A Ca ego y o Blu and he G and Lemma
Aleksanda Pe išić
Augus 2025
Abs ac
We axioma ize blu —a unable neighbou hood/ elaxa ion—as a ca ego ical cons uc ion.
A blu ed objec is a diag am
BX
:
I→ C
indexed by blu scales wi h a eading map
ρX
:
limIBX→X
(sha p limi ) and exhaus ion
colimIBX≃
1. An unce ain y window is
a lax e ac pai
jε
X
:
X→BεX
,
ε
X
:
BεX→X
. We p o e a p ope y- anspo p inciple:
o p ope ies s able unde e ac s and il e ed limi s and mono one o
≾
,P(
X
)holds i
P
ε
(
BεX
)holds e en ually. Blu -mo phisms a e na u al ans o ma ions; he G and Lemma
(Blu – anspo lemma) shows sending sha p equals sending blu ed hen eading, yielding
a simple s ing-diag am calculus and a g aded (co)monad s uc u e o scale composi ion.
Examples include analy ic blu (Ma ko /con olu ion semig oups) and logical blu (quan ale-
en iched nuclei), linking he cons uc ion o domain- heo e ic app oxima ion and p obabilis ic
powe domains.
We ix a base ca ego y
C
(e.g.
Se
,
Top
, o
Meas
). In ui i ely, objec s a e p oposi ion spaces
(o answe spaces), and a blu is a unable neighbou hood/ elaxa ion.
Blu index and blu ed objec s
Le (
I, ⪯,⊕,
0
,∞
)be a di ec ed monoidal pose o blu scales ( hink:
ε∈
(0
,∞
], wi h 0= sha p,
∞
= “no esponse”), whe e
⊕
models composing blu (e.g. a iance addi ion), 0is he neu al
elemen , and ∞is abso bing (ε⊕ ∞ =∞).
De ini ion 1 (Blu ed objec ).Ablu ed objec is a pai X= (
X, BX
)whe e
X∈Ob
(
C
)and
BX:I→ C is a unc o (a il e ed diag am) wi h:
1.
Reading cone (sha p limi ): a uni e sal cone
ρX
:
limIBX→X
. We say he blu is
ai h ul i ρXis an isomo phism (i.e. X≃limIBX).
2.
Exhaus ion a in ini y: a cocone o a e minal/“no-in o” objec 1, i.e.
colimIBX≃
1
(“neighbou hood ends o no esponse”).
W i e BεX o BX(ε)and ε
X:BεX→X o he leg o he limi cone ( eading a scale ε).
Rema k 2 (Rela ion o domain- heo e ic app oxima ion).I we ega d
I
as a di ec ed se
o “ esolu ions”, a ai h ul blu has he la ou o domain heo y:
X
is eco e ed as he limi
o i s app oximan s (
BεX
)
ε∈I
. The exhaus ion condi ion plays he ole o a bo om elemen ,
and he blu diag am is analogous o a di ec ed sys em o ini e app oxima ions whose limi is
he ull objec . The di e ence is ha he e he app oxima ion is explici ly pa ame ized by a
physical/epis emic scale ε.
Rema k 3. Conc e e examples:
•C
=
Meas
,
BεX
=
X
equipped wi h a Gaussian ji e o a iance
ε
(con olu ion on
unc ions, o Gi y- ype andomiza ion on poin s);
1
•C
=
Top
,
BεX
he same se wi h a coa se uni o mi y/en ou age (neighbou hood hicken-
ing);
•C
=
Se
,
BεX
he se o “
ε
-consis en u h ables” o
X
wi h a collapse map o
X
as
ε→0.
Weak con ainmen and he unce ain y window
The poin o blu is ha an objec is s ill p esen while a elling wi h a neighbou hood, bu
only in a elaxed sense. We cap u e his by a lax e ac pai .
De ini ion 4 (Unce ain y window / lax e ac pai ).Fo each scale
ε∈I
, we equip
X
wi h a
pai o na u al maps
jε
X:X−→ BεXand ε
X:BεX−→ X,
called he injec ion ( hickening) and he eading (deblu ing), such ha
ε
X◦jε
X= idXand jε
X◦ ε
X≾idBεX.
