k.

We construct the functors

ι_{X} : Mod(A) ⟶ I(A), ι_{X}F = “ \underrightarrow{lim}_{U⊂⊂X}”F_{U},
α_{X} : I(A) ⟶ Mod(A), α_{X}(“\underrightarrow{lim}_{i}”F_{i}) = \underrightarrow{lim}_{i} F_{i}.

The functor α_{X} admits a left adjoint β_{X} : Mod(A) ⟶ I(A).

These functors satisfy:

(i) ι_{X} and α_{X} are exact, fully faithful, and commute with \underrightarrow{lim} and \underleftarrow{lim},

(ii) β_{X} is right exact, fully faithful and commutes with \underrightarrow{lim}, and if A is left coherent,

β_{X} is exact.

(iii) α_{X} is left adjoint to ι_{X} and is right adjoint to β_{X},

(iv) α_{X} ∘ ι_{X} ≃ id and α_{X} ∘ β_{X} ≃ id.

If Z is locally closed in X and F is a sheaf on X, recall that the sheaf F_{Z} is 0 on X \ Z and is isomorphic to F |_{Z} on Z. We set

Ã_{Z} = β_{X}(A_{Z}).

Since β_{X} is right exact and commutes with \underrightarrow{lim}, it is characterized by its values on

the sheaves A_{U}, U open. If U is open and S is closed, one has

Ã_{U} ≃ “ \underrightarrow{lim}_{V ⊂⊂U}”A_{V} , V open, and  Ã_{S} ≃ “\underrightarrow{lim}_{V ⊃S}”AV̄ , V open.

Note that Ã_{U} ⟶ A_{U} is a monomorphism and Ã_{S} ⟶ A_{S} is an epimorphism.

Example 3.2. Let X be a real manifold of dimension n ≥ 1 and let A = k_{X}. For short, we shall write k_{Z} instead of (k_{X})_{Z}. Let a ∈ X. Define N_{a} ∈ I(k_{X}) by the exact sequence

0 ⟶ N_{a} ⟶ k̃_{{a}} ⟶ k_{{a}} ⟶ 0.  (3.1)

The derived functor RHom_{I(k_{X})}(·, ·) is well-defined and moreover if G and F_{i} are sheaves on X,

H^{k}RHom_{I(k_{X})}(G, “\underrightarrow{lim}_{i}”F_{i}) ≃ \underrightarrow{lim}_{i} H^{k}RHom_{k_{X}}(G, F_{i}).  (3.2)

Since \underrightarrow{lim}_{a∈V} H_{\{a\}}^{n}(X; k\overline{V}) ≠ 0, we find that the morphism k̃_{\{a\}} ⟶ k_{\{a\}} is not an isomor-

phism, hence N_{a} ≠ 0.

On the other hand, for any open neighborhood U of a, we have

RHom_{I(k_{X})}(k_{U}, N_{a}) ≃ 0.

4 Operations on ind-sheaves

We define the functors internal tensor product, denoted ⊗, and internal Hom, denoted Ihom

⊗: I(A) × I(A) ⟶ I(A),
Ihom : I(A)^{op} × I(A) ⟶ I(A),

by the formulas:

“\underrightarrow{lim}_{i}”F_{i} ⊗ “\underrightarrow{lim}_{j}”G_{j} = “\underrightarrow{lim}_{i,j}”(F_{i} ⊗ G_{j}),
Ihom (“\underrightarrow{lim}_{i}”F_{i}, “\underrightarrow{lim}_{j}”G_{j}) = \underleftarrow{lim}_{i} “\underrightarrow{lim}_{j}”Hom (F_{i}, G_{j}).

One has:

α_{X}Ihom (F, G) ≃ Hom (F, G).

