On the category of stratifolds – The Serre-Swan Theorem -

– The Serre-Swan Theorem -

栗林 勝彦 信州大学

2017年度ホモトピー論シンポジウム

高松市生涯学習センター(まなびCAN) (香川県高松市)

Introduction

This is joint work with Toshiki Aoki.

T. Aoki and K. Kuribayashi, On the category of stratifolds, Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, 58 (2017), 131–160. arXiv:1605.04142.

Acknowledgements. The second author thanks Takayoshi Aoki and Wakana Otsuka for considerable discussions on stratifolds.

This research was partially supported by a Grant-in-Aid for challenging

Exploratory Research 16K13753 from Japan Society for the Promotion of Science.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 2 / 20

Introduction

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1 Introduction

Definition of a stratifold Examples of stratifolds

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2 The category of stratifolds

Stfdembeds into the category ofR-algebras (From the talk in Karatsu 2013) The structure sheaf of a stratifold

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3 Vector bundles and the Serre-Swan theorem Vector bundles over stratifolds

The Serre-Swan theorem for stratifolds

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4 Perspective

Toward Diffeology for Homotopy Theory of stratifolds

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 3 / 20

Introduction Definition of a stratifold

Definition of a stratifold

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Definition 1.1 (Differential spaces in the sense of Sikorski (1971))

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Adifferential spaceis a pair(S,C)consisting of a topological spaceS and anR- subalgebraCof theR-algebraC⁰(S)of continuous real-valued functions onS, which is supposed to belocally detectableandC^∞-closed.

I Local detectability : f ∈ C if and only if for anyx∈ S, there exist an open neighborhoodU ofxand an elementg∈ C such that f|U =g|U.

I C^∞-closedness : For eachn ≥ 1, eachn-tuple(f1, ..., fn)of maps inC and each smooth mapg : Rⁿ → R, the compositeh :S → Rdefined by h(x) =g(f1(x), ...., fn(x)) belongs toC.

The tangent spaceatx∈S.

TxS:= The vector space of derivations onCx the germs atx

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 4 / 20

Introduction Definition of a stratifold

Definition 1.2 (Kreck (2010))

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Astratifoldis a differential space(S,C)such that the following four conditions hold:

1. S is a locally compact Hausdorff space with countable basis;

2. theskeletaskk(S) :={x∈S |dimTxS ≤k}are closed inS;

3. for eachx ∈ S and open neighborhoodU ofxin S, there exists abump functionatxsubordinate toU; that is, a non-negative functionρ ∈ Csuch thatρ(x) 6= 0and such that the support suppρ := {p∈S |ρ(p)6= 0} is contained inU;

4. thestrataS^k :=skk(S)−skk−1(S)arek-dimensional smooth manifolds such that restriction alongi:S^k,→S induces an isomorphism of stalks

i^∗ :Cx

∼=

→C^∞(S^k)_x. for eachx∈ S^k.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 5 / 20

Introduction Examples of stratifolds

Examples

I LetM be a manifold. The open cone ofM is defined by

CM^◦:=M ×[0,1)/M × {0} 3[M × {0}] =∗ C:=

{

f :CM^◦→R f_|M×(0,1)is smooth, f_|U is constant for some openU 3 ∗

}

(CM^◦,C)is a stratifold with non–empty strata S^k+1 = M ×(0,1)and S⁰=∗.

I (S,C)a stratifold,W is a manifold with boundary, which has a collarc :

∂W ×[0, )→^∼= W. We have a stratifold (S⁰= S∪fW,C⁰), C⁰ ={

g :S⁰→R g_|S ∈ C, gc(w, t) =gf(w)forw∈ ∂W }

I Moreover, we have a sub stratifold and the product of stratifolds.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 6 / 20

The category of stratifolds Stfdembeds into the category ofR-algebras

The category of stratifolds

We assume that all stratifolds are finite-dimensional;S =sk_n(S)for some n.