He e
≾
is a chosen ambien p eo de on endomo phisms (e.g. poin wise
≤
in
Se
o
Top
when
a ailable, a.s.
≤
in
Meas
, o he en ichmen o de i
C
is Pos/quan ale-en iched). Thus
X
is
con ained while blu ed (exac on he sha p side, elaxed on he blu ed side). We also equi e
na u ali y in
X
and ha
jε
X→idX
and
ε
X→idX
as
ε→
0(in he sense o you eading cone).
Rema k 5 (Bo h–sides neighbou hood and synch onized emo al).The pai (
jε
X, ε
X
)says:
as long as he P oposi ion and he Answe ca y ma ching neighbou hoods, he mechanism
can pass hem wi h con olled elaxa ion, and emo ing he neighbou hood on one side o ces
he co esponding emo al on he o he ia na u ali y o
¯
and he g aded (co)monad maps
Bε⊕δX→Bε(BδX).
Rema k 6 (T ansmission p inciple (AC-analogy)).Think o
ε
as an impedance knob. Sending
X
h ough he ne wo k a scale
ε
uses
jε
X
o load
X
in o
BεX
, anspo s wi h
¯
ε
, hen eads wi h
ε
Y
. Some in a ian s (mass, expec a ions, ze o-mode, conse ed cha ges) pass losslessly because
¯
ε
and
ε
p ese e hem; sha p, agile in a ian s a e eplaced by hei elaxed e sions while he
signal is AC-ca ied (blu ed). As
ε↓
0 he impedance anishes and he elaxa ion collapses back
o sha p equali y.
P oposi ion 7 (P ope y anspo unde unce ain y).Le Pbe a p ope y o objec s ha is (i)
s able unde e ac s and il e ed limi s, and (ii) mono one wi h espec o he ambien p eo de
≾
on endomo phisms. De ine i s
ε
- elaxa ion P
ε
by “
BεX
has Pup o
≾
”. Then o ai h ul blu s:
P(X)⇐⇒ ∃ε0∀ε⪯ε0:Pε(BεX).
In wo ds: an objec has Pi i has he elaxed p ope y h oughou some unce ain y window,
and eading emo es he elaxa ion wi hou loss.
Ske ch.
(
⇒
) I P(
X
)holds and Pis s able unde e ac s, hen
BεX
inhe i s P
ε
ia
jε
X
and
ε
X
(lax e ac ). (
⇐
) I P
ε
(
BεX
)holds e en ually and
ρX
:
limIBεX∼
=
−−→ X
, s abili y unde il e ed
limi s ans e s P o X; mono onici y in ≾ emo es he laxi y in he limi .
Rema k 8 (Blu as a g aded comonad).The amily (
Bε
)
ε∈I
comes wi h canonical compa ison
maps
Bε⊕δX→Bε
(
BδX
), na u al in
X
, making i a g aded comonad (o dually, a g aded
monad) on
C
. In conc e e analy ic models hese maps a e ealized by Ma ko semig oups o
con olu ion semig oups; in logical models hey co espond o i e a ing a nucleus o closu e
ope a o . Thus he ca ego ical s uc u e behind blu is compa ible wi h s anda d cons uc ions in
p obabili y (Gi y/Ma ko ) and in logic (modal/comonadic iewpoin s).
2
Mo phisms ha espec blu
De ini ion 9 (Blu -mo phism).Gi en blu ed objec s X= (
X, BX
)and Y= (
Y, BY
), a
blu -mo phism ¯
:X→Yis a na u al ans o ma ion
¯
:BX⇒ BY,i.e. ∀ε∈I¯
ε:BεX→BεYand ¯
ε⪯ε′commu e wi h eindexing.
I s sha p pa is := ρY◦limI¯
◦ρ−1
X:X→Ywhene e he blu s a e ai h ul.
Composi ion is poin wise: (
¯g◦¯
)
ε
:=
¯gε◦¯
ε
; iden i ies a e (
id
)
ε
=
id
. Thus blu ed
objec s/mo phisms o m a ca ego y Blu (C).