The functor ⊗ is right exact and commutes with “\underrightarrow{lim}” and the functor Ihom is left

exact. The functors ⊗ and Ihom are adjoint:

Hom_{I(A)}(F ⊗_{A} K, G) ≃ Hom_{I(A)}(G, Ihom_{A}(K, G)),

Now consider a continuous map f : X ⟶ Y of locally compact spaces, and assume A = f^{−1}B, for a sheaf of rings B on Y.

If G = “\underrightarrow{lim}_{i}”G_{i} is an ind-sheaf on Y, we define

f^{−1}G = “\underrightarrow{lim}_{i}”(f^{−1}G_{i})_{U}, U ⊂⊂ X, U open.

The functor f^{−1} : I(B) ⟶ I(A) is exact and commutes with “\underrightarrow{lim}” and ⊗. Moreover,

it commutes with the functors ι, α and β.

If F = “\underrightarrow{lim}_{i}”F_{i} is an ind-sheaf on X, we define:

f_{∗}F = \underleftarrow{lim}_{K} “\underrightarrow{lim}_{i}”f_{∗}F_{iK}, K compact.

The functor f_{∗} : I(A) ⟶ I(B) is left exact and commutes with \underleftarrow{lim}. Moreover, it

commutes with the functors ι and α.

The two functors f^{−1} and f_{∗} are adjoint. More precisely, for F ∈ I(A) and G ∈ I(B), we have

Hom_{I(A)}(f^{−1}G, F) ≃ Hom_{I(B)}(G, f_{∗}F).

If F = “\underrightarrow{lim}_{i}”F_{i} is an ind-sheaf on X, we define f_{!!}F by the formula

f_{!!}“\underrightarrow{lim}_{i}”F_{i} = “\underrightarrow{lim}_{i}”f_{!}F_{i}.

Note that the natural morphism f_{!!}ι_{X}F ⟶ ι_{Y} f_{!}F is not an isomorphism in general. The functor f_{!!} is left exact and commutes with “\underrightarrow{lim}”. Moreover, it commutes with

the functor α.

There is a base change formula as well as a projection formula for ind-sheaves.

Example 4.1. For F ∈ I(k_{X}) and x ∈ X set F_{x} = j_{x}^{−1}F, where j_{x} : {x} ↪ X. Let N_{a} be as in Example 3.2. Then (N_{a})_{x} ≃ 0 for all x ∈ X. On the other hand, one shows that the functor F ↦ ∏_{x∈X}(F ⊗ k̃_{{x}}) is faithful.

5 Construction of ind-sheaves

We assume that X is a real analytic manifold and denote by T the family of open relatively compact subanalytic subsets of X. We denote by ℝ-C(k_{X}) the (small) category of ℝ-constructible sheaves of k-vector spaces on X and by ℝ-C^{c}(k_{X}) the full subcategory consisting of ℝ-constructible sheaves with compact support. Note that for any F ∈ ℝ-C^{c}(k_{X}) there is an exact sequence F^{1} ⟶ F^{0} ⟶ F ⟶ 0, with F^{0} and F^{1} finite direct sums of sheaves k_{U} with U ∈ T .

We set

I_{ℝ-c}(k_{X}) = Ind(ℝ-C^{c}(k_{X})).

The natural functor ι : ℝ-C^{c}(k_{X}) ⟶ Mod^{c}(k_{X}) defines the fully faithful exact functor

ι : I_{ℝ-c}(k_{X}) ⟶ I(k_{X}).  (5.1)

On the other hand, the forgetful functor Mod^{c}(k_{X})^{∨,add} ⟶ ℝ-C^{c}(k_{X})^{∨,add} induces a functor

ρ : I(k_{X}) ⟶ I_{ℝ-c}(k_{X}).  (5.2)

The next theorem formulates a previous result of [6] in the language of ind-sheaves.

Theorem 5.1. Let F be a presheaf of k-vector spaces on T . Assume:

(i) F (∅) = 0,

(ii) for any U and V in T , the sequence 0 ⟶ F (U ∪ V) ⟶ F (U) ⊕ F (V) ⟶ F (U ∩ V) is exact.

