Sheaves on the category of periodic observation

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Sheaves on the category of periodic observation

Cho Cho Than

Department of Mathematics, Hokkaido University Department of Mathematics, Yangon University

Toru Tsujishita

Department of Mathematics, Hokkaido University March 28, 2000

Abstract

A Grothendieck topology on the subgroup category of the additive group of integers is defined and the sheafification of the presheaves induced from discrete dynamical systems are determined.

AMS Classification:18F20,18B25.

KeyWords:category, dynamical systems,Grothendieck topology,sheafification

Introduction

Suppose various observers record the activity of one object periodically with their own time units and each obtains his own dynamical model of the object.

How should we obtain a comprehensive model of the object starting from these personal models?

This question may be regarded as a special case of the universal problem of recovering the global information from coherent pieces of local information, which is often analyzed succinctly by the sheaf theory.

In this paper, we introduce a Grothendieck topology on the category of observers with different time units and show that the sheafification procedure gives us an effective method of synthesizing the personal dynamical models of observers whose time units generates the unit ideal of the integer.

1 The category of observers with different time units

1.1 The category N

LetN be the category whose objects are natural integers and whose arrows are generated by©

β_n,m :m→n ¯

¯n|m ª and©

α_n:n → n ¯

¯n∈N ª with the following relations:

β_nn = 1_n β_`,mβ_m,n=β_`,n and

αⁿ_mβ_m,mn=β_m,mnα_mn

form∈N and n >1. The latter relation can be expressed as the commutativity of the diagram

m ^αⁿ^m ^//m

mn

OO

βm,mn

//αmn mn

OO

βm,mn

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The following picture illustrates this category.

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The objectnstands for the observer with time unitn. The arrowα_nis the time flow of the observer nand the arrow β_m,nm :nm→m means that the observer mn can obtain his own data via the observerm.

Alternatively we can defineN as follows: Its objects are natural numbers and for each m and its divisord, there are countable arrows

α^p_dβ_d,m:m→d, p= 0,1,2,· · ·.

We often write them as α^pβ when there is no danger of confusion. When d=m, we writeα^pβ simply as α^p. We also denote α⁰β by β.

For eachn, the identity arrow isα⁰ =α⁰_nβ_n,n. The composition is defined by

(α^pβ_d,dk)◦(α^qβ) =α^p+qkβ.

Lemma 1.1 The composition satisfies the axioms of category.

Proof. The identity axiom is obvious. A simple calculation shows that both (α^p¹βd,dn1◦α^p²βdn1,dn1n2)◦α^p³β and α^p¹βd,dn1◦(α^p²βdn1,dn1n2◦α^p³β) coincides with

α^p¹^+p²ⁿ¹^+p³ⁿ¹ⁿ²β.

QED We note that α^p◦β = α^pβ, which shows our notation for arrows con- forms with the composition and the abbreviation. Hereafter we omit the composition symbol ◦.

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1.2 The categories Q and R

When the multiplicative monoidN^× acts on a set X by x7→n.x (n∈N), we can define a categoryX as follows. Its objects are elements of X. For each elementx ofX, there is an arrow

α_x:x→x

which is aperiodic, namely, for every natural number k,α^k_x 6= 1_x. For each elementx of X and a natural numbern, there is an arrow

β_x,n :n.x→x satisfying

β_x,n◦β_n.x,m=β_x,mn for each x∈X and n, m∈N and

αⁿ_x◦β_x,n =β_x,n◦α_n.x for each x∈X and n∈nat.

Note that when X=N with the actionn.x=nx, the two meanings of N coincide.

When N^× acts on the sets of rational numbers and real numbers by multiplication, we obtain categoriesQ and R respectively.

1.3 Localization of N

We also consider the categoryN^}which can be obtained fromN by invert- ing theα’s. Its objects are natural numbers and arrowsnd→nis described asα^pβ with p ∈ Znow. The composition is defined by the same formulas as before. Hence,N is a full and faithful subcategory of N^}.

1.4 N^} as the subgroup category of the additive group Z Every groupG induces a category Sub(G) called the subgroup category. Its objects are nontrivial subgroups of G and if H₁ ⊆ H₂ then an element h of H₂ gives an arrow (H₁, h, H₂) : H₁→H₂. If (H₁, h, H₂) : H₁→H₂ and (H₂, k, H₃) :H₂→H₃, then its composition is (H₁, kh, H₃) :H₁→H₃ where the productkh is possible sinceh∈H₂ ⊆H₃.