Rema k 10 (Bidi ec ional obse abili y).A ne wo k is obse able in bo h di ec ions i he e
exis s ¯
†:Y→Xwi h
ρX◦lim
I(¯
†◦¯
)◦ρ−1
X= idX, ρY◦lim
I(¯
◦¯
†)◦ρ−1
Y= idY,
i.e.
¯
is an isomo phism in
Blu
(
C
). This o malizes ha we may swi ch P oposi ion/Answe
a will.
Unison emo al and scale composi ion
We assume eindexing is monoidal: o each
ε, δ ∈I
he e is a canonical compa ison
µε,δ
:
Bε⊕δX→Bε
(
BδX
), na u al in
X
, making (
Bε
)
ε∈I
ag aded comonad (o g aded monad,
depending on he conc e e model). Na u ali y o
¯
means emo ing neighbou hoods happens in
unison:
¯
εcommu es wi h he ansi ion maps Bε′X→BεXand Bε′Y→BεY.
S anding con en ion. We ix na u al
jε
X
as in De ini ion 4, so
Xjε
X
−−→ BεX ε
X
−−→ X
is a lax
e ac pai (objec con ained wi h an unce ain y window).
G and Lemma (sequen ial deblu ing = one-sho deblu ing)
Theo em 11 (G and Lemma (Blu – anspo lemma): sending sha p
⇔
sending blu ed and
eading s epwise).Le X¯
−→Y¯g
−→Zbe blu -mo phisms wi h ai h ul blu s. Then o e e y ε∈I,
ε
Z◦(¯gε◦¯
ε) = ε
Z◦¯gε◦ ε
Y◦¯
ε= (g◦ )◦ ε
X,
and in pa icula , a e aking he limi ε→0( eading),
g◦ =ρZ◦lim
I(¯g◦¯
)◦ρ−1
X=ρZ◦lim
I¯g◦ρ−1
Y◦ρY◦lim
I
¯
◦ρ−1
X.
Thus sending a P oposi ion h ough he ne wo k is equi alen o sending i oge he wi h i s
neighbou hood and eading in successi e s eps.
P oo .
Poin wise iden i y
ε
Z◦¯gε◦¯
ε
= (
g◦
)
◦ ε
X
ollows om na u ali y o he limi cones:
ε
Y
and
ε
Z
a e he legs o
ρY, ρZ
, and
¯
, ¯g
a e na u al ans o ma ions. Taking limi s in
I
yields
he equali ies wi h ρX, ρY, ρZby he uni e sal p ope y o limi s.
Reading a in ini y (no esponse)
I
colimIBX≃
1and
colimIBY≃
1, hen any blu -mo phism
¯
induces he unique a ow 1
→
1
a
∞
; i.e. as neighbou hood g ows wi hou bound bo h P oposi ion and Answe collapse o “no
esponse” in unison.
3
Passing neighbou hoods ai h ully
Gi en any neighbou hood
ε
on he P oposi ion, he ne wo k passes i o he Answe ia
¯
ε
;
he amily
{¯
ε}ε∈I
ensu es he Answe e ains he co esponding neighbou hood. Rescaling o
neighbou hoods is handled by eindexing
I
(e.g.
ε7→ αε
) and he g aded (co)monad cohe ence.
Two canonical models
1.
Analy ic blu (con olu ion).
C
=
Meas
,
BεX
ac s on measu able unc ions by
con olu ion wi h a posi i e ke nel
Kε
(Gaussian/Poisson/compac molli ie ), and on poin s
ia andomiza ion wi h law
Kε
. Then
¯
is a Ma ko ke nel amily commu ing wi h
con olu ion; ρis he decon olu ion limi as ε→0.
2.
Logical blu ( uzzy/en iched).
C
en iched o e he quan ale ([0
,
1]
,≤,·,
1): a p oposi ion
has a neighbou hood o u h alues;
Bε
hickens u h ia a nucleus (closu e ope a o ).
Mo phisms a e [0
,
1]-nonexpansi e maps. Limi s eco e c isp u h; colimi s a
∞
gi e
he i ial u h.