Then there exists F_{T}^{+} ∈ I_{ℝ-c}(k_{X}) such that for any U ∈ T ,

Hom_{I_{ℝ-c}(k_{X})}(k_{U}, F_{T}^{+}) ≃ F (U).  (5.3)

We denote by F^{+} the image of F_{T}^{+} in I(k_{X}) by the functor ι of (5.1). If D is a sheaf of (not necessarily commutative) k-algebras and F a presheaf of D-modules, then F^{+} belongs to Mod(D, I(k_{X})), the subcategory of I(k_{X}) of ind-sheaves endowed with a D-action. Recall that F ∈ Mod(D, I(k_{X})) means that F is endowed with a morphism of sheaves of unitary rings D ⟶ End_{I(k_{X})}(F).

6 Some classical ind-sheaves

In this section, the base field k is ℂ. We denote by C_{X}^{∞} the sheaf of complex valued functions of class C^{∞}, and by D_{X} the sheaf of real analytic finite-order differential operators.

(a) Let f ∈ C^{∞}(U). One says that f has polynomial growth at p ∈ X if for a local coordinate system (x_{1}, ⋯, x_{n}) around p, there exist a sufficiently small compact neighborhood K of p and a positive integer N such that

sup_{x∈K∩U} (dist(x, X \ U))^{N}|f(x)| < ∞ .  (6.1)

We say that f is tempered at p if all of its derivatives are of polynomial growth at p. We say that f is tempered if it is tempered at any point of X. One denotes by C^{∞,t}(U) the ℂ-vector subspace of C^{∞}(U) consisting of tempered functions. By a theorem of Lojaciewicz, the contravariant functor C^{∞,t}(·) defined on the category T of open relatively compact subanalytic sets is exact. Hence, we may apply Theorem 5.1, and we get an ind-sheaf:

C_{X}^{∞,t} ∈ Mod(D_{X}, I(ℂ_{X})).

(b) If S is closed in X, one denotes by I_{S}^{∞}(X) the ideal of C^{∞}(X) consisting of functions which vanish on S with infinite order, and for an open subset U in X, we set Γ(X; ℂ_{U} ⊗^{w} C_{X}^{∞}) = I_{X\U}^{∞}(X). Again by a theorem of Lojaciewicz, the (covariant)

functor Γ(X; · ⊗^{w} C_{X}^{∞}) defined on the category T is exact. It extends to an exact functor Γ(X; · ⊗^{w} C_{X}^{∞}) on the category ℝ-C^{c}(ℂ_{X}) and we may define

C_{X}^{∞,w} ∈ Mod(D_{X}, I(ℝ-C(ℂ_{X})))

by the formula

C_{X}^{∞,w}(F) = Γ(X; H^{0}(D′F) ⊗^{w} C_{X}^{∞})

where F ∈ ℝ-C^{c}(ℂ_{X}) and D′F = RHom_{ℂ_{X}}(F, ℂ_{X}).

(c) Similarly, replacing ⊗^{w} by ⊗ in the above formula, we find an ind-sheaf C_{X}^{∞,ω} which, in fact, is isomorphic to β_{X}C_{X}^{∞}.

(d) Now assume that X is a complex manifold with structure sheaf O_{X}, denote by \overline{X} the complex conjugate manifold and by X^{ℝ} the underlying real manifold, identified with the diagonal of X × \overline{X}. We define the objects O_{X}^{t}, O_{X}^{w} and by O_{X}^{ω} considering the Dolbeault complexes with coefficients in C_{X}^{∞,t}, C_{X}^{∞,w} and C_{X}^{∞,ω}:

O_{X}^{λ} = RHom_{D\overline{X}}(O\overline{X}, C_{X}^{∞,λ}), λ = t, w, ω.