If G = Z, then the subgroups are ©

nZ ¯¯n∈N ª

and kZ ⊆ nZ iff n|k and arrows kZ→nZ are ©

n` ¯

¯`∈Z ª

. Hence there is an isomorphic

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functor J : Sub(Z)→N^} defined by

J(nZ) = n, J((nkZ, np, nZ)) = α^pβ_n,nk.

2 Some properties of N

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2.1 Comma category N ↓p

The remarkable property ofN is that the comma categories are all isomorphic. In fact, for eachp∈N, there is an isomorphic functor

I_p :N →N ↓p defined by

I_p(n) = np, I_p(α_n) = α_np, I_p(β_n,nk) = β_np,nkp. 2.2 Relation with N^÷

Let N^÷ denotes the thin category whose objects are natural numbers and n→m if and only if nis divided by m. There is a functor

ι:N^÷→N

which is identity on objects and mapsn→m toβ_m,n. There is a right inverse toι

π:N →N^÷

which is the identity map on objects and mapsβ_m,n ton→m andα_n to the identity arrow ofn.

2.3 Properties of arrows

Proposition 2.1 Every arrow is both monic and epic.

Proof.Letf :nd→n. Supposef◦h1=f◦h2for somehi:nde→nd(i= 1, 2).

Iff =α^sβ,hi =α^tⁱ (i= 1, 2), thenf ◦hi =α^dtⁱ^+sβ(i= 1, 2). Hencedt1+s= dt2+swhich implies h1=h2.

Similar arguments show that every arrow is an epic. QED

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Proposition 2.2 n

nd₁ ^α→^p¹^βn^α←^p²^βnd₂ o

has a pull-back in N^} if and only if p₁−p₂ is divisible by the greatest common divisor of d₁, d₂.

Proof. Let n

nd1 α^p¹β

→ n^α←^p²^βnd2

o

. If there is its pull-back, it must be of the form n

nd1α^q¹β

← nd^α→^q²^βnd2

o

with d the common least multiplier of d1 and d2. The necessary condition for this is the commutativity of the square, which means

d1q1+p1=d2q2+p2. (∗) Hencep1−p2must be divisible byGCD(d1, d2). Suppose now that this condition is satisfied. Then there areq1, q2 which satisfy (∗).

Let n

nd1α^r¹β

← ndu^α→^r²^βnd2

o

be any arrow which makes the square commu- tative, namely

d1r1+p1=d2r2+p2. Then we have

d1(q1−r1) =d2(q2−r2).

We have to show the existence ofrsuch that

α^qⁱβ◦α^rβ=α^rⁱβ(i= 1, 2), which means

eir=ri−qi(i= 1, 2), (∗∗) whereei =d/di(i= 1, 2). Sincee1, e2 are mutually prime, we can solve (∗∗). The

uniqueness ofris obvious. QED

We note that pull-backs may not exist inN since the equation (∗∗) has no positive solutions when the right hand side is negative.

Proposition 2.3 The category N does not have the followings:

1. initial objects, 2. terminal objects, 3. products,

4. pull-backs, 5. coproduct, 6. equalizers, 7. coequalizers.

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Proof. We do have no arrows from mto 2m for anym, whence there are no initial objects.

The only candidate for the terminal object is 1, but 1 has non trivial endoarrows αⁿ₁.

The products do not exist in general. For example, the product cone of 2 and 3 if existed must be of the form

2←−^α^k^β6−→^α^`^β 3.

Letα^pβ : 12→6. Then

α^kβ2,6α^pβ6,12=α^k+3pβ2,12, α^`β3,6α^pβ6,12=α^`+2pβ3,12.

Hence, if we take f :=α^k+1β2,12: 12→2 and anyg: 12→3, there are noh: 12→6 withα^kβ◦h=f.

Letf, g : m→n be parallel arrows. If f ◦h= g◦h for some h: p→m, then f =g. Hence there are no equalizers except for the trivial casef =g.

Similarly parallel arrowsf, g withf 6=ghave no coequalizers.

We can show similarly that coproducts and coequalizers do not exist in general.