Rema k 12 (S ochas ic uni e sali y).Ins ead o ixed ke nels, le Uεbe andom pe u ba ions
wi h
P
(
|Uε|> δ
)
→
0as
ε→
0. Then
Bε
(
x
) :=
E
[
(
x
+
Uε
)] de ines a blu ed diag am; all
s a emen s abo e hold e ba im. “Gaussian” is me ely a maximally symme ic ins ance.
S ing-diag am iew o blu
We now depic blu mo phisms as s ing diag ams. Wi es a e objec s, small boxes a e p ocesses,
iangles a e eading ( emo ing he blu ).
Symbols
Bε
Blu a scale εRead ε
G and Lemma as a diag am
Sending a p oposi ion
X
h ough a ne wo k
and hen eading is equal o blu ing i s , sending
h ough he blu ed ne wo k ¯
, hen eading. This is he commu a i i y o he diag am:
XBεX BεYY
ε
X¯
ε ε
Y
This s ing-diag am encapsula es he G and Lemma (Blu – anspo lemma):
ε
Y◦¯
ε◦ ε
X= ◦ ε
X.
Bidi ec ionali y
I
¯
is an isomo phism in he blu ca ego y, he diag am wo ks bo h ways: we may slide he
eading iangle ac oss he ne wo k in ei he di ec ion, exp essing he obse abili y in bo h oles
o P oposi ion and Answe .
4
Conclusion: The G and Pu pose o Blu ing
The cen al ole o blu is no me ely echnical bu concep ual.
G and pu pose. Blu ing allows an objec o expand in o a neighbou hood, becoming a bulk
a he han a sha p poin . In his elaxed s a e, he objec has mo e “ene gy” o in e ac wi h
i s en i onmen : i collec s in o ma ion ha may no s ic ly belong o he objec i sel bu o
he ne wo k i a e ses. The neighbou hood unc ions as a p obe o he medium, cap u ing how
he objec and he ne wo k esona e oge he .
In his sense, blu ing is ca aly ic. I does no damage he objec no dis o he ne wo k; ins ead,
i enhances hei in e ac ion, allowing la en s uc u e o su ace. The messages ha su i e
his join e olu ion a e hen a ached back o he objec when he neighbou hood is emo ed
( eading).
Thus, he philosophy o blu is:
P ecision is achie ed no by esis ing elaxa ion, bu by passing h ough i . Blu ing
e eals, in e ac ion e ines, and eading es o es.
In his sense, he applicabili y o blu ing is uni e sal.
Blu and Undecidabili y ( oy bu sha p)
We b ie ly show ha blu ing does no make ha d decision p oblems easy: e en a posi i e,
no malized blu p ese es he hal ing/non–hal ing gap.
Disc e e ime as a blu ed objec
Wo k in
C
=
Meas
. Le
P og
be he se o p og ams (o ini ial s a es) and, o
e∈P og
, le
(
F e
)
∈N
be i s e olu ion wi h a hal ing p edica e
h
:
P og → {
0
,
1
}
( ue exac ly on hal ing
s a es). Fo λ∈(0,1) de ine he ime–blu ke nel Kλ( ) = (1 −λ)λ on N.
The blu ed hal ing mass is he ead-ou
Hλ(e) := X
≥0
Kλ( )h(F e) = ET∼Geom(1−λ)
h(FTe),
i.e. con olu ion in ime wi h a posi i e, no malized ke nel (a Ma ko /semig oup blu in he
sense o ou analy ic model).
Theo em 13 (Undecidabili y su i es blu ).Fo e e y ixed
λ∈
(0
,
1) and e e y p og am
e
wi h hal ing ime τ(e)∈N∪ {∞},
Hλ(e) =
λτ(e)i τ(e)<∞,
0i τ(e) = ∞.Hence {e:Hλ(e)>0} ≡ HALT.
In pa icula , deciding he posi i i y o his blu ed ead-ou is undecidable.
