text: E → T Deﬁnition : We will say that a geometric morphism f : is ∆-separated E → E × E if the diagonal map ∆ : T is a separated geometric morphism. f Proposition : • Delta-separated morphisms are stable under compositions and pull-back. • ◦ If f g is separated and f is ∆-separated then g is separated. • ◦ If f g is ∆-separated then g is ∆-separated. Proof : This follows from proposition and the well known fact that proper geometric morphism are stable under composition and pullback (see for example [8] section I). For the last property one also need to use proposition B3.3.8 of [6] which say that if f is any geometric morphism then ∆ is localic, that if f is localic ∆ is f f an inclusion and if f is an inclusion then ∆ is an isomorphism, which implies f that in particular, as any isomorphism is proper, that any geometric morphism (cid:3) has its diagonal map ∆-separated. Proposition : A quasi-decidable geometric morphism is ∆-separated. Proof : We will work internally in the target of the morphism and show that a quasi- decidable topos is ∆-separated. T Let be a quasi-decidable topos, it has a generating family (D ) ∈ of decidable L i i I objects. There exists a locale whose map to the terminal locale is an open L N surjection and such that, internally in , I is covered by a sub-object of (for L example, take to be the classifying locale of the theory of partial surjections N from to I, see [7] V.3 just after proposition 2. for the details ). L N Hence, internally in one has a (partially) indexed generating family of de- L T cidable object, hence internally in , admit a decidable bound, hence it is lo- calic over the Schanuel topos5 . Now the Schanuel topos is ∆-separated because N the localic group of automorphism of is separated and any localic geometric morphism is ∆-separated (the diagonal of a localic geometric morphism is an inclusion and inclusions are always separated because their diagonal is an iso- L T morphism). Hence internally in , the (pullback of the) topos is ∆-separated. But ∆-separated maps descend along open surjection because separated maps T (cid:3) do (see [6, C5.1.7]), hence is ∆-separated. 5The topos of set endowed with a continuous action of the localic group of permutation of N , see [6, C5.4.4] 13 ##LTLine##

text: E

text: T

text: Deﬁnition

text: We

text: will

text: say

text: that

text: a

text: geometric

text: morphism

text: f

text: is

text: ∆-separated

text: E

text: E

text: E

text: if

text: the

text: diagonal

text: map

text: T

text: is

text: a

text: separated

text: geometric

text: morphism.

text: f

text: Proposition

text: Delta-separated

text: morphisms

text: are

text: stable

text: under

text: compositions

text: and

text: pull-back.

text: If

text: f

text: g

text: is

text: separated

text: and

text: f

text: is

text: ∆-separated

text: then

text: g

text: is

text: separated.

text: If

text: f

text: g

text: is

text: ∆-separated

text: then

text: g

text: is

text: ∆-separated.

text: Proof

text: This

text: follows

text: from

text: proposition

text: and

text: the

text: well

text: known

text: fact

text: that

text: proper

text: geometric

text: morphism

text: are

text: stable

text: under

text: composition

text: and

text: pullback

text: (see

text: for

text: example

text: section

text: I).

text: For

text: the

text: last

text: property

text: one

text: also

text: need

text: to

text: use

text: proposition

text: of

text: which

text: say

text: that

text: if

text: f

text: is

text: any

text: geometric

text: morphism

text: then

text: is

text: localic,

text: that

text: if

text: f

text: is

text: localic

text: is

text: f

text: f

text: an

text: inclusion

text: and

text: if

text: f

text: is

text: an

text: inclusion

text: then

text: is

text: an

text: isomorphism,

text: which

text: implies

text: f

text: that

text: in

text: particular,

text: as

text: any

text: isomorphism

text: is

text: proper,

text: that

text: any

text: geometric

text: morphism

text: has

text: its

text: diagonal

text: map

text: ∆-separated.

text: Proposition

text: A

text: quasi-decidable

text: geometric

text: morphism

text: is

text: ∆-separated.

text: Proof

text: We

text: will

text: work

text: internally

text: in

text: the

text: target

text: of

text: the

text: morphism

text: and

text: show

text: that

text: a

text: quasi-

text: decidable

text: topos

text: is

text: ∆-separated.

text: T

text: Let

text: be

text: a

text: quasi-decidable

text: topos,

text: it

text: has

text: a

text: generating

text: family

text: (D

text: of

text: decidable

text: L

text: i

text: i

text: I

text: objects.

text: There

text: exists

text: a

text: locale

text: whose

text: map

text: to

text: the

text: terminal

text: locale

text: is

text: an

text: open

text: L

text: N

text: surjection

text: and

text: such

text: that,

text: internally

text: in

text: I

text: is

text: covered

text: by

text: a

text: sub-object

text: of

text: (for

text: L

text: example,

text: take

text: to

text: be

text: the

text: classifying

text: locale

text: of

text: the

text: theory

text: of

text: partial

text: surjections

text: N

text: from

text: to

text: I,

text: see

text: just

text: after

text: proposition

text: for

text: the

text: details

text: L

text: N

text: Hence,

text: internally

text: in

text: one

text: has

text: a

text: (partially)

text: indexed

text: generating

text: family

text: of

text: de-

text: L

text: T

text: cidable

text: object,

text: hence

text: internally

text: in

text: admit

text: a

text: decidable

text: bound,

text: hence

text: it

text: is

text: lo-

text: calic

text: over

text: the

text: Schanuel

text: Now

text: the

text: Schanuel

text: topos

text: is

text: ∆-separated

text: because

text: N

text: the

text: localic

text: group

text: of

text: automorphism

text: of

text: is

text: separated

text: and

text: any

text: localic

text: geometric

text: morphism

text: is

text: ∆-separated

text: (the

text: diagonal

text: of

text: a

text: localic

text: geometric

text: morphism

text: is

text: an

text: inclusion

text: and

text: inclusions

text: are

text: always

text: separated

text: because

text: their

text: diagonal

text: is

text: an

text: iso-

text: L

text: T

text: morphism).

text: Hence

text: internally

text: in

text: the

text: (pullback

text: of

text: the)

text: topos

text: is

text: ∆-separated.

text: But

text: ∆-separated

text: maps

text: descend

text: along

text: open

text: surjection

text: because

text: separated

text: maps

text: T

text: do

text: (see

text: hence

text: is

text: ∆-separated.

text: topos

text: of

text: set

text: endowed

text: with

text: a

text: continuous

text: action

text: of

text: the

text: localic

text: group

text: of

text: permutation

text: of

text: N

text: see

text: ##LTLine##


text: Proposition : Assuming the axiom of dependant choice, for every absolutely locally compact topos if the ring A contains the ring of Dedekind real numbers then A is admissible and every separating object is A-separating. If we do not assume the axiom of dependant choice one needs to assume ad- ditionally that there is a generating family of separating objects X such that T i the localic reﬂection of is completely regular in addition of being locally /Xi compact and Hausdorﬀ, which become automatic by Urysohn’s lemma if one assumes the axiom of dependant choice. Proof : In this case, every integer is invertible in A hence the second condition of deﬁni- tion is automatic because of . The ﬁrst conditions follow easily from complete regularity (which comes from Urysohn lemma, see [9, XIV.5 and 6]) that allows (cid:28) to construct for any U V two subobjects of X a positive real valued functions f which is 1 on U and zero outside of V . then any section deﬁned on V can be (cid:3) extended to X by multiplying by f and extending by 0 outside of V . T If X is a separating object of a topos , or more generally any object such that T → is locally compact, an arrow s : X M for M some abelian group object /XT in can be said to be compactly supported on X if it is compactly supported T T when seen as a section in Loc( ) of the restriction of M the Loc( ), or to ∪/X (cid:28) | /X put it more explicitly if X = U W with U X and s = 0. W T Lemma : Let be an absolutely locally compact topos, A an admissible ring T T object over and M an A-module in and X a A-separating object. → Let m be a compactly supported section of M over X and let (U (cid:44) X) i i=1...n a covering of the support of m by sub-object of X. Then there exists (λ ) i i=1...n compactly supported sections of A on X such that: • (cid:88) for all i, λ has support in U , i i • n λ m = m. i i=1 (cid:83) Proof : (cid:28) Let V be a sub-object of X such that V U and V contains the support (cid:28) (cid:28) i of m. Pick any W such that V W X, as A is c-soft over X, one has by ∈ (cid:80)| lemma a section λ A(X) such that λ = 1 and λ has support in W (in V particular λ is compactly supported), and by lemma one can ﬁnd λ , . . . , λ (cid:80) ∧ 1 n such that λ has support in W U and λ = λ. i i i As λ is equal to 1 on V and m has support in V one has λm = m and hence (cid:3) λ m = m which concludes the proof. i 18

text: Proposition

label: paragraph

text: Assuming

text: the

text: axiom

text: of

text: dependant

text: choice,

text: for

text: every

text: absolutely

text: locally

text: compact

text: topos

text: if

text: the

text: ring

text: A

text: contains

text: the

text: ring

text: of

text: Dedekind

text: real

text: numbers

text: then

text: A

text: is

text: admissible

text: and

text: every

text: separating

text: object

text: is

text: A-separating.

text: If

text: we

text: do

text: not

text: assume

text: the

text: axiom

text: of

text: dependant

text: choice

text: one

text: needs

text: to

text: assume

text: ad-

text: ditionally

text: that

text: there

text: is

text: a

text: generating

text: family

text: of

text: separating

text: objects

text: X

text: such

text: that

text: T

text: i

text: the

text: localic

text: reﬂection

text: of

text: is

text: completely

text: regular

text: in

text: addition

text: of

text: being

text: locally

text: /Xi

text: compact

text: and

text: Hausdorﬀ,

text: which

text: become

text: automatic

text: by

text: Urysohn’s

text: lemma

text: if

text: one

text: assumes

text: the

text: axiom

text: of

text: dependant

text: choice.