QED

3 Presheaves on N

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3.1 Presheaves

A presheaf on N is a family of discrete dynamical systems with different time units plus comparison morphisms from one with time unitkto another with time unit nk. More precisely, a presheaf over N is given by the following data:

• A family of sets© X_n ¯

¯n∈N ª

indexed by natural numbers,

• a family of endomaps τ_n:X_n→X_n forn∈N,

• a family of mapsσ_n,mn:X_n→X_mn, form, n∈N, satisfying

(P_A) σ_mn,`mn◦σ_n,mn =σ_n,`mn, (P_B) σ_n,kn◦τ_n^k=τ_kn◦σ_n,kn.

Hence, for each n ∈ N, we have a discrete dynamical system ¹ (X_n, τ_n), which we regard as the model conceived by the observern.

1A pair (X, τ) is called a discrete dynamical system, ifX is a set andτ :X→X is an endomap. X is calledthe state spaceandτ the transition map.

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Note that P_B means that σ_n,kn : X_n→X_kn induces a morphism of dynamical systems²

(X_n, τ_n^k)→(X_kn, τ_kn).

This morphism compares the model of the observer nwith that of the observer kn, which is possible because we can extract, from the model of the observern, the information at the time intervalsnk,2nk,3nk, . . . and com- pare them with the information extractable from the model of the observer nk.

For example, the periodic points of (X_n, τ_n) with periods dividingk are mapped to fixed points of (X_nk, τ_nk) by σ_n,kn.

3.2 Presheaf induced by a discrete dynamical systems Suppose we know a dynamical system model of an object. Then we obtain a presheaf as follows: LetD= (X, τ) be the discrete dynamical system. For each natural number n, put P_D(n) = X and P_D(α_n) = τⁿ. Furthermore define P_D(β_x,n) = id for every x, n. Then P_D is a presheaf on N , called the presheaf induced by the dynamical systemP_D.

3.3 Fixed point functor

Each presheafX= (X_n, τ_n, σ_n,kn) overN induces the presheaf F ix(X) = (F ix(X_n, τ_n), σ_n,kn) over N^÷, whereF ix(X_n, τ_n) :=©

x∈X_n ¯

¯τ_nx=x ª .

2When (Xi, τi) (i= 1, 2) are discrete dynamical systems, a map f:X1→X2 is called a morphism of dynamical systems whenf◦τ1=τ2◦f.

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4 The category of presheaves on N

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4.1 Topos structure

The presheaves on N form a category Set^N ^op, which is the functor category from N ^op to Set. An arrow F : X→Y is a family of morphisms F_n : X_n→Y_n of dynamical systems which commute with the comparison operators, i.e.,

F_nm◦X(β_nm,n) =Y(β_nm,n)◦F_n. The categorySet^N ^op has the following properties.

1. It is complete and cocomplete, with pointwise limit and colimit opera- tions. For example, a product ofXandY is defined as (X_n×Y_n, τ_n^X× τ_n^Y).

2. It has an exponentiation.

3. It has a subobject classifier.

Hence it is a topos. See [2] for generalities on topos.

4.2 Yoneda embedding

We first write explicitly the Yoneda embeddingy:N →Set^N ^op, which we need to describe the subobject classifier. The presheafy(n) is defined by

y(n)_m :=N (m, n) =

½ ∅© ifn6 |m

α^p_nβ_n,m ¯

¯p= 0,1,2,· · · ª

ifn|m Since N is a small category, we identify

y(n) =N (−, n).

The arrows with codomainn are written uniquely asα^p_nβ_n,nk with (p, k)∈ Z₊×N:=Z₊×N. Denote the bijectionZ₊×N→N (−, n) by Γ_n:

Γ_n(p, k) :=α^pβ_n,nk.

We will identifyy(n) withZ₊×Nby the bijection Γ_n(p, k).

Lemma 4.1 For (p, k)∈y(n), we have

(p, k)◦α_nk = (p+k, k)), (p, k)◦β_nk,nk`= (p, k`).

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Proof. These are just the following identities:

α^p_nβn,nkαnk=α^p+k_n βn,nk, α^p_nβn,nkβnk,nk`=α_n^pβn,nk`.

QED Define transformations on Z₊×Nas follows:

A:Z₊×N3(p, k)7→(p+k, k), B_`:Z₊×N3(p, k)7→(p, k`) (`∈N)

Then the composition of α from the right is described by A and that of β_nk,nk` from the right is by B_`.

The functoriaity of the Yoneda embeddingy is described by

Lemma 4.2 1. β_n,ns^∗ ((p, k)) = (ps, ks), where the map β_n,ns^∗ : y(β_n,ns : y(ns)→y(n) is the induced map.