Ske ch in ou language. Kλ
is a posi i e-de ini e molli ie on ime (a blu scale). The e olu ion
e7→
(
F e
)
is a blu -mo phism in o he pa h objec ; he obse able
h
is a nonnega i e map. I
e
hal s a ime
τ
, hen
h
(
F e
) = 0 o
<τ
and 1 he ea e , so
Hλ
(
e
) = (1
−λ
)
P ≥τλ
=
λτ>
0;
i i ne e hal s, he sum is 0. Thus he posi i i y ead-ou a e blu ing is many–one equi alen
o HALT.
5
Rema k 14 (Fi s he amewo k and links o compu able analysis).This is a special case o
ou analy ic blu model:
Bλ
is con olu ion on he ime axis,
¯
λ
is he (Ma ko ) blu –mo phism
induced by he dynamics, and
λ
is he nume ical ead-ou . The undecidabili y hinges only on
posi i i y, no maliza ion, and a s ic posi i i y gap on hal ing—exac ly he bloo in a ian s.
Concep ually his is close in spi i o classical encodings o hal ing in o analy ic p ope ies o
eal- alued unc ions in compu able analysis: blu does no elimina e undecidabili y; i packages
i in o a obus posi i i y gap.
Con inuous ime and spa ial blu (one-line s)
•Con inuous ime. Wi h Ky( ) = ye−y on R≥0and hal ing ime τ(e)∈[0,∞],
Hy(e) = Z∞
0
Ky( )h(F e)d =1{τ(e)<∞} e−yτ(e).
Again, Hy(e)>0i ehal s.
•
Spa ial blu . I hal ing emi s a uni -mass bump a a known loca ion
xe
while non-hal ing
emi s 0, hen any nonnega i e ke nel
ϕy
wi h
ϕy
(0)
>
0yields (
ϕy∗ e
)(
xe
)
>
0i
e
hal s.
Mo al. Blu ing simpli ies analysis by p ese ing posi i i y and beha ing well wi h limi s;
hose same ea u es p ese e he hal ing/non–hal ing dicho omy as a s ic posi i i y gap. In he
ca ego ical language o blu , undecidabili y is he e o e bloo – obus : blu cla i ies, i doesn’
conju e answe s.
1 Two guiding examples o blu
We b ie ly ske ch wo conc e e examples ha can be ead di ec ly in he language o blu ed
objec s and blu –mo phisms: (1) addi ion and mul iplica ion as a blu ed pai o channels; (2)
Weyl equidis ibu ion on he ci cle unde Poisson blu . In bo h cases, he ole o blu is no
deco a i e: i is o ced by he unde lying ha monic analysis, and he ca ego ical pic u e simply
packages ha necessi y.
1.1 Addi ion and mul iplica ion as a blu ed pai
A a e y classical le el, addi ion and mul iplica ion li e on di e en backg ounds:
•addi ion ac s na u ally on (R, dy) ia ansla ions Ta (y) = (y+a);
•mul iplica ion ac s na u ally on ((0,∞), dx/x) ia dila ions Dc (x) = (cx).
A s anda d de ice is o mo e o he log-line: wi h
x
=
ey
he mul iplica i e g oup (0
,∞
)wi h
Haa measu e
dx/x
becomes (
R, dy
), and he Mellin ans o m on unc ions o
x
becomes he
Fou ie ans o m on unc ions o
y
. This is he p ecise sense in which mul iplica i e s uc u e is
ablu ed copy o addi i e s uc u e.
Fix he base ca ego y
C
=
Meas
(o , i p e e ed, he co esponding
L2
Hilbe spaces).
Conside wo blu ed objec s:
X×:= (0,∞),B×,X+:= R,B+,
whe e:
6
•B×
ε
is log–Gaussian blu on he mul iplica i e line: i
U
:
L2
((0
,∞
)
, dx/x
)
→L2
(
R, dy
)is
he log-change isome y (U )(y) = ey/2 (ey), and Gεis he Gaussian o a iance ε, hen
B×
ε := U−1Gε∗(U ),
i.e. blu in log-coo dina es, pulled back o he x-line;
•B+
τis o dina y Gaussian blu on he addi i e line:
B+
τg:= Gτ∗g, g :R→C.
The so addi ion / so mul iplica ion pic u e can hen be ead as ollows.