text: Proof

text: In

text: this

text: case,

text: every

text: integer

text: is

text: invertible

text: in

text: A

text: hence

text: the

text: second

text: condition

text: of

text: deﬁni-

text: tion

text: is

text: automatic

text: because

text: of

text: The

text: ﬁrst

text: conditions

text: follow

text: easily

text: from

text: complete

text: regularity

text: (which

text: comes

text: from

text: Urysohn

text: lemma,

text: see

text: and

text: that

text: allows

text: to

text: construct

text: for

text: any

text: U

text: V

text: two

text: subobjects

text: of

text: X

text: a

text: positive

text: real

text: valued

text: functions

text: f

text: which

text: is

text: on

text: U

text: and

text: zero

text: outside

text: of

text: V

text: then

text: any

text: section

text: deﬁned

text: on

text: V

text: can

text: be

text: extended

text: to

text: X

text: by

text: multiplying

text: by

text: f

text: and

text: extending

text: by

text: outside

text: of

text: V

text: T

text: If

text: X

text: is

text: a

text: separating

text: object

text: of

text: a

text: topos

text: or

text: more

text: generally

text: any

text: object

text: such

text: that

text: T

text: is

text: locally

text: compact,

text: an

text: arrow

text: s

text: X

text: M

text: for

text: M

text: some

text: abelian

text: group

text: object

text: /XT

text: in

text: can

text: be

text: said

text: to

text: be

text: compactly

text: supported

text: on

text: X

text: if

text: it

text: is

text: compactly

text: supported

text: T

text: T

text: when

text: seen

text: as

text: a

text: section

text: in

text: Loc(

text: of

text: the

text: restriction

text: of

text: M

text: the

text: Loc(

text: or

text: to

text: ∪/X

text: /X

text: put

text: it

text: more

text: explicitly

text: if

text: X

text: U

text: W

text: with

text: U

text: X

text: and

text: s

text: W

text: T

text: Lemma

text: Let

text: be

text: an

text: absolutely

text: locally

text: compact

text: topos,

text: A

text: an

text: admissible

text: ring

text: T

text: T

text: object

text: over

text: and

text: M

text: an

text: A-module

text: in

text: and

text: X

text: a

text: A-separating

text: object.

text: Let

text: m

text: be

text: a

text: compactly

text: supported

text: section

text: of

text: M

text: over

text: X

text: and

text: let

text: (U

text: X)

text: i

text: a

text: covering

text: of

text: the

text: support

text: of

text: m

text: by

text: sub-object

text: of

text: X.

text: Then

text: there

text: exists

text: (λ

text: i

text: compactly

text: supported

text: sections

text: of

text: A

text: on

text: X

text: such

text: that:

text: for

text: all

text: i,

text: λ

text: has

text: support

text: in

text: U

text: i

text: i

text: n

text: λ

text: m

text: m.

text: i

text: Proof

text: Let

text: V

text: be

text: a

text: sub-object

text: of

text: X

text: such

text: that

text: V

text: U

text: and

text: V

text: contains

text: the

text: support

text: i

text: of

text: m.

text: Pick

text: any

text: W

text: such

text: that

text: V

text: W

text: X,

text: as

text: A

text: is

text: c-soft

text: over

text: X,

text: one

text: has

text: by

text: lemma

text: a

text: section

text: λ

text: A(X)

text: such

text: that

text: λ

text: and

text: λ

text: has

text: support

text: in

text: W

text: (in

text: V

text: particular

text: λ

text: is

text: compactly

text: supported),

text: and

text: by

text: lemma

text: one

text: can

text: ﬁnd

text: λ

text: λ

text: n

text: such

text: that

text: λ

text: has

text: support

text: in

text: W

text: U

text: and

text: λ

text: λ.

text: i

text: i

text: i

text: As

text: λ

text: is

text: equal

text: to

text: on

text: V

text: and

text: m

text: has

text: support

text: in

text: V

text: one

text: has

text: λm

text: m

text: and

text: hence

text: λ

text: m

text: m

text: which

text: concludes

text: the

text: proof.

text: i


text: T Let be an absolutely locally compact topos, and A be an admissible ring T T object over . Let V be a sheaf of A-module over . T For any A-separating object X of we deﬁne Γ (X; V ) to be the set of sections c of V over X which are compactly supported. → If f : V W is a map between two sheaves of A-modules, then post-composing → with f induces a map f : Γ (X, V ) Γ (X, W), which makes Γ (X, ) into a c c c functor. Proposition : Let A be an admissible sheaf of ring over an absolutely locally compact topos. Then for any A-separating object X, the functor Γ (X, ) from T c the category of A-modules in to the category of commutative groups commute to all colimits and all ﬁnite limits. Proof : • As Γ (X, ) is an additive functor between additive categories, it commutes c to the zero object and to bi-products, hence both to ﬁnite co-products and ﬁnite products. • → Γ (X, ) commute to kernel (and hence to all ﬁnite limite): let f : V W c → be a morphism of sheaf of A-module and let P (cid:44) V be the kernel of f. A map with compact support from X to P is exactly a map a from X to ◦ V which has compact support and such that f a = 0, hence Γ (X, P) is → c exactly the kernel of f : Γ (X, V ) Γ (X, W). c c • Γ (X, ) commutes to all ﬁnite colimits: as we already know that it is a c left exact functor between abelian categories it is enough, for example by [1, 1.11.2 and 1.11.4], to show that it send epimorphisms to epimorphisms. (cid:16) Let f : V W be an epimorphism between two sheaves of A-modules, and → let s : X W be a map with compact support, our goal is to lift it into a compactly supported function from X to V . As f is an epimorphisms, → → there exists a covering v : X X with for each i a map s : X V ◦ ◦ i i i i such that f s = s v . Because X is A-separating one can assume that i i for each i, X is quasi-ﬁnite of cardinal k , that its support is relatively i i compact in X and that k is invertible in A and moreover there is a ﬁnite i family X , . . . , X of such object that cover the support of s. 1 n One can then (be lemma ) ﬁnd functions λ , . . . , λ from X to A such 1 n that for each i, λ is supported(cid:88)in the support of X and: i i n λ s = s i i=1  T(cid:88) Finally, we deﬁne inter(cid:88)nally in /X:  ∃ ∈ s(cid:48) := n λkii ∈ si(v) if x Xi v Xi i=1 0 if λi = 0 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T

text: Let

text: be

text: an

text: absolutely

text: locally

text: compact

text: topos,

text: and

text: A

text: be

text: an

text: admissible

text: ring

text: T

text: T

text: object

text: over

text: Let

text: V

text: be

text: a

text: sheaf

text: of

text: A-module

text: over

text: T

text: For

text: any

text: A-separating

text: object

text: X

text: of

text: we

text: deﬁne

text: Γ

text: (X;

text: V

text: to

text: be

text: the

text: set

text: of

text: sections

text: c

text: of

text: V

text: over

text: X

text: which

text: are

text: compactly

text: supported.

text: If

text: f

text: V

text: W

text: is

text: a

text: map

text: between

text: two

text: sheaves

text: of

text: A-modules,

text: then

text: post-composing

text: with

text: f

text: induces

text: a

text: map

text: f

text: Γ

text: (X,

text: V

text: Γ

text: (X,

text: W),

text: which

text: makes

text: Γ

text: (X,

text: into

text: a

text: c

text: c

text: c

text: functor.

text: Proposition

text: Let

text: A

text: be

text: an

text: admissible

text: sheaf

text: of

text: ring

text: over

text: an

text: absolutely

text: locally

text: compact

text: topos.

text: Then

text: for

text: any

text: A-separating

text: object

text: X,

text: the

text: functor

text: Γ

text: (X,

text: from

text: T

text: c

text: the

text: category

text: of

text: A-modules

text: in

text: to

text: the

text: category

text: of

text: commutative

text: groups

text: commute

text: to

text: all

text: colimits

text: and

text: all

text: ﬁnite

text: limits.

text: Proof

text: As

text: Γ

text: (X,

text: is

text: an

text: additive

text: functor

text: between

text: additive

text: categories,

text: it

text: commutes

text: c

text: to

text: the

text: zero

text: object

text: and

text: to

text: bi-products,

text: hence

text: both

text: to

text: ﬁnite

text: co-products

text: and

text: ﬁnite

text: products.

text: Γ

text: (X,

text: commute

text: to

text: kernel

text: (and

text: hence

text: to

text: all

text: ﬁnite

text: limite):

text: let

text: f

text: V

text: W

text: c

text: be

text: a

text: morphism

text: of

text: sheaf

text: of

text: A-module

text: and

text: let

text: P

text: V

text: be

text: the

text: kernel

text: of

text: f.

text: A

text: map

text: with

text: compact

text: support

text: from

text: X

text: to

text: P

text: is

text: exactly

text: a

text: map

text: a

text: from

text: X

text: to

text: V

text: which

text: has

text: compact

text: support

text: and

text: such

text: that

text: f

text: a

text: hence

text: Γ

text: (X,

text: P)

text: is

text: c

text: exactly

text: the

text: kernel

text: of

text: f

text: Γ

text: (X,

text: V

text: Γ

text: (X,

text: W).

text: c

text: c

text: Γ

text: (X,

text: commutes

text: to

text: all

text: ﬁnite

text: colimits:

text: as

text: we

text: already

text: know

text: that

text: it

text: is

text: a

text: c

text: left

text: exact

text: functor

text: between

text: abelian

text: categories

text: it

text: is

text: enough,

text: for

text: example

text: by

text: and

text: to

text: show

text: that

text: it

text: send

text: epimorphisms

text: to

text: epimorphisms.

text: Let

text: f

text: V

text: W

text: be

text: an

text: epimorphism

text: between

text: two

text: sheaves

text: of

text: A-modules,

text: and

text: let

text: s

text: X

text: W

text: be

text: a

text: map

text: with

text: compact

text: support,

text: our

text: goal

text: is

text: to

text: lift

text: it

text: into

text: a

text: compactly

text: supported

text: function

text: from

text: X

text: to

text: V

text: As

text: f

text: is

text: an

text: epimorphisms,

text: there

text: exists

text: a

text: covering

text: v

text: X

text: X

text: with

text: for

text: each

text: i

text: a

text: map

text: s

text: X

text: V

text: i

text: i

text: i

text: i

text: such

text: that

text: f

text: s

text: s

text: v

text: Because

text: X

text: is

text: A-separating

text: one

text: can

text: assume

text: that

text: i

text: i

text: for

text: each

text: i,

text: X

text: is

text: quasi-ﬁnite

text: of

text: cardinal

text: k

text: that

text: its

text: support

text: is

text: relatively

text: i

text: i

text: compact

text: in

text: X

text: and

text: that

text: k

text: is

text: invertible

text: in

text: A

text: and

text: moreover

text: there

text: is

text: a

text: ﬁnite

text: i

text: family

text: X

text: X

text: of

text: such

text: object

text: that

text: cover

text: the

text: support

text: of

text: s.