Let(S,C)and(S⁰,C⁰)be stratifolds. We call a continuous map f :S →S⁰ a morphism of the stratifolds, denoted

f : (S,C)→(S⁰,C⁰)

iff induces a mapf^∗:C⁰→ C; that is,ϕ◦f ∈ C for eachϕ∈ C⁰. Thus we define a category

Stfd

of stratifolds.

Theorem 2.1

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The categoryStfdfully faithfully embeds into the category ofR-algebras.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 7 / 20

The category of stratifolds Stfdembeds into the category ofR-algebras

Sketch of the proof

For anR-algebraF, we define

|F|:=the set of all morphisms of R-algebras from F toR

Moreover, we define a mapfe:|F| →Rby fe(x) =x(f)for any f ∈ F. Let Fe be theR-algebra of maps from|F|toRof the formfeforf ∈ F. Then we consider the Gelfand topology on|F|; that is,|F|is regarded as the topological space with the open basis

{fe⁻¹(U)|U :open inR,fe∈F}e

Thus the assignment of a topological space to anR-algebra gives rise to a contravariant functor

| | :R-Alg→Top

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 8 / 20

The category of stratifolds Stfdembeds into the category ofR-algebras

Lemma 2.2 (Using a bump function)

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Let(S,C)be a stratifold. Then the mapθ : S → |C|defined byθ(p)(f) = f(p)is a homeomorphism.

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Proposition 2.3

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The mapθ :S → |C| gives rise to an isomorphism of continuous spaces θ : (S,C)→(|C|,Ce)

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Theorem 2.4

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The forgetful functorF : Stfd → R-Algdefined byF(S,C) = Cis fully faithful; that is, the induced map

F :HomStfd((S,C),(S⁰,C⁰))→Hom_R-Alg(C⁰,C) is a bijection.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 9 / 20

The category of stratifolds The structure sheaf of a stratifold

The structure sheaf of a stratifold

I A maximal idealmof Creal⇐⇒^def the quotient C/mis isomorphic toRas anR-algebra.

I Spec_r C : thereal spectrum, i.e. the subset of the prime spectrum SpecC ofC consisting of real ideals. We consider Spec_r C the subspace of SpecC with the Zariski topology.

I A mapu:|C| →Spec_r C defined byu(ϕ) =Kerϕis bijective. Moreover, the mapuis continuous. In fact,

for an open baseD(f) = {m ∈ Spec_r C | f /∈ m}for somef ∈ C, we see thatu⁻¹(D(f)) =fe⁻¹(R\{0}).

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Proposition 2.5

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The bijectionu:|C|→^∼= Spec_r C is a homeomorphism.

S ∼=|C| ∼=Spec_r C ⊂SpecC.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 10 / 20

The category of stratifolds The structure sheaf of a stratifold

Theorem 2.6

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Let(S,OS)be a ringed space which comes from a stratifold(S,C)andi : Spec_rOS(S) → SpecOS(S)the inclusion. Then(S,OS)is isomorphic to i^∗(SpecOS(S),O^S(S))as a ringed space, where(SpecOS(S),O^S(S)) is the affine scheme associated with the ringOS(S).

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Sketch of the proof.

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Letm : S → |^∼= S| →^∼= Spec_rOS(S). It suffices to show that(S,OS)is isomor- phic to the structure sheaf(Spec_rOS(S),O\S(S)). To this end, we construct an isomorphism fromO\S(S)tom_∗OS. For an open setU of Spec_rOS(S), we define

α_U :M_U⁻¹OS(S)→(m_∗OS)(U) byα([f /s]) =f ·¹_s, whereM_U :=∩

m∈U m^c.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 11 / 20

Vector bundles and the Serre-Swan theorem Vector bundles over stratifolds

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Definition 3.1 (A vector bundle over a stratifold)

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Let(S,CS)be a stratifold and(E,CE)a differential space. A morphism of dif- ferential spacesπ : (E,CE) → (S,CS)is a vector bundleover(S,CS)if the following conditions are satisfied.