•
On he addi i e side, +is sha p: we a e ee o wo k wi h
g1⊕g2
de ined by (
g1⊕g2
)(
y
) =
g1(y) + g2(y)o wi h con olu ion in y, depending on he le el o s uc u e we ack.
•
On he mul iplica i e side,
×
is sha p: we wo k wi h poin wise p oduc in
x
o wi h
mul iplica i e con olu ion
( 1∗× 2)(x) = Z∞
0
1
x
u 2(u)du
u.
•
The link be ween he wo channels is he isome y
U
, which in e wines mul iplica ion on
(0,∞)wi h ansla ion in he spec al a iable on he log-line.
The c ucial ac is ha Fou ie on he log-line sa is ies a genuine Heisenbe g- ype unce ain y:
i
b
h
(
ξ
)is he Fou ie ans o m o
h
(
y
), hen one canno simul aneously ha e bo h
h
and
b
h
sha ply localized. Conc e ely, he e is a lowe bound o he o m
Va y(h) Va ξ(b
h)≥1
4
(up o he usual no maliza ion choices). On he mul iplica i e side his eads: you canno
make he blu in log-
x
a bi a ily small while also keeping he mul iplica i e spec al con en
a bi a ily sha p.
In he language o blu ed objec s:
• he amilies (B×
ε)ε>0and (B+
τ)τ>0a e blu unc o s on X×and X+;
• he log-change Uinduces a blu -mo phism
Swi ch×→+:X×−→ X+,Swi ch×→+ε:= B+
τ(ε)◦U◦B×
ε,
whe e he blu indices a e coupled so ha ε·τ(ε)≥c > 0 o a uni e sal cons an .
The Heisenbe g inequali y is exac ly he s a emen ha we canno make bo h
ε
and
τ
(
ε
)
a bi a ily small: he e is a ha d lowe bound on he blu budge equi ed o swi ch be ween he
addi i e and mul iplica i e lenses. In o he wo ds, any blu -mo phism ha genuinely in e wines
he wo channels mus pay wi h a nonze o unce ain y window.
So addi ion and so mul iplica ion hen appea as ope a ions de ined a he blu ed le el:
• o add mul iplica i e da a “as i ” i we e addi i e, we
1. blu in he mul iplica i e channel a some ε,
2. swi ch o he addi i e channel ia Swi ch×→+,
3. pe o m he na i e addi i e ope a ion he e,
7
4. and ead back sha ply.
•
o mul iply addi i e da a “as i ” i we e mul iplica i e, we do he symme ic cons uc ion
using he in e se blu -mo phism Swi ch+→×.
The G and Lemma hen says p ecisely: sending sha p da a h ough his ne wo k is equi alen
o sending i oge he wi h i s neighbou hood and eading in successi e s eps. The analy ic
con en (Fou ie –Mellin duali y and Heisenbe g unce ain y) ensu es ha such a ne wo k canno
exis wi h ze o blu ; he ca ego ical con en ensu es ha , once we ix a blu budge , p ope ies
ha a e s able unde e ac s and il e ed limi s can be anspo ed be ween he wo channels.
Thus he addi ion/mul iplica ion pai is a pa icula ly i id example o a blu ed objec and
blu –mo phism ha a e no op ional: blu is no some hing we choose o add; i is some hing
he ha monic analysis o ces us o accoun o .
1.2 Weyl equidis ibu ion unde Poisson blu
As a second example, we ake a classical heo em whe e blu appea s almos una oidably in
s anda d p oo s: Weyl’s equidis ibu ion o {nα}modulo 1.
Le
T
=
R/
2
πZ
be he ci cle wi h Lebesgue p obabili y measu e
m
, and ix
α∈R
. Conside
he o bi xn:= nα (mod 2π)and he empi ical measu es
µN:= 1
N
N
X
n=1
δxn.
Weyl’s heo em says: i
α/
2
π
is i a ional, hen
µN
con e ges weak-* o
m
, i.e. he o bi is
equidis ibu ed.