text: n

text: One

text: can

text: then

text: (be

text: lemma

text: ﬁnd

text: functions

text: λ

text: λ

text: from

text: X

text: to

text: A

text: such

text: n

text: that

text: for

text: each

text: i,

text: λ

text: is

text: the

text: support

text: of

text: X

text: and:

text: i

text: i

text: n

text: λ

text: s

text: s

text: i

text: Finally,

text: we

text: deﬁne

text: in

text: /X:

text: n

text: λkii

text: si(v)

text: if

text: x

text: Xi

text: v

text: Xi

text: if

text: λi

text: ##LTLine##

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text: of the topos. When the object is “A-separating” it corresponds to the subset of sections of M over X which are compactly supported on X, but for a general X compactly supported sections on X are not necessarily a special case of sections and they are deﬁned by the fact that for a ﬁxed M the compactly supported T of M on objects of form a cosheaf, with summation along the ﬁber as the functoriality. Section contains some preliminary on locally compact Hausdorﬀ (regular) lo- cales and compactly supported sections (in the usual sense) over these. Sec- tion contains preliminary on the “topological” assumption (separation, quasi- decidability, absolute local compactness etc...) that we will need on toposes and some of their consequences. Section is the heart of the article, it contains the more general deﬁnition of compactly supported sections and all the main result of the paper. Finally, section contains various example of toposes to which this construction applies and which gives back certain classical algebras. We ﬁnish this introduction be some general mathematical preliminary. This paper in written in framework of constructive mathematics, we allow unbounded quantiﬁcation but we do not use the unbounded replacement axiom. So it corresponds for example to the internal logic of an elementary topos as extended S in [10]. Object of this base topos are called sets, and in the rest of the article S by “topos” we mean bounded topos over , so “Grothendieck toposes” internally S S T in , in the sense of categories of -valued sheaves over an internal site. If T T is a topos a topos is a Grothendieck topos in the internal logic of , in the T sense that it is described by an internal site. The 2-category of -toposes is T naturally equivalent to the 2-category of toposes bounded over . C |C| C If is a category, is the class of object of . By pseudo category we mean a category possibly without identity elements, all the pseudo categories we will consider are enriched in abelian group, hence they are the several objects gen- eralization of non-unital non-commutative rings. C If is a site, the category of co-sheaf of sets or of abelian groups is the opposite of the category of sheaf with value in the opposite of the category of sets or of C C(cid:48) T abelian group. If two sites and deﬁnes the same topos then the category C C(cid:48) of cosheaf over and are equivalent, this is what we call the category of T cosheaf (of sets or of abelian group) over . T A generating family of a topos is a family of object X such that equivalently i every object can be covered by the X or the X form a site of deﬁnition of i i the topos for the induced topology. A bound of a topos is an object whose sub-object form a generating family. 2 Locally compact locales and compactly sup- ported functions Locales are the object of study of point-free topology and are very close to O topological spaces. A locale X is deﬁned formally by the data of a poset (X) 3

text: of

text: the

text: topos.

text: When

text: the

text: object

text: is

text: “A-separating”

text: it

text: corresponds

text: to

text: the

text: subset

text: of

text: sections

text: of

text: M

text: over

text: X

text: which

text: are

text: compactly

text: supported

text: on

text: X,

text: but

text: for

text: a

text: general

text: X

text: compactly

text: supported

text: sections

text: on

text: X

text: are

text: not

text: necessarily

text: a

text: special

text: case

text: of

text: sections

text: and

text: they

text: are

text: deﬁned

text: by

text: the

text: fact

text: that

text: for

text: a

text: ﬁxed

text: M

text: the

text: compactly

text: supported

text: T

text: of

text: M

text: on

text: objects

text: of

text: form

text: a

text: cosheaf,

text: with

text: summation

text: along

text: the

text: ﬁber

text: as

text: the

text: functoriality.

text: Section

text: contains

text: some

text: preliminary

text: on

text: locally

text: compact

text: Hausdorﬀ

text: (regular)

text: lo-

text: cales

text: and

text: compactly

text: supported

text: sections

text: (in

text: the

text: usual

text: sense)

text: over

text: these.

text: Sec-

text: tion

text: contains

text: preliminary

text: on

text: the

text: “topological”

text: assumption

text: (separation,

text: quasi-

text: decidability,

text: absolute

text: local

text: compactness

text: etc...)

text: that

text: we

text: will

text: need

text: on

text: toposes

text: and

text: some

text: of

text: their

text: consequences.

text: Section

text: is

text: the

text: heart

text: of

text: the

text: article,

text: it

text: contains

text: the

text: more

text: general

text: deﬁnition

text: of

text: compactly

text: supported

text: sections

text: and

text: all

text: the

text: main

text: result

text: of

text: the

text: paper.

text: Finally,

text: section

text: contains

text: various

text: example

text: of

text: toposes

text: to

text: which

text: this

text: construction

text: applies

text: and

text: which

text: gives

text: back

text: certain

text: classical

text: algebras.

text: We

text: ﬁnish

text: this

text: introduction

text: be

text: some

text: general

text: mathematical

text: preliminary.

text: This

text: paper

text: in

text: written

text: in

text: framework

text: of

text: constructive

text: mathematics,

text: we

text: allow

text: unbounded

text: quantiﬁcation

text: but

text: we

text: do

text: not

text: use

text: the

text: unbounded

text: replacement

text: axiom.

text: So

text: it

text: corresponds

text: for

text: example

text: to

text: the

text: internal

text: logic

text: of

text: an

text: elementary

text: topos

text: as

text: extended

text: S

text: in

text: Object

text: of

text: this

text: base

text: topos

text: are

text: called

text: sets,

text: and

text: in

text: the

text: rest

text: of

text: the

text: article

text: S

text: by

text: “topos”

text: we

text: mean

text: bounded

text: topos

text: over

text: so

text: “Grothendieck

text: toposes”

text: internally

text: S

text: S

text: T

text: in

text: in

text: the

text: sense

text: of

text: categories

text: of

text: -valued

text: sheaves

text: over

text: an

text: internal

text: site.

text: If

text: T

text: T

text: is

text: a

text: topos

text: a

text: topos

text: is

text: a

text: Grothendieck

text: topos

text: in

text: the

text: internal

text: logic

text: of

text: in

text: the

text: T

text: sense

text: that

text: it

text: is

text: described

text: by

text: an

text: internal

text: site.

text: The

text: of

text: -toposes

text: is

text: T

text: naturally

text: equivalent

text: to

text: the

text: of

text: toposes

text: bounded

text: over

text: C

text: |C|

text: C

text: If

text: is

text: a

text: category,

text: is

text: the

text: class

text: of

text: object

text: of

text: By

text: pseudo

text: category

text: we

text: mean

text: a

text: category

text: possibly

text: without

text: identity

text: elements,

text: all

text: the

text: pseudo

text: categories

text: we

text: will

text: consider

text: are

text: enriched

text: in

text: abelian

text: group,

text: hence

text: they

text: are

text: the

text: several

text: objects

text: gen-

text: eralization

text: of

text: non-unital

text: non-commutative

text: rings.

text: C

text: If

text: is

text: a

text: site,

text: the

text: category

text: of

text: co-sheaf

text: of

text: sets

text: or

text: of

text: abelian

text: groups

text: is

text: the

text: opposite

text: of

text: the

text: category

text: of

text: sheaf

text: with

text: value

text: in

text: the

text: opposite

text: of

text: the

text: category

text: of

text: sets

text: or

text: of

text: C

text: T

text: abelian

text: group.

text: If

text: two

text: sites

text: and

text: deﬁnes

text: the

text: same

text: topos

text: then

text: the

text: category

text: C

text: of

text: cosheaf

text: over

text: and

text: are

text: equivalent,

text: this

text: is

text: what

text: we

text: call

text: the

text: category

text: of

text: T

text: cosheaf

text: (of

text: sets

text: or

text: of

text: abelian

text: group)

text: over

text: T

text: A

text: generating

text: family

text: of

text: a

text: topos

text: is

text: a

text: family

text: of

text: object

text: X

text: such

text: that

text: equivalently

text: i

text: every

text: object

text: can

text: be

text: covered

text: by

text: the

text: X

text: or

text: the

text: X

text: form

text: a

text: site

text: of

text: deﬁnition

text: of

text: i

text: i

text: the

text: topos

text: for

text: the

text: induced

text: topology.

text: A

text: bound

text: of

text: a

text: topos

text: is

text: an

text: object

text: whose

text: sub-object

text: form

text: a

text: generating

text: family.

text: Locally

text: compact

text: locales

text: and

text: compactly

text: sup-

text: ported

text: functions

text: Locales

text: are

text: the

text: object

text: of

text: study

text: of

text: point-free

text: topology

text: and

text: are

text: very

text: close

text: to

text: O

text: topological

text: spaces.

text: A

text: locale

text: X

text: is

text: deﬁned

text: formally

text: by

text: the

text: data

text: of

text: a

text: poset

text: (X)


text: (cid:88) F(x) = i (γ (d)) p2(d) 0 p1(d)=x ∈ Where the i for y Y are the structural maps from A to Y y A This theorem is one of the key result of this paper. It should be understood as a description of compactly supported map from X to Y by matrix elements × → A A X Y A, but with the subtleties that matrix elements are not a function × from X Y to A as one should expect, but a compactly supported section, × and that in the case where X Y is not decidable, those compactly supported sections are not “sections which are compactly supported”. Proof : The isomorphism of gives us directly that: (cid:39) × Γ (X; AY ) Γ (X Y ; A) c c We just need to show that it is indeed as described in the theorem, which amount to understand the composite: → × →(cid:39) Γ (D; A) Γ (X Y ; A) Γ (X; AY ) c c c → × × For p : D X Y be a separating object over X Y as in the theorem. The isomorphism corresponds to the one of when X is A-separating, hence it is T essentially the isomorphism of applied to the the topos and to the object × /XT X Y by cosheaf extension from the separating objects of , but D is one T /X of the separating objects of and hence one has a diagram (all the Γ of the /X T c left square being computed in the topos ): /X ∗ × ∗ (cid:39) × Γ (D; p A) Γ (X Y ; p A) Γ (X Y ; A) c X c X c (cid:39) (cid:39) (cid:39) ∗ ∗ × (cid:39) Γ (X; p (A)D) Γ (X, p (A)(X Y )) Γ (X; Y ) c X c X c A But the explicit description of the left most vertical arrow given in allows to give an explicit description of the total diagonal map which is exactly the one presented in the theorem. (cid:3) T Deﬁnition : We ﬁx some set (X ) ∈ of A-separating object of , such that the i i I T C T X and their sub-object form a generating family of . And we deﬁne ( ; A) i c to be the additive pseudo-category whose objects are the separating objects X of T i and whose morphisms are the compactly supported map between the AX . i If the ground topos is locally decidable, one can ﬁnd such a family formed of a C T single object X, in which case ( ; A) will simply be a (non-unital) algebra. c Because of the result above, this algebra should be thought of as the algebra of 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F(x)

text: i

text: (γ

text: (d))

text: Where

text: the

text: i

text: for

text: y

text: Y

text: are

text: the

text: structural

text: maps

text: from

text: A

text: to

text: Y

text: y

text: A

text: This

text: theorem

text: is

text: one

text: of

text: the

text: key

text: result

text: of

text: this

text: paper.