2. α^∗(p, k)) = (p+ 1, k), where the map α^∗ : yα_n : y(n)→y(n) is the induced map.

4.3 Sieves

We describe the subobject classifier Ω of the presheaf toposSet^N ^op using the Yoneda lemma:

Ω_n ' Set^N ^op(y_n,Ω) ' Sub(y(n)).

A subobjectSofy_nis a subset ofN (−, n) =Z₊×Nclosed by compositions from the right, which is calleda sieve on ninN .

Proposition 4.3 Sieves on n are the subsets of Z₊×N which are closed under the transformations A, B_`.

Proof. Obvious from Lemma4.1 QED

By Lemma 4.2, the action of arrows on sieves can be described as follows:

Lemma 4.4 1. Ω_β : Ω_n→Ω_ns induced by β :ns→n is given by Ω_β(S) =©

(n, k) ¯

¯(ns, ks)∈S ª ).

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2. Ω_α: Ω_n→Ω_n induced by α_n:n→n is given by Ω_α(S) =©

(n, k) ¯

¯(n+ 1, k)∈S ª . Define mapsM_s, σ:Z₊×N→Z₊×Nby

M_s(n, k) := (sn, sk) σ(n, k) = (n+ 1, k).

Then the above lemma can be written as

Lemma 4.5 1. Ω_β : Ω_n→Ω_ns induced by β :ns→nis given by Ω_β(S) =M_s⁻¹(S).

2. Ω_α: Ω_n→Ω_n induced by α_n:n→n is given by Ω_α(S) =σ⁻¹(S).

LetT ∈Ω_n⊆ P(Z₊×N). We have a smallest subsieveTb containingT. In fact we add to T those elements obtained by A and B_` (`∈N)actions.

This operation is a closure operator[1] T 7→TbonP(Z₊×N) and its closed sets are precisely the sieves. Hence the set of sieves forms a complete meet sublattice ofP(Z₊×N).

Proposition 4.6 The lattice structure of Ω_n is given by 1. S₁ ≤ S₂ ⇐⇒ S₁ ⊆S₂,

2. S₁ V

S₂ = S₁ T S₂, 3. S₁ W

S₂ = S₁\S S₂. 4.4 Canonical sieves

For a finite subsetK⊆N, we denote byS(n;K)∈J(n) the sieve generated by the arrows ©

α^sβ_n,nkt ¯

¯s, t∈N, k∈K ª

. This can be written also as S(n;K) =©

(p, `) ¯

¯`∈N, `∈K^∗ ª .

HereK^∗denotes the multipliers of elements ofK. A sieve is calledcanonical if it can be expressed asS(n;K) with a finiteK ⊆N.

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Lemma 4.7 S(n, K₁)T

S(n, K₂) =S(n, K₁V

K₂), where K₁^

K₂ :=

n k₁^

k₂ ¯

¯k_i∈K_i(i= 1, 2) o

withk₁V

k₂ denoting the least common multiplier.

Proof. The right hand side obviously is contained in the left hand side. Sup- pose (m, k) is in the left side hand. Then there are ki with ki|k and ki ∈Ki for i= 1,2. Hencek1

Vk2|kand (m, k) is in the right hand side. QED We describe the action of α_n and β_n,nk on canonical sieves.

From Lemma4.2, we have obviously the following

Proposition 4.8 The endoarrow α_n leaves the canonical sieves invariant, namely,α^∗_nS(n;K) =S(n;K).

Similarly, we have

Proposition 4.9 The arrow β_n,ns maps S(n, K) toS(ns, K/s), where

K/s:=

( k k∨s

¯¯k∈K )

,

withk∨s denoting the greatest common divisor ofk and s.

Proof. Since

β_n,ns^∗ S(n, K) =© (p, `) ¯

¯(ps, `s)∈S(n, K)ª

and (ps, `s)∈S(n, K) is equivalent tok|`sfor somek∈K. The assertion follows from

k|`s ⇐⇒ (k/k∧s)|`.

QED

5 A Grothendieck topology on N

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5.1 Definition

LetS be a sieve on n identified with a subset ofZ₊×N. Define µ(S) :=©

k∈N ¯¯(p, k)∈S for all p∈Z₊ ª . A sieveS is calleddense ifW

µ(S) = 1, i.e., the greatest common divisor of µ(S) is 1. LetJ(n) be the set of dense sieves on n.