1. E_x:=π⁻¹(x)is a vector space overRforx∈ S.

2. There exist an open cover{Uα}α∈J ofS and an isomorphismφα : π⁻¹(Uα) → Uα × R^nα of differential spaces for eachα ∈ J. Here π⁻¹(Uα)is regarded as a differential subspace of(E,CE)andUα×R^nα is considered the product of the substratifold(U_α,CUα)of(S,CS)and the manifold(R^nα, C^∞(R^nα)).

3. The diagram π⁻¹(Uα) ^φα //

πSSSSS))

SS U_α×R^nα

pr₁

uujjjjjjjj

Uα

is commutative,

wherepr₁ is the projection onto the first factor.

4. The compositepr2 ◦φα|E_x : Ex → Uα ×R^nα → R^nα is a linear isomorphism, wherepr2 : Uα×R^nα → R^nα denotes the projection onto the second factor.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 12 / 20

Vector bundles and the Serre-Swan theorem Vector bundles over stratifolds

Proposition 3.2

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The transition functionsgαβ :Uα∩Uβ→GLn(R)are morphisms of stratifolds.

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Proposition 3.3

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Letπ : (E,CE) → (S,CS)be a vector bundle in the sense of Definition 3.1.

Then the differential space(E,CE)admits a stratifold structure for which πis a morphism of stratifolds.

By virtue of Proposition 3.2, we see thatπ:π⁻¹(Sⁱ)→Sⁱis a smooth vector bundle.

(CE)_x ^i∗ //

res^∗ ∼=

C^∞(π⁻¹(Sⁱ))x res^∗

∼=

(Cπ⁻¹(U_α))x i^∗ //C^∞(π⁻¹(Sⁱ∩Uα))x

(CUα×Rⁿ)φ_α(x)

(i×1_Rn)^∗//

φ^∗_αOO∼=

C(Sⁱ∩U_α×Rⁿ)_φα(x)

φ^∗_α

∼=OO

SinceUα×Rⁿ is a stratifold, we see that(i×1_Rⁿ)^∗is an isomorphism.

Vector bundles and the Serre-Swan theorem The Serre-Swan theorem for stratifolds

The Serre-Swan theorem for stratifolds

We denote byVBb_(S,C) the category of vector bundles over(S,C)of bounded rank.

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Theorem 3.4

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Let(S,C)be a stratifold. Then the global section functor Γ(S,−) :VBb_(S,C) →Fgp(C)

gives rise to an equivalence of categories, whereFgp(C)denotes the category of finitely generated projective modules overC.

LetLfb(S)be the full subcategory ofOS-Modconsisting of locally free OS-modules of bounded rank. We define a functorL:VBb(S,C) →Lfb(S)by LE :U ;Γ(U, E), which is fully faithful and essentially surjective.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 14 / 20

Vector bundles and the Serre-Swan theorem The Serre-Swan theorem for stratifolds

Theorem 3.5 (Morye (2013))

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Let(X,OX)be a locally ringed space such thatX is a paracompact Hausdorff space of finite covering dimension, andOX is a fine sheaf of rings. Then the Serre-Swan theorem holds for(X,OX); that is, the global section functor in- duces an equivalence of categories betweenLfb(X)andFgp(Γ(X,OX)).

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Corollary 3.6

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Let(S,C)be a stratifold andOS the structure sheaf. Then the global sections functorΓ(S,−) :Lfb(S)→Fgp(C)is an equivalence.

VBb_(S,C) ^Γ(S,−)

Theorem 3.4 //

L 'QQQQ((

QQ

QQ Fgp(C)

Lfb(S)

Γ(S,−) 'nnnnn66 nn n

Lfb(SpecC)

' Serre

iiRRRRRRRRR

Perspective Toward Diffeology for Homotopy Theory of stratifolds

Perspective –

LetDiffeologybe the category of diffeological spaces. We define a functor

k:Stfd→Diffeology

byk(S,C) = (S,D_C)andk(φ) =φfor a morphismφ:S →S⁰ of stratifolds, where

D_C :=

{

u:U →S U :open inR^q, q≥0,

φ◦u∈C^∞(U)for anyφ∈ C }

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The functorkis faithful, but not full; that is, for a continuous mapf :S →S⁰, it is more restrictive to be a morphism of stratifolds(S,C)→(S⁰,C⁰)than to be a morphism of diffeological spaces(S,DC)→(S⁰,DC⁰).