text: It

text: should

text: be

text: understood

text: as

text: a

text: description

text: of

text: compactly

text: supported

text: map

text: from

text: X

text: to

text: Y

text: by

text: matrix

text: elements

text: A

text: A

text: X

text: Y

text: A,

text: but

text: with

text: the

text: subtleties

text: that

text: matrix

text: elements

text: are

text: not

text: a

text: function

text: from

text: X

text: Y

text: to

text: A

text: as

text: one

text: should

text: expect,

text: but

text: a

text: compactly

text: supported

text: section,

text: and

text: that

text: in

text: the

text: case

text: where

text: X

text: Y

text: is

text: not

text: decidable,

text: those

text: compactly

text: supported

text: sections

text: are

text: not

text: “sections

text: which

text: are

text: compactly

text: supported”.

text: Proof

text: The

text: isomorphism

text: of

text: gives

text: us

text: directly

text: that:

text: Γ

text: (X;

text: AY

text: Γ

text: (X

text: Y

text: A)

text: c

text: c

text: We

text: just

text: need

text: to

text: show

text: that

text: it

text: is

text: indeed

text: as

text: described

text: in

text: the

text: theorem,

text: which

text: amount

text: to

text: understand

text: the

text: composite:

text: Γ

text: (D;

text: A)

text: Γ

text: (X

text: Y

text: A)

text: Γ

text: (X;

text: AY

text: c

text: c

text: c

text: For

text: p

text: D

text: X

text: Y

text: be

text: a

text: separating

text: object

text: over

text: X

text: Y

text: as

text: in

text: the

text: theorem.

text: The

text: isomorphism

text: corresponds

text: to

text: the

text: one

text: of

text: when

text: X

text: is

text: A-separating,

text: hence

text: it

text: is

text: T

text: essentially

text: the

text: isomorphism

text: of

text: applied

text: to

text: the

text: the

text: topos

text: and

text: to

text: the

text: object

text: /XT

text: X

text: Y

text: by

text: cosheaf

text: extension

text: from

text: the

text: separating

text: objects

text: of

text: but

text: D

text: is

text: one

text: T

text: /X

text: of

text: the

text: separating

text: objects

text: of

text: and

text: hence

text: one

text: has

text: a

text: diagram

text: (all

text: the

text: Γ

text: of

text: the

text: /X

text: T

text: c

text: left

text: square

text: being

text: computed

text: in

text: the

text: topos

text: /X

text: Γ

text: (D;

text: p

text: A)

text: Γ

text: (X

text: Y

text: p

text: A)

text: Γ

text: (X

text: Y

text: A)

text: c

text: X

text: c

text: X

text: c

text: Γ

text: (X;

text: p

text: (A)D)

text: Γ

text: (X,

text: p

text: (A)(X

text: Y

text: Γ

text: (X;

text: Y

text: c

text: X

text: c

text: X

text: c

text: A

text: But

text: the

text: explicit

text: description

text: of

text: the

text: left

text: most

text: vertical

text: arrow

text: given

text: in

text: allows

text: to

text: give

text: an

text: explicit

text: description

text: of

text: the

text: total

text: diagonal

text: map

text: which

text: is

text: exactly

text: the

text: one

text: presented

text: in

text: the

text: theorem.

text: T

text: Deﬁnition

text: We

text: ﬁx

text: some

text: set

text: (X

text: of

text: A-separating

text: object

text: of

text: such

text: that

text: the

text: i

text: i

text: I

text: T

text: C

text: T

text: X

text: and

text: their

text: sub-object

text: form

text: a

text: generating

text: family

text: of

text: And

text: we

text: deﬁne

text: A)

text: i

text: c

text: to

text: be

text: the

text: additive

text: pseudo-category

text: whose

text: objects

text: are

text: the

text: separating

text: objects

text: X

text: of

text: T

text: i

text: and

text: whose

text: morphisms

text: are

text: the

text: compactly

text: supported

text: map

text: between

text: the

text: AX

text: i

text: If

text: the

text: ground

text: topos

text: is

text: locally

text: decidable,

text: one

text: can

text: ﬁnd

text: such

text: a

text: family

text: formed

text: of

text: a

text: C

text: T

text: single

text: object

text: X,

text: in

text: which

text: case

text: A)

text: will

text: simply

text: be

text: a

text: (non-unital)

text: algebra.

text: c

text: Because

text: of

text: the

text: result

text: above,

text: this

text: algebra

text: should

text: be

text: thought

text: of

text: as

text: the

text: algebra

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: T Let now G = (G , G , s, t, µ) be an ´etale topological groupoid. Let be the 0 1 topos of G-equivariant sheaves, i.e. sheaves over G endowed with an action of 0 G and G-equivariant maps between them. ∗ The forgetful functor from G-equivariant sheaves to sheaves over G is the f → T 0 part of an ´etale surjective geometric morphism f : G which corresponds 0 to the object X which is G over G endowed with its multiplication action on 1 0 itself. × Moreover G can be described as G T G and the groupoid structure on G 1 0 0 can be recovered as the obvious “pair groupoid” structure coming from this description of G . Any topos admitting an ´etale covering by a locale G is of 1 0 this form, such topos is called an ´etendu. T If G is locally compact and Hausdorﬀ, then any ring object A of which is 0 c-soft when seen as a sheaf over G is admissible with X as a A-separating 0 T object. The sub-object of X form a generating family of , hence one can take X as the single element of the family (X ) ∈ . i i I By theorem , compactly supported endomorphisms of X are the same as com- × T A pactly supported sections of A on X X but × is exactly the locale G X X 1 hence we need to distinguishes two cases: Either G is Hausdorﬀ in this case “compactly supported sections” of X do 1 mean sections which are compactly supported. Or G is not Hausdorﬀ, in 1 which case compactly supported sections on G are computed using the co- 1 sheaf construction of . As G is always locally Hausdorﬀ (it is ´etale over G 1 0 which is Hausdorﬀ), compactly supported sections on G can be computed using 1 the cosheaf property on the covering of G by some Hausdorﬀ open sub-objects, 1 and will be exactly linear combinations of compactly supported functions on Hausdorﬀ open subspaces as it is usual in non-comutative geometry. In both case it is not very hard to see that multiplication and involution are the usual multiplication and involution of ´etale groupoid algebra. In the special case of an ´etale groupoid whose space of object is locally compact Hausdorﬀ and totally disconnected (i.e. with a basis of compact open subspaces stable under intersection) then any sheaf over the space of objects is c-soft because any open can be covered by compact clopen subspaces. Hence in this case any sheaf of ring is admissible. Applying the above machinery to such a groupoid with a constant sheaf of rings gives exactly the algebra constructed by B.Steinberg in [11] and the equivalence between modules on the algebra and sheaf of module over the topos is the main result of his second paper on the subject [12]. Let now look at a simple example where the divisibility condition in the deﬁni- tion of admissible sheaf of ring is not vacuous. Let G be a pro-ﬁnite group, in Z T fact, for simplicity, take G = the additive group of p-adic integer. Let be p the topos of smooth G-set, i.e. the category of sets endowed with a continuous Z (i.e. smooth, or locally constant) action of . This is a absolutely compact T p topos and 1 is separating. A ring object in will be admissible if and only if p is 33

text: T

text: Let

text: now

text: G

text: (G

text: G

text: s,

text: t,

text: µ)

text: be

text: an

text: ´etale

text: topological

text: groupoid.

text: Let

text: be

text: the

text: topos

text: of

text: G-equivariant

text: sheaves,

text: i.e.

text: sheaves

text: over

text: G

text: endowed

text: with

text: an

text: action

text: of

text: G

text: and

text: G-equivariant

text: maps

text: between

text: them.

text: The

text: forgetful

text: functor

text: from

text: G-equivariant

text: sheaves

text: to

text: sheaves

text: over

text: G

text: is

text: the

text: f

text: T

text: part

text: of

text: an

text: ´etale

text: surjective

text: geometric

text: morphism

text: f

text: G

text: which

text: corresponds

text: to

text: the

text: object

text: X

text: which

text: is

text: G

text: over

text: G

text: endowed

text: with

text: its

text: multiplication

text: action

text: on

text: itself.

text: Moreover

text: G

text: can

text: be

text: described

text: as

text: G

text: T

text: G

text: and

text: the

text: groupoid

text: structure

text: on

text: G

text: can

text: be

text: recovered

text: as

text: the

text: obvious

text: “pair

text: groupoid”

text: structure

text: coming

text: from

text: this

text: description

text: of

text: G

text: Any

text: topos

text: admitting

text: an

text: ´etale

text: covering

text: by

text: a

text: locale

text: G

text: is

text: of

text: this

text: form,

text: such

text: topos

text: is

text: called

text: an

text: ´etendu.

text: T

text: If

text: G

text: is

text: locally

text: compact

text: and

text: Hausdorﬀ,

text: then

text: any

text: ring

text: object

text: A

text: of

text: which

text: is

text: c-soft

text: when

text: seen

text: as

text: a

text: sheaf

text: over

text: G

text: is

text: admissible

text: with

text: X

text: as

text: a

text: A-separating

text: T

text: object.

text: The

text: sub-object

text: of

text: X

text: form

text: a

text: generating

text: family

text: of

text: hence

text: one

text: can

text: take

text: X

text: as

text: the

text: single

text: element

text: of

text: the

text: family

text: (X

text: i

text: i

text: I

text: By

text: theorem

text: compactly

text: supported

text: endomorphisms

text: of

text: X

text: are

text: the

text: same

text: as

text: com-

text: T

text: A

text: pactly

text: supported

text: sections

text: of

text: A

text: on

text: X

text: X

text: but

text: is

text: exactly

text: the

text: locale

text: G

text: X

text: X

text: hence

text: we

text: need

text: to

text: distinguishes

text: two

text: cases:

text: Either

text: G

text: is

text: Hausdorﬀ

text: in

text: this

text: case

text: “compactly

text: supported

text: sections”

text: of

text: X

text: do

text: mean

text: sections

text: which

text: are

text: compactly

text: supported.