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Proposition 5.1 J is a Grothendieck topology onN .

Proof. Obviouslytn =y(n) =Z+×Nis dense since µ(tn) =N.

Let f : ns→n and S ∈ J(n). We show that f^∗S ∈ J(ns). Since f is the composition of α^k_n : n→n and βn,ns : ns→n, it suffices to show thatα^∗_nS ∈ J(n) andβ^∗_n,nsS ∈J(ns).

By Lemma 4.4, we have obviously µ(α^∗_nS) = µ(S), whence α^∗_nS ∈J(n). By the same lemma,

µ(β_n,ns^∗ S)⊇© k ¯

¯ks∈µ(S) ª

⊇µ(S), sincesµ(S)⊆µ(S) obviously. Hence fromW

µ(S) = 1, we haveW

µ(β_n,ns^∗ S) = 1.

Finally, we have to show the transitivity of J. Let S ∈ J(n) and R be a sieve on n. Suppose, for every f ∈ S, f^∗R ∈ J(dom(f)). Let s1,· · ·, sm ∈ µ(S) with W

isi = 1. For each i and ` ∈ {0,1,· · ·, si−1} we have α^`_nβn,nsi ∈ S, whence µ(β_n,ns^∗ _iα^`∗_nR) has greatest common divisor 1. This means there areti`j ∈ µ(β_n,ns^∗ _iα_n^`∗R) (j ∈I_i`) such thatW

j∈Ii`t_i`j = 1. Since (p, nt_i`j)∈β^∗_n,ns_iα^`∗_nR for allp∈Z+, we have

(∗) (psi+`, ti`jsi)∈Rfor allp.

Let Ii := Q_s_i₋₁

`=0 Ii` and for J = (j0, j1,· · ·, jsi−1) ∈ Ii, define tiJ := V_s_i₋₁

`=0 ti`j`. Then by the distributivity of the posetN^÷, we have

_

J∈Ii

tiJ = 1.

From (*), we have

(ps_i+`, t_iJs_i)∈Rfor allpandJ ∈I_i and`, since for all` there is aj withtiJ|ti`j. Hence we have

(p, tiJsi)∈R for allp, which implies tiJsi∈µ(R) for all iandJ ∈Ii. Since

_

i

_

J∈Ii

tijsi=_

i

si= 1,

we conclude thatR∈J(n). QED

5.2 Canonical dense sieves

Since µ(S(n, K)) is generated multiplicatively by K, the canonical sieve S(n;K) is dense if and only ifW

K= 1.

Lemma 5.2 Every dense sieve contains a canonical dense sieve.

Proof. LetS be a dense sieve. Since W

µ(S) = 1, there are finite K ⊆µ(S) withW

K= 1. Hence S contains the canonical dense sieveS(n, K). QED

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6 Sheaves

6.1 Matching family

LetP be a presheaf over N . Let S ∈Ω_n be a sieve. A matching familyx is described as follows. It is a family (x_i,k)_(i,k)∈S satisfying, for i∈ Nand k, `∈K,

(M1) x_i,k ∈P(nk), (M2) x_i,k·α_nk =x_i+k,k, (M3) x_i,k·β_nk,nkp=x_i,kp.

Each x ∈ P(n) defines a matching family κx := (x_i,k)_(i,k)∈S, where x_i,k :=x·αⁱ_nβ_n,nk, whence we have

(∗) κ_S:P(n)→Match(S, P).

A presheaf P is called separated if and only if κ_S is injective for every n and for every dense sieve S on n. A presheaf P is called a sheaf for the Grothendieck topologyJ if and only ifκ is bijective for everynand for every dense sieve S on n.

Lemma 6.1 A presheaf P is a sheaf if κ_S is bijective for canonical sieves S.

Proof. In fact, ifS contains a denseS(n, K), then we have PD(n)−→^f Match(S, PD)−→^g Match(S(n, K), PD).

Sincef and g are obviously injective, if g◦f is bijective then f is surjective and

hence bijective. QED

6.2 Presheaves PD

LetD = (X, τ) be a discrete dynamical system and let P_D be the induced presheaf defined in§??.

Then P_D(n) =X for every nand β’s act as the identity and α_n:n→n acts as τⁿby definition.

We have the following description of matching families.