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 16 / 20

Sets^2op

Mfd fully faithful

j //

`:fully faithful

%%

Stfd _k //Diffeology

D

⊥ //

S₂

OO

S^D a

Top,

C

oo

Sets^∆op

| |D

OO

I Haraguchi and Shimakawa are considering a model structure ofDiffeology with the adjoint pair(D, S). (2013 – )

I Christensen and Wu have studied a model structure ofDiffeologywith the adjoint pair(| |D, S^D), where S^D(X) :={Aⁿ→X :smooth}and Aⁿ := {(t0, ..., tn) ∈ Rⁿ | ∑

ti = 1}, the “non-compactn-simplex”.

(2014)

I Kihara has given a model structure toDiffeologywith an adjoint pair given by modifying(| |D, S^D), more precisely, changing the diffeological structure of∆ⁿ. (2016, 2017)

Perspective Toward Diffeology for Homotopy Theory of stratifolds

I Iwase and Izumida (2015) have considerd de Rham theorem inDiffeology usingS₂ and the cubical differential forms

LetΩ^∗_DR(X)be the de Rham complex of a diffeological space(X,D^X)in the sense of Iglesias-Zemmour.

Ω^p_DR(X) :=



 Open

D^X **

∧^p

44 ^ω Sets natural trans.



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∧^∗(U) ={U ^smooth−→ ∧^∗(⊕^dimi=1^URdxi)}: the usual de Rham complex onU

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Theorem 4.1 (Iwase - Izumida (2015))

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For a CW complexX, one has isomorphisms

H^∗(X;R)∼=H^∗(S₂(X))∼=H(“a cubical de Rhma complex” ofX)

∼=H(Ω^∗_DR(X))

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 18 / 20

Perspective Toward Diffeology for Homotopy Theory of stratifolds

I For a simplicial setK,C^∗(K;R)denotes the normalized cochain algebra.

I We have two simplicial DGAC^∆ := C^∗(∆[•])andΩ^∆ := Ω^∗_DR(A^•).

Define cochain algebraA(K) := Sets^∆op(K, A_•)for a simplicial setK and a simplicial DGAA_•.

Assertion 4.2 (Emoto - K. (Work in progress))

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For a diffeology(X,DX), one has a commutative diagram

C^∗(S^D(X))

=SSSSSSS)) SS

SS

S ^'ϕ //_(C^∆_⊗Ω^∆_)(S^D_(X))

mult◦(1⊗R )

Ω^∆(S^D(X))

' ψ

oo ΩDR(X)^αoo

“integration”R

ssfffffffffffffffffffffff

C^∗(S^D(X))

in whichϕandψ are quasi-isomorphism of DGAs andαis a DGA map. More- over, if(X,D)comes from a stratifolds, thenαis a quasi-iso. and hence∫

is an isomorphism of graded algebras on the cohomology. We get the “de Rham theorem” for stratifolds.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 19 / 20

Perspective Toward Diffeology for Homotopy Theory of stratifolds

A little more perspective

I Rational homotopy theory uses(A^∗_{P L})_• the simplicial DG algebra of polyno- mial (rational) differential forms.

I Real homotopy theory in the sense of Brown and Szczarba usesΩ^∗_{de Rham}(∆^•) the usual de Rham complex on the standard simplexes, which is regarded as the simplicial DGtopologicalalgebra.

I Smooth homotopy theorymay useΩ^∆ := Ω^∗_DR(A^•),Ω^∗_DR(∆^•_sub)or Ω^∗_DR(∆^•_Kihara), which is considered a simplicial DGdiffeologicalalgebra.

E. Wu, Homological algebra for diffeological vector spaces, Homology Homotopy Appl. 17(2015), 339–376.

Katsuhiko Kuribayashi (Shinshu University) On the category of stratifolds Symposium on Homotopy Theory 2017 20 / 20