text: Or

text: G

text: is

text: not

text: Hausdorﬀ,

text: in

text: which

text: case

text: compactly

text: supported

text: sections

text: on

text: G

text: are

text: computed

text: using

text: the

text: co-

text: sheaf

text: construction

text: of

text: As

text: G

text: is

text: always

text: locally

text: Hausdorﬀ

text: (it

text: is

text: ´etale

text: over

text: G

text: which

text: is

text: Hausdorﬀ),

text: compactly

text: supported

text: sections

text: on

text: G

text: can

text: be

text: computed

text: using

text: the

text: cosheaf

text: property

text: on

text: the

text: covering

text: of

text: G

text: by

text: some

text: Hausdorﬀ

text: open

text: sub-objects,

text: and

text: will

text: be

text: exactly

text: linear

text: combinations

text: of

text: compactly

text: supported

text: functions

text: on

text: Hausdorﬀ

text: open

text: subspaces

text: as

text: it

text: is

text: usual

text: in

text: non-comutative

text: geometry.

text: In

text: both

text: case

text: it

text: is

text: not

text: very

text: hard

text: to

text: see

text: that

text: multiplication

text: and

text: involution

text: are

text: the

text: usual

text: multiplication

text: and

text: involution

text: of

text: ´etale

text: groupoid

text: algebra.

text: In

text: the

text: special

text: case

text: of

text: an

text: ´etale

text: groupoid

text: whose

text: space

text: of

text: object

text: is

text: locally

text: compact

text: Hausdorﬀ

text: and

text: totally

text: disconnected

text: (i.e.

text: with

text: a

text: basis

text: of

text: compact

text: open

text: subspaces

text: stable

text: under

text: intersection)

text: then

text: any

text: sheaf

text: over

text: the

text: space

text: of

text: objects

text: is

text: c-soft

text: because

text: any

text: open

text: can

text: be

text: covered

text: by

text: compact

text: clopen

text: subspaces.

text: Hence

text: in

text: this

text: case

text: any

text: sheaf

text: of

text: ring

text: is

text: admissible.

text: Applying

text: the

text: above

text: machinery

text: to

text: such

text: a

text: groupoid

text: with

text: a

text: constant

text: sheaf

text: of

text: rings

text: gives

text: exactly

text: the

text: algebra

text: constructed

text: by

text: B.Steinberg

text: in

text: and

text: the

text: equivalence

text: between

text: modules

text: on

text: the

text: algebra

text: and

text: sheaf

text: of

text: module

text: over

text: the

text: topos

text: is

text: the

text: main

text: result

text: of

text: his

text: second

text: paper

text: on

text: the

text: subject

text: Let

text: now

text: look

text: at

text: a

text: simple

text: example

text: where

text: the

text: divisibility

text: condition

text: in

text: the

text: deﬁni-

text: tion

text: of

text: admissible

text: sheaf

text: of

text: ring

text: is

text: not

text: vacuous.

text: Let

text: G

text: be

text: a

text: pro-ﬁnite

text: group,

text: in

text: Z

text: T

text: fact,

text: for

text: simplicity,

text: take

text: G

text: the

text: additive

text: group

text: of

text: p-adic

text: integer.

text: Let

text: be

text: p

text: the

text: topos

text: of

text: smooth

text: G-set,

text: i.e.

text: the

text: category

text: of

text: sets

text: endowed

text: with

text: a

text: continuous

text: Z

text: (i.e.

text: smooth,

text: or

text: locally

text: constant)

text: action

text: of

text: This

text: is

text: a

text: absolutely

text: compact

text: T

text: p

text: topos

text: and

text: is

text: separating.

text: A

text: ring

text: object

text: in

text: will

text: be

text: admissible

text: if

text: and

text: only

text: if

text: p

text: is


text: interested in compactly supported section in the rest of the paper we will not need this extension. Proof : (cid:28) (cid:28) (cid:28) (cid:28) Assume ﬁrst that U V , then let Y and Z such that U Y Z V , let (cid:48) ∧ ∧ (cid:48) ∅ ∪ ∪ (cid:48) also D and D be such that Y D = Z D = and Z D = V D = X which (cid:48) ∧ ∅ ∪ exists because Y (cid:47) Z and Z (cid:47) V . In particular as D Z = and Z D = X (cid:48) one has D (cid:47) D. ∈ F ∪ F ×F One can then take a s (Y D) = (Y ) (D) being equal to the restriction F 1 ∪ (cid:48) ∪ of s in (Y ) and 0 one D. Moreover U D (cid:47)Y D and D is a neighborhood of ∪ (cid:48) ∈ F inﬁnity because D Z = X, hence by lemma , there exists a section s (X) ∪ (cid:48) (cid:48) which is equal to s on U D , hence is equal to s on U and 0 on D . As ∪ (cid:48) 1∪ (cid:48) Z D = X hence V D = X this concludes the proof in the ﬁrst special case. (cid:3) F Lemma : Let X be a locally compact regular locale and a c-soft sheaf of F abelian group over X. Let s be a compactly supported section of , and let (U ) i be a covering of the support of s. Then there exists U , . . . , U a ﬁnite sub-cover 1 n (cid:88) of U of the support of s and a decomposition: i n s = s i i=1 such that for all i, s has support in U . i i Proof : The fact that s has compact support immediately gives the ﬁnite sub-cover. (cid:83) (cid:83) We then proceed by induction on the cardinality of the covering. If n = 0 or n = 1 the results is obvious. Otherwise, let W containing the support of s and (cid:28) (cid:28) (cid:54) ∪ − W U , then one can ﬁnd V such that V U and W V n 1 U . i n n n n i=1 i Using the above lemma one can ﬁnd a global section s which agree with s on − n ∪ · · · ∪ V and has support in U . Then s s has support included in U U − n n n − · · ·1 n 1 hence by induction admit a decomposition into s s = s + + s − with · · · n 1 n 1 (cid:3) each s having support in U . Writing s = s + + s concludes the proof. i i 1 n 3 Absolutely separated and absolutely locally compact toposes 8

text: interested

text: in

text: compactly

text: supported

text: section

text: in

text: the

text: rest

text: of

text: the

text: paper

text: we

text: will

text: not

text: need

text: this

text: extension.

text: Proof

text: Assume

text: ﬁrst

text: that

text: U

text: V

text: then

text: let

text: Y

text: and

text: Z

text: such

text: that

text: U

text: Y

text: Z

text: V

text: let

text: also

text: D

text: and

text: D

text: be

text: such

text: that

text: Y

text: D

text: Z

text: D

text: and

text: Z

text: D

text: V

text: D

text: X

text: which

text: exists

text: because

text: Y

text: Z

text: and

text: Z

text: V

text: In

text: particular

text: as

text: D

text: Z

text: and

text: Z

text: D

text: X

text: one

text: has

text: D

text: D.

text: F

text: F

text: ×F

text: One

text: can

text: then

text: take

text: a

text: s

text: (Y

text: D)

text: (Y

text: (D)

text: being

text: equal

text: to

text: the

text: restriction

text: F

text: of

text: s

text: in

text: (Y

text: and

text: one

text: D.

text: Moreover

text: U

text: D

text: D

text: and

text: D

text: is

text: a

text: neighborhood

text: of

text: F

text: inﬁnity

text: because

text: D

text: Z

text: X,

text: hence

text: by

text: lemma

text: there

text: exists

text: a

text: section

text: s

text: (X)

text: which

text: is

text: equal

text: to

text: s

text: on

text: U

text: D

text: hence

text: is

text: equal

text: to

text: s

text: on

text: U

text: and

text: on

text: D

text: As

text: 1∪

text: Z

text: D

text: X

text: hence

text: V

text: D

text: X

text: this

text: concludes

text: the

text: proof

text: in

text: the

text: ﬁrst

text: special

text: case.

text: F

text: Lemma

text: Let

text: X

text: be

text: a

text: locally

text: compact

text: regular

text: locale

text: and

text: a

text: c-soft

text: sheaf

text: of

text: F

text: abelian

text: group

text: over

text: X.

text: Let

text: s

text: be

text: a

text: compactly

text: supported

text: section

text: of

text: and

text: let

text: (U

text: i

text: be

text: a

text: covering

text: of

text: the

text: support

text: of

text: s.

text: Then

text: there

text: exists

text: U

text: U

text: a

text: ﬁnite

text: sub-cover

text: n

text: of

text: U

text: of

text: the

text: support

text: of

text: s

text: and

text: a

text: decomposition:

text: i

text: n

text: s

text: s

text: i

text: such

text: that

text: for

text: all

text: i,

text: s

text: has

text: support

text: in

text: U

text: i

text: i

text: Proof

text: The

text: fact

text: that

text: s

text: has

text: compact

text: support

text: immediately

text: gives

text: the

text: ﬁnite

text: sub-cover.

text: We

text: then

text: proceed

text: by

text: induction

text: on

text: the

text: cardinality

text: of

text: the

text: covering.

text: If

text: n

text: or

text: n

text: the

text: results

text: is

text: obvious.

text: Otherwise,

text: let

text: W

text: containing

text: the

text: support

text: of

text: s

text: and

text: W

text: U

text: then

text: one

text: can

text: ﬁnd

text: V

text: such

text: that

text: V

text: U

text: and

text: W

text: V

text: n

text: U

text: i

text: n

text: n

text: n

text: n

text: i

text: Using

text: the

text: above

text: lemma

text: one

text: can

text: ﬁnd

text: a

text: global

text: section

text: s

text: which

text: agree

text: with

text: s

text: on

text: n

text: V

text: and

text: has

text: support

text: in

text: U

text: Then

text: s

text: s

text: has

text: support

text: included

text: in

text: U

text: U

text: n

text: n

text: n

text: ·1

text: n

text: hence

text: by

text: induction

text: admit

text: a

text: decomposition

text: into

text: s

text: s

text: s

text: s

text: with

text: n

text: n

text: each

text: s

text: having

text: support

text: in

text: U

text: Writing

text: s

text: s

text: s

text: concludes

text: the

text: proof.