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Proposition 6.2 If K ={n₁, n₂,· · · , n_k}, then matching families x∈Match(S(n, k), P_D)

correspond bijectively to the sequences (x_i)_i∈N∈X^N satisfying

τⁿⁿ^jx_i=x_i+n_j for all i∈N and j∈ {1,2,· · ·, k}, by the correspondence x_i =x_i,^V_K fori∈N, whereV

K is the least common multiplier. Moreover the κ_S(n,K):P_D(n)→Match(S(n, K), P_D) is given by

x7→(x, τⁿx, τ²ⁿx,· · ·, τ^knx,· · ·).

Hence it is obviously injective and we have the following proposition.

Proposition 6.3 The presheaf P_D is separated.

We introduce an equivalence relation∼_n on X by x∼y⇐⇒^def τ^nmx=τ^nmy for somem∈N.

It is obvious that ∼_n is in fact an equivalence relation.

Lemma 6.4 Let S(n,{n₁,· · · , n_k}) ∈ J(n). If a sequence (x_i)_i∈N ∈X^N satisfies

τⁿⁿ^jx_i =x_i+n_j for all i∈Z₊ and j= 1,· · · , k, then

τⁿx_i ∼_nx_i+1 ∀i∈N.

Proof. Since 1 =P

1≤i≤k`ini, with`i∈Z, we have 1 + X

`i<0

|`i|ni= X

`i≥0

`ini,

which we denote bym. Then, for allp∈N, τ^nmxp = (Y

`i>0

τ^`ⁱⁿⁿⁱ)xp=xp+P

`i>0`ini=xp+m

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and

τ^nm−nxp+1= Y

`i<0

τ^|`ⁱ^|nnⁱxp+1=x_p+^P

`i<0τ^|^`i^|ⁿⁱ =xp+m. Hence

τ^nm−n(τⁿxp) =τ^mn−nxp+1,

which impliesτⁿxp∼nxp+1. QED

When τ is injective, the equivalence relation∼_n is the identity relation.

Hence we have the following theorem.

Theorem 6.5 The presheaf P_D induced from a discrete dynamical system D= (X, τ) is a sheaf if τ is injective.

Proof. In fact we show that

PD(n)−→Match(S, PD)

is a bijection forn∈N andS∈J(n). By Lemma 6.1, we may also assume that S=S(n, K)∈J(n).

By Proposition 6.2, it suffices to show that if a sequence (xi)i∈N∈X^N satisfies τⁿⁿ^jxi=xi+nj

for alli∈Z+ andj= 1,· · · , k, then

xi =τⁿⁱx0 ∀i∈N,

which is valid by Lemma 6.4. Hence we conclude that the matching familyxcomes

fromx0∈PD(1). QED

7 Sheafification of discrete dynamical systems

7.1 Sheafification operation

There is a general method of converting presheaves to sheaves.

For a presheafP, we can define another presheaf P⁺ by P⁺(n) := colim_S∈J(n)Match(S, P).

Note that ifS⊆T, then there is a natural restriction map Match(T, P)→Match(S, P),

and the colimit is taken with respect to the poset of sieves onnordered by the inclusion order.

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Theκ_S’s induce

κ(n) :P(n)→P⁺(n).

IfP is separated, thenκ(n) is injective for allnand if P is a sheaf then κ is bijection for alln. In fact the converse is true.

Proposition 7.1 [2] A presheaf is separated ifκ is injective and a sheaf for the Grothendieck topology J, if κ is bijective.

Theorem 7.2 [2, Lemma4,Lemma5, p131] The presheaf P⁺ is separated.

If P is already separated, then P⁺ is a sheaf.

7.2 Discrete dynamical systems

Let Dbe a discrete dynamical system. SinceP_D is separated, the presheaf P_D⁺ is a sheaf.

In this section, we examine the sheafification of the presheafP_D induced from some concrete discrete dynamical systems D3.2.

The following lemma gives us a method of calculating the matching family. We note that the arrow α_n leaves the canonical sieves invariant and whence induces an endomap of Match(S(n, K), P).

Obviously we have the following.

Lemma 7.3 LetK={p, q}withp < q, then a matching family in Match(S(n, K), P_D) is determined by the sequence hx₀, x₁,· · ·, x_p−1i which satisfies

τ^nqx_i = (τ^p)^sx_t where i+q≡t modp with 0≤t < p and s= i+q

p . The arrowα_n acts on Match(S(n, K), P) by

(x₀, x₁,· · · , x_p−1).α_n= (x₁, x₂,· · · , x_p−1, τ^px₀).