Then, O_{X}^{t}, O_{X}^{w} and O_{X}^{ω} belong to D^{b}(Mod(D_{X}, I(ℂ_{X}))). Note that if F is ℝconstructible,

RHom (F, O_{X}^{t}) ≃ T Hom (F, O_{X})
RHom (F, O_{X}^{w}) ≃ D′F ⊗^{w} O_{X}
RHom (F, O_{X}^{ω}) ≃ D′F ⊗ O_{X},

where T Hom (·, O_{X}) and · ⊗^{w} O_{X} are the functors of tempered and formal cohomology of [4] and [6], respectively.

In particular, let M be a real analytic manifold and assume that X is a complexification of M. We find:

RHom (D′ℂ_{M}, O_{X}) ≃ B_{M} (Sato’s hyperfunctions),
RHom (D′ℂ_{M}, O_{X}^{t}) ≃ Db_{M} (Schwartz’s distributions),
RHom (D′ℂ_{M}, O_{X}^{w}) ≃ C_{M}^{∞} (C^{∞}-functions),
RHom (D′ℂ_{M}, O_{X}^{ω}) ≃ A_{M} (real analytic functions).

Replacing Hom by Ihom we find new ind-sheaves. For example, we can define the ind-sheaf Db_{M}^{t} of tempered distributions by setting

Db_{M}^{t} = RIhom (D′ℂ_{M}, O_{X}^{t}).

Note that α_{X}(Db_{M}^{t}) ≃ Db_{M}.

Remark 6.1. The object O_{X}^{t} is not concentrated in degree 0 if dim X > 1. In fact if C is an abelian category, a complex A′ ⟶_{f} A ⟶_{g} A′′ in Ind(C) is exact if and only if

the dotted arrows in the diagram below with X ∈ C may be completed with Y ∈ C in such a way that the morphism α : Y ⟶ X is an epimorphism:

[figure: A′; f; A; g; A′′; Y; t; α; X; s; 0]

Let us apply this result to the complex

C_{X}^{∞,t,(p−1)} ⟶_{\overline{∂}} C_{X}^{∞,t,(p)} ⟶_{\overline{∂}} C_{X}^{∞,t,(p+1)}

and choose F = ℂ_{U}, U open in X. Assume O_{X}^{t} is concentrated in degree 0. Then, for any s ∈ C^{∞,t,(p)}(U) solution of \overline{∂}s = 0, there exists an epimorphism α : G↠F and t ∈ Hom (G, C^{∞,t,(p−1)}) such that s ∘ α = \overline{∂}t. We may assume G is a finite direct sum of sheaves ℂ_{U_{j}}, with U = ∪_{j}U_{j}. We thus find that s|_{U_{j}} = \overline{∂}t_{j}, which is in general not possible.

Note that the same argument hold with the sheaf O_{X}, which shows that the functor ρ in (5.2) is not exact.

7 Microlocalization

In this section, X denotes a real analytic manifold. We denote by p_{1} and p_{2} the first and second projection defined on X × X and by Δ the diagonal. We denote by τ : T X ⟶ X and π : T^{∗}X ⟶ X its tangent and cotangent bundles, respectively. We denote by T_{Y} X and T_{Y}^{∗}X the normal and conormal bundle to a closed submanifold Y of X. In particular, T_{X}X and T_{X}^{∗}X are the zero-sections of these bundles. We shall identify T_{Δ}(X × X) with T X by the first projection on T X × T X. If S is a subset of X, one denotes by C_{Y} (S) the Whitney normal cone of S along Y, a closed cone in T_{Y} X.

We denote by ω_{X} the dualizing complex on X. (Hence, ω_{X} ≃ or_{X}[dim X].) The micro-support SS(F) of a sheaf F and the functor μhom are defined in [5].