text: i

text: i

text: n

text: Absolutely

text: separated

text: and

text: absolutely

text: locally

text: compact

text: toposes


text: to the best of our knowledge, no such explicit conformal mapping formulae exist to these slit domains. Crowdy [6] found explicit formulae for the ideal ﬂow due to any ﬁnite number of arbitrarily-shaped ﬂuid stirrers, and he presented several examples of the induced ﬂow ﬁeld. He did not undertake computations for stirrer domains of the same variety as we have considered in this paper due to computational restrictions related to the Schottky- Klein prime function deﬁned over highly multiply connected circular domains at the time; however, new eﬀective software is now available to compute this special function if it is required in problems where the domains are highly multiply connected [7]. The formulae in [6] also require knowledge of conformal maps from multiply connected circular domains to complicated target domains, and these are not always easy or even possible to establish (e.g. those comprised of boundary curves of diﬀering shapes, like those we presented in Figures 3, 4, 9 and 11). There is no doubt that having the explicit formulae of [6] for the problem of ﬂuid stirrers is extremely valuable, but what we have oﬀered in this paper is an eﬀective alternative approach which can be used to generate accurate numerical solutions to this problem. It has the particular advantage of being computationally inexpensive and can be used with minimal geometrical restrictions on the target ﬂuid domain, in addition to being especially useful when dealing with ﬂow domains with many ﬂuid stirrers. Acknowledgements MMSN and CCG both acknowledge ﬁnancial support from Qatar University grant \ QUUG-CAS-DMSP-15 16-27. CCG acknowledges support from Australian Research Coun- cil Discovery Project DP140100933; he is also grateful for the hospitality of the Department of Mathematics, Statistics & Physics at Qatar University where this work was completed. References [1] M J Ablowitz, A S Fokas, Complex variables: introduction and applications, 2nd edition, Cambridge University Press, 2003. [2] N. Aoyama, T. Sakajo, H. Tanaka, A computational theory for spiral point vortices in multiply connected domains with slit boundaries, Japan J. Indust. Appl. Math. 30 (2013) 485–509. [3] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge, 1997. [4] P L Boyland, H Aref, M A Stremler, Topological ﬂuid mechanics of stirring, J. Fluid Mech. 403 (2000) 277–304. [5] D.A. Burton, J. Gratus, R.W. Tucker, Hydrodynamic forces on two moving discs, Theor. Appl. Mech. 31 (2004) 153–187. [6] D.G. Crowdy, Explicit solution for the potential ﬂow due to an assembly of stirrers in an inviscid ﬂuid, J. Eng. Math. 62 (2008) 333–344. [7] D.G. Crowdy, E.H. Kropf, C.C. Green, M.M.S. Nasser, The Schottky-Klein prime function: a theoretical and computational tool for applications, IMA J. Appl. Math. 81(3) (2016) 589–628. [8] D.G. Crowdy, J. Marshall, Conformal mappings between canonical multiply con- nected domains, Comput. Methods Funct. Theory 6(1) (2006) 59–76. [9] D.G. Crowdy, A. Surana, K.Y. Yick, The irrotational motion generated by two planar stirrers in inviscid ﬂuid, Phys. Fluids, 19(1) (2007), 018103. 20

text: to

text: the

text: best

text: of

text: our

text: knowledge,

text: no

text: such

text: explicit

text: conformal

text: mapping

text: formulae

text: exist

text: to

text: these

text: slit

text: domains.

text: Crowdy

text: found

text: explicit

text: formulae

text: for

text: the

text: ideal

text: ﬂow

text: due

text: to

text: any

text: ﬁnite

text: number

text: of

text: arbitrarily-shaped

text: ﬂuid

text: stirrers,

text: and

text: he

text: presented

text: several

text: examples

text: of

text: the

text: induced

text: ﬂow

text: ﬁeld.

text: He

text: did

text: not

text: undertake

text: computations

text: for

text: stirrer

text: domains

text: of

text: the

text: same

text: variety

text: as

text: we

text: have

text: considered

text: in

text: this

text: paper

text: due

text: to

text: computational

text: restrictions

text: related

text: to

text: the

text: Schottky-

text: Klein

text: prime

text: function

text: deﬁned

text: over

text: highly

text: multiply

text: connected

text: circular

text: domains

text: at

text: the

text: time;

text: however,

text: new

text: eﬀective

text: software

text: is

text: now

text: available

text: to

text: compute

text: this

text: special

text: function

text: if

text: it

text: is

text: required

text: in

text: problems

text: where

text: the

text: domains

text: are

text: highly

text: multiply

text: connected

text: The

text: formulae

text: in

text: also

text: require

text: knowledge

text: of

text: conformal

text: maps

text: from

text: multiply

text: connected

text: circular

text: domains

text: to

text: complicated

text: target

text: domains,

text: and

text: these

text: are

text: not

text: always

text: easy

text: or

text: even

text: possible

text: to

text: establish

text: (e.g.

text: those

text: comprised

text: of

text: boundary

text: curves

text: of

text: diﬀering

text: shapes,

text: like

text: those

text: we

text: presented

text: in

text: Figures

text: and

text: There

text: is

text: no

text: doubt

text: that

text: having

text: the

text: explicit

text: formulae

text: of

text: for

text: the

text: problem

text: of

text: ﬂuid

text: stirrers

text: is

text: extremely

text: valuable,

text: but

text: what

text: we

text: have

text: oﬀered

text: in

text: this

text: paper

text: is

text: an

text: eﬀective

text: alternative

text: approach

text: which

text: can

text: be

text: used

text: to

text: generate

text: accurate

text: numerical

text: solutions

text: to

text: this

text: problem.

text: It

text: has

text: the

text: particular

text: advantage

text: of

text: being

text: computationally

text: inexpensive

text: and

text: can

text: be

text: used

text: with

text: minimal

text: geometrical

text: restrictions

text: on

text: the

text: target

text: ﬂuid

text: domain,

text: in

text: addition

text: to

text: being

text: especially

text: useful

text: when

text: dealing

text: with

text: ﬂow

text: domains

text: with

text: many

text: ﬂuid

text: stirrers.

text: Acknowledgements

text: MMSN

text: and

text: CCG

text: both

text: acknowledge

text: ﬁnancial

text: support

text: from

text: Qatar

text: University

text: grant

text: CCG

text: acknowledges

text: support

text: from

text: Australian

text: Research

text: Coun-

text: cil

text: Discovery

text: Project

text: he

text: is

text: also

text: grateful

text: for

text: the

text: hospitality

text: of

text: the

text: Department

text: of

text: Mathematics,

text: Statistics

text: Physics

text: at

text: Qatar

text: University

text: where

text: this

text: work

text: was

text: completed.

text: References

text: M

text: J

text: Ablowitz,

text: A

text: S

text: Fokas,

text: Complex

text: variables:

text: introduction

text: and

text: applications,

text: edition,

text: Cambridge

text: University

text: Press,

text: N.

text: Aoyama,

text: T.

text: Sakajo,

text: H.

text: Tanaka,

text: A

text: computational

text: theory

text: for

text: spiral

text: point

text: vortices

text: in

text: multiply

text: connected

text: domains

text: with

text: slit

text: boundaries,

text: Japan

text: J.

text: Indust.

text: Appl.

text: Math.

text: K.E.

text: Atkinson,

text: The

text: Numerical

text: Solution

text: of

text: Integral

text: Equations

text: of

text: the

text: Second

text: Kind.

text: Cambridge

text: University

text: Press,

text: Cambridge,

text: P

text: L

text: Boyland,

text: H

text: Aref,

text: M

text: A

text: Stremler,

text: Topological

text: ﬂuid

text: mechanics

text: of

text: stirring,

text: J.

text: Fluid

text: Mech.

text: D.A.

text: Burton,

text: J.

text: Gratus,

text: R.W.

text: Tucker,

text: Hydrodynamic

text: forces

text: on

text: two

text: moving

text: discs,

text: Theor.

text: Appl.

text: Mech.

text: D.G.

text: Crowdy,

text: Explicit

text: solution

text: for

text: the

text: potential

text: ﬂow

text: due

text: to

text: an

text: assembly

text: of

text: stirrers

text: in

text: an

text: inviscid

text: ﬂuid,

text: J.

text: Eng.

text: Math.

text: D.G.

text: Crowdy,

text: E.H.

text: Kropf,

text: C.C.

text: Green,

text: M.M.S.

text: Nasser,

text: The

text: Schottky-Klein

text: prime

text: function:

text: a

text: theoretical

text: and

text: computational

text: tool

text: for

text: applications,

text: IMA

text: J.

text: Appl.

text: Math.

text: D.G.

text: Crowdy,

text: J.

text: Marshall,

text: Conformal

text: mappings

text: between

text: canonical

text: multiply

text: con-

text: nected

text: domains,

text: Comput.

text: Methods

text: Funct.

text: Theory

text: D.G.

text: Crowdy,

text: A.

text: Surana,

text: K.Y.

text: Yick,

text: The

text: irrotational

text: motion

text: generated

text: by

text: two

text: planar

text: stirrers

text: in

text: inviscid

text: ﬂuid,

text: Phys.

text: Fluids,


text: (cid:88) m − χ | − | j γ(t) = Re[ iU(t)η(t)] + log η(t) a , (29) j 2π j=0 U(t) = (U , U , . . . , U ), (30) 0 1 m h(t) = (h , h , . . . , h ). (31) 0 1 m The function f(z) will be computed by solving the integral equation (17) as explained § in . The complex potential w(z) then follows from (27). We shall now use the boundary integral equation (17) to compute the streamline distri- butions for some bounded and unbounded multiply connected domains whose boundaries consist of various piecewise smooth Jordan curves. To check that our numerical scheme recovers some existing results, we computed for Figure 2 some streamline distributions presented in both Crowdy [6] and Finn et al. [10] for the ﬂow in two bounded multiply connected circular domains (see Figure 2). We see that ours are in very good qualitative agreement with theirs. In Figure 3, we show the streamlines of the ﬂow due to ﬂuid stirrers in a bounded and unbounded domain each of connectivity ﬁfteen. To demonstrate that our numerical scheme can be used for domains with piecewise smooth Jordan curves, we show in Figure 4 the streamlines of the ﬂow generated by forty-four stirrers inside a square. Further, to show that our numerical method can also be used eﬀectively for domains with very high connectivity, we show in Figure 5 the streamlines of the ﬂow due to one thousand circular disk stirrers in an unbounded domain. In Figures 2–4, stirrers with arrows inside them have complex velocity of modulus 1 in the directions indicated by the arrows. Stirrers with no arrows are stationary. All stirrers in Figure 5 have complex velocities of modulus 1 in arbitrary directions. The directions are indicated by the arrows in Figure 6 which is a magniﬁed sub-section of the lower left-hand domain of the ﬂow domain of Figure 5 showing twenty circular stirrers. The ﬂuid stirrers in Figures 2, 4 (left) and 5 have zero circulation around them whilst the ﬂuid stirrers in Figures 3 and 4 (right) have been allocated random − circulations between 1 and +1. The following table shows the computation times (in seconds) for solving the integral equation (17) for the domains in Figures 2–5. Computation times were measured using the MATLAB tic toc command on a standard laptop computer. Domain n Total number of nodes Time (s) Figure 2 (left) 1024 3072 0.2 Figure 2 (right) 1024 5120 0.3 Figure 3 (left) 1024 15360 1.2 Figure 3 (right) 1024 15360 1.8 Figure 4 2048 92160 6.2 Figure 5 1024 1024000 36.0 It is important to note that Crowdy [6] found explicit formulae for the ideal ﬂow produced by any ﬁnite number of ﬂuid stirrers, circular or otherwise. These analytical formulae are written in terms of the Schottky-Klein prime function which should be com- puted using software based on the numerical schemes of [7]. The fastest method presented in [7] is based on a boundary integral equation of the form in (17) and computing the Schottky-Klein prime function requires solving at least m + 2 of these integral equations; however, the method presented in this paper requires solving only one such integral equa- tion. Thus, if one were to use the formulae of [6] to compute the streamlines in Figure 5, the total computational cost would be considerably greater compared to that due to our method. Furthermore, the formulae in [6] can be used for domains with arbitrarily-shaped ﬂuid stirrers (such as the domains in Figures 3 and 4) provided a conformal map from a 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: m