Example SupposeD is as in Figure 1.

When K = {2,3}, then x ∈ Match(S(1, K), P_D) is determined by (x, y)∈X² with

τ³x₀ =τ²x₁,

whence τ²(τ x₀) = τ²(x₁). It is easy to show that the discrete dynamical system (Match(S(1, K), P_D), α₁) is given by D₂ in Figure 1.

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0 1 2 3 a_-1

a_-2

b_-1 b_-2

0 1 2 3

a_-1 a_-2

a_-2

b_-1 b_-2

b_-2

0 1

-2 -1 2 3

D

D₂ D₁

Figure 1: Example of sheafification

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7.3 Sheafification of P_D

Let D= (X, τ) be a discrete dynamical system.

Define firstits reduced dynamical system Das follows. Let π:X→Xbe the quotient map of the equivalence relation∼₁ introduced in §6.2. Thenτ inducesτ :X→X byτ([x]) := [τ x].

Define a discrete dynamical system Db as follows. Let Xb be the set of sequences (x₀, x₁,· · ·)∈X^N which satisfy the following two conditions:

τ[x_i] = [x_i+1] for all i∈N, and there is a natural number N such that

(∗) τⁱx_j =x_i+j for alli, j with i+j > N . Defineτb(x₀, x₁, x₂,· · ·) = (x₁, x₂, x₃,· · ·).

Example LetD be as in Figure 1. Then D₁ =D andD₂ =D.b

Theorem 7.4 LetD= (X, τ)be a discrete dynamical system, thenP_D⁺(n) = (X, τ\ⁿ).

Proof. We showP_D⁺(1) =(X, τ\). The general case can be shown similarly.

Let x = (xi) ∈ P_D⁺(1). Then x ∈ Match(S(1,{n1,· · ·, nk}, PD)) for some n1 < · · · < nk. Then by Lemma 6.4, τ xi ∼1 xi+1 for all i. Since the second condition (∗) is obvious if we takeN =n1, we havex∈D.b

Conversely supposex∈D. Letb N be an integer such that (∗) holds. Letp, q >

N be relatively prime integers so thatS(1,{p, q})∈J(1). Then, by (∗), we have τ^pxi = xi+p and τ^qxi =xi+q for all i. This shows x∈Match(S(1,{p, q}), PD).

QED

8 Concluding remarks

We considered the problem of reconstructing the dynamic behavior of an object from the data of observers who observe it periodically with mutually prime periods. We analyzed this problem by introducing the base category N with a natural Grothendieck topology.

It turned out that when the original dynamics has no states which merge, then the original structure is recovered from the observations. If the observed system has merging states, then the presheaf P_D is not a sheaf, but the

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sheafification procedure recovers the structure of the quotient dynamical system obtained by identifying two states which eventually coincides.

We will consider in future the general case when the comparison maps β are not identities. Then the sheafification procedure gives rise to the new state spaces which are fibred products of the local observers.

Finally we note that the Grothendieck topologyJ is not the unique one.

We show another natural Grothendieck topology in the appendix, whose sheafification operator however destroys the transition information among the transient states.

References

[1] G. Birkoff. Lattice Theory. AMS Colloquium Publications, Vol. 25, 1948.

[2] S. Maclane and L. Moerdijk. Sheaves in Geometry and Logic. A first Introduction to Topos Theory. Second corrected printing. Springer 1994.

A Another Topology on S et

^N ^op

There is another natural Grothendieck topology onN , which we define as a Lawvere-Tierney topologyj on the presheaf toposSet^N ^op.

Recall[2, p219] that a Lawvere-Tierney topology j is an endo arrow of the subobject classifier Ω satisfying

LT1 j◦true = true, LT2 j◦j=j,

LT3 j◦V

=V

◦(j×j).

Here true : 1→Ω is the arrow classifying the identity arrow 1₁. The

arrow ^

: Ω×Ω→Ω

is the meet operation andj×j: Ω×Ω→Ω×Ω is the product ofj.

Define now j_n: Ω_n→Ω_n by

j_n(S) =S,

whereS ⊆Z₊×Nis a sieve andS is defined as follows: First

|S|:=©

n∈Z₊ ¯

¯(n, p)∈S for some p ª .

Sheaves on the category of periodic observation