In ou language, we de ine a blu on T ia he Poisson ke nel
P (θ) = 1− 2
1−2 cos θ+ 2,0< < 1,
and se
B
:=
P ∗
o
∈L1
(
T
). The amily (
B
)
0< <1
is a s anda d app oxima e iden i y:
lim
↑1B = in Lp(T)
o 1
≤p < ∞
, and
B
annihila es high Fou ie modes exponen ially: i
(
θ
) =
Pk∈Zˆ
(
k
)
eikθ
,
hen
B (θ) = X
k∈Z
|k|ˆ
(k)eikθ.
De ine blu ed empi ical measu es
µN, := µN∗P ,0< < 1,
so ha o a con inuous ,
ZT
dµN, =ZT
(B )dµN=1
N
N
X
n=1
(B )(xn).
Now se up he blu ed objec
XT:= (T,B), B := B( ) : L1(T)→L1(T),
wi h
∈I
:= (0
,
1), pa ially o de ed by
⪯ ′
i
≤ ′
. The sha p limi
ρ
is
7→ lim ↑1B
=
(when i exis s), and he exhaus ion a “in ini y” is he collapse o cons an s as ↓0.
8
The p ope y o in e es is:
P(µ•) : µNw∗
−−→ mas N→ ∞ (equidis ibu ion).
I s blu ed e sion a scale is:
P (µ•) : µN, w∗
−−→ mas N→ ∞,
i.e. equidis ibu ion holds a e Poisson blu a ixed < 1.
Classically, Weyl’s c i e ion educes P(µ•) o he decay o exponen ial sums:
1
N
N
X
n=1
eikxn−→ 0 (N→ ∞) o all k∈Z {0}.
In he blu ed pic u e, we es agains
B
ins ead. On Fou ie coe icien s, his inse s he
ex a ac o |k|, making he es ima es s ic ly easie :
1
N
N
X
n=1
(B )(xn) = X
k∈Z
|k|ˆ
(k)1
N
N
X
n=1
eikxn.
Fo each ixed
<
1, he ail o e la ge
|k|
is exponen ially supp essed, and he analysis educes
o ini ely many equencies wi h a buil -in damping ac o
|k|
. In o he wo ds, P
(
µ•
)is s ic ly
easie o e i y han P(µ•).
F om he iewpoin o he p ope y- anspo P oposi ion ( anspo unde unce ain y
window), equidis ibu ion is:
•
s able unde con olu ion wi h an app oxima e iden i y (Poisson blu ): i
µN, →m
o a
ixed < 1and B →id as ↑1, hen µN→m;
•
s able unde il e ed limi s in
and mono one wi h espec o he na u al p eo de on
ke nels (P ≾P ′i ≤ ′in he sense o poin wise domina ion).
Thus, in his example, we may legi ima ely wo k a any ixed blu le el
<
1, p o e P
(
µ•
)
in he blu ed wo ld (whe e he Fou ie side is be e beha ed), and hen le
↑
1 o eco e he
sha p s a emen P(
µ•
). The Poisson blu is no an a bi a y smoo hing ick: i is p ecisely a
blu unc o B wi h a eading map ρ ha sa is ies he hypo heses o ou anspo p inciple.
In summa y:
•
addi ion s. mul iplica ion illus a es how blu is o ced by Fou ie –Mellin unce ain y
when we y o swi ch be ween algeb aic channels;
•
Weyl equidis ibu ion on he ci cle illus a es how blu is a na u al echnical lens: we p o e
a p ope y in a blu ed egime whe e he analysis is easie , and hen ead back o he
sha p egime using s abili y unde limi s.
Bo h examples i cleanly in o he ca ego ical amewo k de eloped abo e and show ha blu is
no me ely a philosophical deco a ion, bu a ma hema ically ine i able media o in si ua ions
whe e di e en s uc u es mus be made o communica e.
Re e ences
[1]
F. W. Law e e, Me ic spaces, gene alized logic, and closed ca ego ies, Rendicon i del
Semina io Ma ema ico e Fisico di Milano 43 (1973), 135–166.
[2]
G. M. Kelly, Basic Concep s o En iched Ca ego y Theo y, London Ma hema ical Socie y
Lec u e No e Se ies, ol. 64, Camb idge Uni e si y P ess, 1982.
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