text: χ

text: j

text: γ(t)

text: Re[

text: iU(t)η(t)]

text: log

text: η(t)

text: a

text: j

text: 2π

text: U(t)

text: (U

text: U

text: U

text: m

text: h(t)

text: (h

text: h

text: h

text: m

text: The

text: function

text: f(z)

text: will

text: be

text: computed

text: by

text: solving

text: the

text: integral

text: equation

text: as

text: explained

text: in

text: The

text: complex

text: potential

text: w(z)

text: then

text: follows

text: from

text: We

text: shall

text: now

text: use

text: the

text: boundary

text: integral

text: equation

text: to

text: compute

text: the

text: streamline

text: distri-

text: butions

text: for

text: some

text: bounded

text: and

text: unbounded

text: multiply

text: connected

text: domains

text: whose

text: boundaries

text: consist

text: of

text: various

text: piecewise

text: smooth

text: Jordan

text: curves.

text: To

text: check

text: that

text: our

text: numerical

text: scheme

text: recovers

text: some

text: existing

text: results,

text: we

text: computed

text: for

text: Figure

text: some

text: streamline

text: distributions

text: presented

text: in

text: both

text: Crowdy

text: and

text: Finn

text: et

text: al.

text: for

text: the

text: ﬂow

text: in

text: two

text: bounded

text: multiply

text: connected

text: circular

text: domains

text: (see

text: Figure

text: We

text: see

text: that

text: ours

text: are

text: in

text: very

text: good

text: qualitative

text: agreement

text: with

text: theirs.

text: In

text: Figure

text: we

text: show

text: the

text: streamlines

text: of

text: the

text: ﬂow

text: due

text: to

text: ﬂuid

text: stirrers

text: in

text: a

text: bounded

text: and

text: unbounded

text: domain

text: each

text: of

text: connectivity

text: ﬁfteen.

text: To

text: demonstrate

text: that

text: our

text: numerical

text: scheme

text: can

text: be

text: used

text: for

text: domains

text: with

text: piecewise

text: smooth

text: Jordan

text: curves,

text: we

text: show

text: in

text: Figure

text: the

text: streamlines

text: of

text: the

text: ﬂow

text: generated

text: by

text: forty-four

text: stirrers

text: inside

text: a

text: square.

text: Further,

text: to

text: show

text: that

text: our

text: numerical

text: method

text: can

text: also

text: be

text: used

text: eﬀectively

text: for

text: domains

text: with

text: very

text: high

text: connectivity,

text: we

text: show

text: in

text: Figure

text: the

text: streamlines

text: of

text: the

text: ﬂow

text: due

text: to

text: one

text: thousand

text: circular

text: disk

text: stirrers

text: in

text: an

text: unbounded

text: domain.

text: In

text: Figures

text: stirrers

text: with

text: arrows

text: inside

text: them

text: have

text: complex

text: velocity

text: of

text: modulus

text: in

text: the

text: directions

text: indicated

text: by

text: the

text: arrows.

text: Stirrers

text: with

text: no

text: arrows

text: are

text: stationary.

text: All

text: stirrers

text: in

text: Figure

text: have

text: complex

text: velocities

text: of

text: modulus

text: in

text: arbitrary

text: directions.

text: The

text: directions

text: are

text: indicated

text: by

text: the

text: arrows

text: in

text: Figure

text: which

text: is

text: a

text: magniﬁed

text: sub-section

text: of

text: the

text: lower

text: left-hand

text: domain

text: of

text: the

text: ﬂow

text: domain

text: of

text: Figure

text: showing

text: twenty

text: circular

text: stirrers.

text: The

text: ﬂuid

text: stirrers

text: in

text: Figures

text: (left)

text: and

text: have

text: zero

text: circulation

text: around

text: them

text: whilst

text: the

text: ﬂuid

text: stirrers

text: in

text: Figures

text: and

text: (right)

text: have

text: been

text: allocated

text: random

text: circulations

text: between

text: and

text: The

text: following

text: table

text: shows

text: the

text: computation

text: times

text: (in

text: seconds)

text: for

text: solving

text: the

text: integral

text: equation

text: for

text: the

text: domains

text: in

text: Figures

text: Computation

text: times

text: were

text: measured

text: using

text: the

text: MATLAB

text: tic

text: toc

text: command

text: on

text: a

text: standard

text: laptop

text: computer.

text: Domain

text: n

text: Total

text: number

text: of

text: nodes

text: Time

text: (s)

text: Figure

text: (left)

text: Figure

text: (right)

text: Figure

text: (left)

text: Figure

text: (right)

text: Figure

text: Figure

text: It

text: is

text: important

text: to

text: note

text: that

text: Crowdy

text: found

text: explicit

text: formulae

text: for

text: the

text: ideal

text: ﬂow

text: produced

text: by

text: any

text: ﬁnite

text: number

text: of

text: ﬂuid

text: stirrers,

text: circular

text: or

text: otherwise.

text: These

text: analytical

text: formulae

text: are

text: written

text: in

text: terms

text: of

text: the

text: Schottky-Klein

text: prime

text: function

text: which

text: should

text: be

text: com-

text: puted

text: using

text: software

text: based

text: on

text: the

text: numerical

text: schemes

text: of

text: The

text: fastest

text: method

text: presented

text: in

text: is

text: based

text: on

text: a

text: boundary

text: integral

text: equation

text: of

text: the

text: form

text: in

text: and

text: computing

text: the

text: Schottky-Klein

text: prime

text: function

text: requires

text: solving

text: at

text: least

text: m

text: of

text: these

text: integral

text: equations;

text: however,

text: the

text: method

text: presented

text: in

text: this

text: paper

text: requires

text: solving

text: only

text: one

text: such

text: integral

text: equa-

text: tion.

text: Thus,

text: if

text: one

text: were

text: to

text: use

text: the

text: formulae

text: of

text: to

text: compute

text: the

text: streamlines

text: in

text: Figure

text: the

text: total

text: computational

text: cost

text: would

text: be

text: considerably

text: greater

text: compared

text: to

text: that

text: due

text: to

text: our

text: method.

text: Furthermore,

text: the

text: formulae

text: in

text: can

text: be

text: used

text: for

text: domains

text: with

text: arbitrarily-shaped

text: ﬂuid

text: stirrers

text: (such

text: as

text: the

text: domains

text: in

text: Figures

text: and

text: provided

text: a

text: conformal

text: map

text: from

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 4: Streamlines of the ﬂow generated by forty-four stirrers of various shape and size inside a square. All internal stirrers have zero circulation (left) and random circulations − between 1 and +1 (right). Figure 5: Streamlines of the ﬂow generated by one thousand circular stirrers in an un- bounded domain. The radii of these stirrers are random real numbers between 0.04 and 0.24. All stirrers have complex velocities of modulus 1 in random directions. 10 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Streamlines

label: caption

text: of

text: the

text: ﬂow

text: generated

text: by

text: forty-four

text: stirrers

text: of

text: various

text: shape

text: and

text: size

text: inside

text: a

text: square.

text: All

text: internal

text: stirrers

text: have

text: zero

text: circulation

text: (left)

text: and

text: random

text: circulations

text: between

text: and

text: (right).

text: Figure

text: Streamlines

text: of

text: the

text: ﬂow

text: generated

text: by

text: one

text: thousand

text: circular

text: stirrers

text: in

text: an

text: un-

text: bounded

text: domain.

text: The

text: radii

text: of

text: these

text: stirrers

text: are

text: random

text: real

text: numbers

text: between

text: and

text: All

text: stirrers

text: have

text: complex

text: velocities

text: of

text: modulus

text: in

text: random

text: directions.

text: ##LTFigure##

label: figure

font: default

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text: A rigourous demonstration of the validity of Boltzmann’s scenario for the spatial homogenization of a freely expanding gas and the equilibration of the Kac ring ∗ S. De Bi`evre Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev´e F-59000 Lille, France Equipe-Projet MEPHYSTO, Centre de Recherche INRIA Futurs, Parc Scientiﬁque de la Haute Borne, 40, avenue Halley B.P. 70478, F-59658 Villeneuve d’Ascq cedex, France and † P. E. Parris Department of Physics Missouri University of Science & Technology, Rolla, MO 65409, USA January 14 2017 Abstract Boltzmann provided a scenario to explain why individual macroscopic systems composed of a large number N of microscopic constituents are inevitably (i.e., with overwhelming probability) observed to approach a unique macroscopic state of thermodynamic equilibrium, and why after having done so, they are then observed to remain in that state, appar- ently forever. We provide here rigourous new results that mathematically prove the basic features of Boltzmann’s scenario for two classical models: a simple boundary-free model for the spatial homogenization of a non- interacting gas of point particles, and the well-known Kac ring model. Our results, based on concentration inequalities that go back to Hoeﬀd- ing, and which focus on the typical behavior of individual macroscopic systems, improve upon previous results by providing estimates, exponen- tial in N, of probabilities and time scales involved. ∗ Stephan.De-Bievre@math.univ-lille1.fr † parris@mst.edu 1 ##LTLine##

text: A

label: title

text: rigourous

text: demonstration

text: of

text: the

text: validity

text: of

text: Boltzmann’s

text: scenario

text: for

text: the

text: spatial

text: homogenization

text: of

text: a

text: freely

text: expanding

text: gas

text: and

text: the

text: equilibration

text: of

text: the

text: Kac

text: ring

label: author

text: S.

text: De

text: Bi`evre

text: Univ.