For a section s : X ⟶ T^{∗}X of π, one can construct an object L_{s} ∈ D^{b}(I(k_{X×X})) with the properties that supp L_{s} = Δ and such that

L_{s} ⊗ ℂ̃_{s^{−1}(T_{X}^{∗}X)} ≃ ℂ̃_{s^{−1}(T_{X}^{∗}X)}

L_{s} ≃ “\underrightarrow{lim}_{U,V}”k_{U∩\overline{V}} ⊗ p_{2}^{−1}ω_{X} over X \ s^{−1}(T_{X}^{∗}X),

where V ranges through the family of open neighborhoods of Δ and U through the family of open subsets of X × X such that:

C_{Δ}(U) ⊂ {v ∈ T_{x}X; ⟨v, s(x)⟩ < 0} ∪ T_{X}X.

One defines

L_{s} ∘ G = Rp_{1}_{!!}(L_{s} ⊗ p_{2}^{−1}G),

and one proves that for F, G ∈ I(k_{X})

s^{−1}μhom(F, G) ≃ RHom (F, L_{s} ∘ G).

Now assume that s : T^{∗}X ⟶ T^{∗}(T^{∗}X) is a section. Then s^{−1}μhom(π^{−1}F, π^{−1}G) ≃ μhom(F, G). Taking as s the canonical 1-form α_{X} on T^{∗}X, one can then construct an object K_{X} ∈ D^{b}(I(k_{T^{∗}X×T^{∗}X})) and define the microlocalization functor

μ_{X} : D^{b}(I(k_{X})) ⟶ D^{b}(I(k_{T^{∗}X})),
μ_{X}(F) = K_{X} ∘ π^{−1}F.

Theorem 7.1. Let F, G ∈ D^{b}(k_{X}). Then

SS(F) = supp μ_{X}(F),
μhom(F, G) ≃ RHom (μ_{X}(F), μ_{X}(G)),
≃ RHom (π^{−1}F, μ_{X}(G)).

8 Applications

Let X denote a complex manifold of complex dimension n. On T^{∗}X there are some classical sheaves associated with a sheaf F on X: the sheaf μhom(F, O_{X}), or (F being ℝ-constructible) the sheaves tμhom(F, O_{X}) of [1] and wμhom(F, O_{X}) of [2]. We can obtain all these sheaves in a unified way:

μhom(F, O_{X}) ≃ RHom (π^{−1}F, μ_{X}(O_{X})),
tμhom(F, O_{X}) ≃ RHom (π^{−1}F, μ_{X}(O_{X}^{t})),
wμhom(F, O_{X}) ≃ RHom (π^{−1}F, μ_{X}(O_{X}^{w})).

In particular, the sheaf E_{X}^{ℝ} of microlocal operators of [9] is isomorphic to the sheaf RHom (π^{−1}ℂ_{Δ}, μ_{X×X}(O_{X×X}^{(0,n)}))[n].

Theorem 8.1. The complex μ_{X}(O_{X})[n] is concentrated in degree 0 on Ṫ^{∗}X := T^{∗}X \ T_{X}^{∗}X. Moreover, μ_{X}(O_{X})[n] ∈ Mod(E_{X}^{ℝ}, I(ℂ_{Ṫ^{∗}X})).

Corollary 8.2. Let F ∈ D^{b}(ℂ_{X}). The object μhom(F, O_{X}) is well-defined in the derived category D^{b}(E_{X}^{ℝ}).

References

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[7] M. Kashiwara and P. Schapira Ind-sheaves, to appear.

[8] S. MacLane Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1971)

[9] M. Sato, T. Kawai and M. Kashiwara Hyperfunctions and pseudodifferential equations, in LN in Math 287, Springer-Verlag 265-529 (1973).

[10] SGA 4 Sém. Géom. Algébrique 1963-64, by M. Artin, A. Grothendieck and J-L. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math. 269, 270, 305 (1972/73)

M-K Research Institute for Mathematical Sciences Kyoto University, Kyoto 606 Japan

P-S Institut de Mathématiques, Analyse Algébrique Université P & M Curie, Case 82 4, place Jussieu F-75252, Paris Cedex 05, France

schapira@math.jussieu.fr   http://www.math.jussieu.fr/~schapira/