text: Lille,

text: CNRS,

text: UMR

text: Laboratoire

text: Paul

text: Painlev´e

text: Lille,

text: France

text: Equipe-Projet

text: MEPHYSTO,

text: Centre

text: de

text: Recherche

text: INRIA

text: Futurs,

text: Parc

text: Scientiﬁque

text: de

text: la

text: Haute

text: Borne,

text: avenue

text: Halley

text: B.P.

text: Villeneuve

text: d’Ascq

text: cedex,

text: France

text: and

text: P.

text: E.

text: Parris

text: Department

text: of

text: Physics

text: Missouri

text: University

text: of

text: Science

text: Technology,

text: Rolla,

text: MO

text: USA

text: January

label: date

text: Abstract

text: Boltzmann

label: abstract

text: provided

text: a

text: scenario

text: to

text: explain

text: why

text: individual

text: macroscopic

text: systems

text: composed

text: of

text: a

text: large

text: number

text: N

text: of

text: microscopic

text: constituents

text: are

text: inevitably

text: (i.e.,

text: with

text: overwhelming

text: probability)

text: observed

text: to

text: approach

text: a

text: unique

text: macroscopic

text: state

text: of

text: thermodynamic

text: equilibrium,

text: and

text: why

text: after

text: having

text: done

text: so,

text: they

text: are

text: then

text: observed

text: to

text: remain

text: in

text: that

text: state,

text: appar-

text: ently

text: forever.

text: We

text: provide

text: here

text: rigourous

text: new

text: results

text: that

text: mathematically

text: prove

text: the

text: basic

text: features

text: of

text: Boltzmann’s

text: scenario

text: for

text: two

text: classical

text: models:

text: a

text: simple

text: boundary-free

text: model

text: for

text: the

text: spatial

text: homogenization

text: of

text: a

text: non-

text: interacting

text: gas

text: of

text: point

text: particles,

text: and

text: the

text: well-known

text: Kac

text: ring

text: model.

text: Our

text: results,

text: based

text: on

text: concentration

text: inequalities

text: that

text: go

text: back

text: to

text: Hoeﬀd-

text: ing,

text: and

text: which

text: focus

text: on

text: the

text: typical

text: behavior

text: of

text: individual

text: macroscopic

text: systems,

text: improve

text: upon

text: previous

text: results

text: by

text: providing

text: estimates,

text: exponen-

text: tial

text: in

text: N,

text: of

text: probabilities

text: and

text: time

text: scales

text: involved.

text: parris@mst.edu

text: ##LTLine##


text: and a summary is provided in Section . Acknowledgements S.D.B. is supported by the Labex CEMPI (ANR- 11-LABX-0007-01) and by the Nord-Pas de Calais Regional Council and the Fonds Europ´een de D´eveloppement E´conomique R´egional (grant CPER Photonics for Society). P.E.P. thanks the University of Lille and the Labex CEMPI, where part of this work was performed, for their hospitality. The authors thank H. Spohn for bringing the work of J. Beck to their atten- tion, and the latter for communicating his recent unpublished work to them. 2 A freely expanding gas T Consider particles moving on a d-dimensional torus d, identiﬁed with T ×R the cube [0, 1]d in what follows. The phase space of one particle is d d (cid:16) (cid:17) and that of N particles is Γ = Td × Rd N . N A point in the phase space of the N-particle system will be written (X, P) = (x , p ) . The dynamics is that of uniform rectilinear i i i=1,...N motion: the particles are not subject to any force and do not interact. Initial conditions are constructed by ﬁrst choosing an a priori initial probability measure µ on Γ , and then drawing, for each of the N parti- 1 cles, initial phase space points (x , p ) independently from this probability i i measure. The only randomness in the model is in this choice of the initial condition (X, P) drawn from the product measure induced by µ on Γ . N The focus of our analysis will be on the following coarse-grained observ- ables for the system. Given any measurable subset I of the conﬁguration T space d, we consider the fraction (cid:88) 1 N f (X, P, t) = χ (x + p t), (2.1) I N I i i i=1 of particles inside I, which is expressed above in terms of the peri- odicized characteristic function χ of I; the latter vanishes everywhere { }I ∈ except at points y such that y I, where it is equal to unity, and { } where y denotes the fractional part of y taken componentwise. Letting { | } I α = 1, . . . , L be a partition of the conﬁguration space, we associate a α | | coarse-grained time-dependent density proﬁle ρ (t) = Nf (X, P, t))/ I ∈ α Iα α to each initial state (X, P) Γ . N Our main result is as follows: given an initial probability measure µ, satisfying suitable conditions spelled out in the next section, we show that for an overwhelming majority of initial conditions, there exists a relatively short time t depending only on µ, and a comparatively long time t 0 N that grows exponentially with N, such that, for all α, the fraction f of ∈ Iα particles that ﬁnds itself in I at times t [t , t ] is close to the measure | | α 0 N I of that subset. Such initial conditions, the probability of which we α will show to diﬀer from unity by an amount that is exponentially small in N, we will call “typical”. 5 ##LTLine##

text: and

text: a

text: summary

text: is

text: provided

text: in

text: Section

text: Acknowledgements

text: S.D.B.

text: is

text: supported

text: by

text: the

text: Labex

text: CEMPI

text: (ANR-

text: and

text: by

text: the

text: Nord-Pas

text: de

text: Calais

text: Regional

text: Council

text: and

text: the

text: Fonds

text: Europ´een

text: de

text: D´eveloppement

text: E´conomique

text: R´egional

text: (grant

text: CPER

text: Photonics

text: for

text: Society).

text: P.E.P.

text: thanks

text: the

text: University

text: of

text: Lille

text: and

text: the

text: Labex

text: CEMPI,

text: where

text: part

text: of

text: this

text: work

text: was

text: performed,

text: for

text: their

text: hospitality.

text: The

text: authors

text: thank

text: H.

text: Spohn

text: for

text: bringing

text: the

text: work

text: of

text: J.

text: Beck

text: to

text: their

text: atten-

text: tion,

text: and

text: the

text: latter

text: for

text: communicating

text: his

text: recent

text: unpublished

text: work

text: to

text: them.

label: section

text: A

text: freely

text: expanding

text: gas

text: T

text: Consider

text: particles

text: moving

text: on

text: a

text: d-dimensional

text: torus

text: d,

text: identiﬁed

text: with

text: T

text: ×R

text: the

text: cube

text: in

text: what

text: follows.

text: The

text: phase

text: space

text: of

text: one

text: particle

text: is

text: d

text: d

text: and

text: that

text: of

text: N

text: particles

text: is

text: Γ

text: Td

text: Rd

text: N

text: N

text: A

text: point

text: in

text: the

text: phase

text: space

text: of

text: the

text: N-particle

text: system

text: will

text: be

text: written

text: (X,

text: P)

text: (x

text: p

text: The

text: dynamics

text: is

text: that

text: of

text: uniform

text: rectilinear

text: i

text: i

text: motion:

text: the

text: particles

text: are

text: not

text: subject

text: to

text: any

text: force

text: and

text: do

text: not

text: interact.

text: Initial

text: conditions

text: are

text: constructed

text: by

text: ﬁrst

text: choosing

text: an

text: a

text: priori

text: initial

text: probability

text: measure

text: µ

text: on

text: Γ

text: and

text: then

text: drawing,

text: for

text: each

text: of

text: the

text: N

text: parti-

text: cles,

text: initial

text: phase

text: space

text: points

text: (x

text: p

text: independently

text: from

text: this

text: probability

text: i

text: i

text: measure.

text: The

text: only

text: randomness

text: in

text: the

text: model

text: is

text: in

text: this

text: choice

text: of

text: the

text: initial

text: condition

text: (X,

text: P)

text: drawn

text: from

text: the

text: product

text: measure

text: induced

text: by

text: µ

text: on

text: Γ

text: N

text: The

text: focus

text: of

text: our

text: analysis

text: will

text: be

text: on

text: the

text: following

text: coarse-grained

text: observ-

text: ables

text: for

text: the

text: system.

text: Given

text: any

text: measurable

text: subset

text: I

text: of

text: the

text: conﬁguration

text: T

text: space

text: d,

text: we

text: consider

text: the

text: fraction

text: N

text: f

text: (X,

text: P,

text: t)

text: χ

text: (x

text: p

text: t),

text: I

text: N

text: I

text: i

text: i

text: of

text: particles

text: inside

text: I,

text: which

text: is

text: expressed

text: above

text: in

text: terms

text: of

text: the

text: peri-

text: odicized

text: characteristic

text: function

text: χ

text: of

text: I;

text: the

text: latter

text: vanishes

text: everywhere

text: }I

text: except

text: at

text: points

text: y

text: such

text: that

text: y

text: I,

text: where

text: it

text: is

text: equal

text: to

text: unity,

text: and

text: where

text: y

text: denotes

text: the

text: fractional

text: part

text: of

text: y

text: taken

text: componentwise.

text: Letting

text: I

text: α

text: L

text: be

text: a

text: partition

text: of

text: the

text: conﬁguration

text: space,

text: we

text: associate

text: a

text: α

text: coarse-grained

text: time-dependent

text: density

text: proﬁle

text: ρ

text: (t)

text: Nf

text: (X,

text: P,

text: t))/

text: I

text: α

text: Iα

text: α

text: to

text: each

text: initial

text: state

text: (X,

text: P)

text: Γ

text: N

text: Our

text: main

text: result

text: is

text: as

text: follows:

text: given

text: an

text: initial

text: probability

text: measure

text: µ,

text: satisfying

text: suitable

text: conditions

text: spelled

text: out

text: in

text: the

text: next

text: section,

text: we

text: show

text: that

text: for

text: an

text: overwhelming

text: majority

text: of

text: initial

text: conditions,

text: there

text: exists

text: a

text: relatively

text: short

text: time

text: t

text: depending

text: only

text: on

text: µ,

text: and

text: a

text: comparatively

text: long

text: time

text: t

text: N

text: that

text: grows

text: exponentially

text: with

text: N,

text: such

text: that,

text: for

text: all

text: α,

text: the

text: fraction

text: f

text: of

text: Iα

text: particles

text: that

text: ﬁnds

text: itself

text: in

text: I

text: at

text: times

text: t

text: [t

text: t

text: is

text: close

text: to

text: the

text: measure

text: α

text: N

text: I

text: of

text: that

text: subset.

text: Such

text: initial

text: conditions,

text: the

text: probability

text: of

text: which

text: we

text: α

text: will

text: show

text: to

text: diﬀer

text: from

text: unity

text: by

text: an

text: amount

text: that

text: is

text: exponentially

text: small

text: in

text: N,

text: we

text: will

text: call

text: “typical”.

text: ##LTLine##