A Categorical Approach to Rough Equality Algebras via Approximation Functors

A CATEGORICAL APPROACH TO ROUGH EQUALITY
ALGEBRAS VIA APPROXIMATION FUNCTORS
JOAQUIM REIZI HIGUCHI
Abs ac .
In his pape we de elop a ca ego ical amewo k o cong ue-
nce-based ough se heo y in he se ing o equali y algeb as. We
in oduce he ca ego y o app oxima ion equali y algeb as, deno ed by
AppEqAlg
, whose objec s a e pai s (
E, θ
) consis ing o an equali y
algeb a
E
and a cong uence ela ion
θ
on
E
, and whose mo phisms a e
cong uence-p ese ing homomo phisms. Fo a ixed (
E, θ
) we analyse
he associa ed ough app oxima ions on
P
(
E
), showing ha he uppe
and lowe app oxima ions gi e ise, espec i ely, o a closu e ope a o
and an in e io ope a o on he powe se la ice, and ha he uppe
app oxima ion es ic s o a closu e ope a o on he subalgeb a la ice
Sub
(
E
). Wi hin his se ing we iden i y he exac (i.e.
θ
-de inable)
subalgeb as and p o e ha hei comple e la ice is isomo phic o he
subalgeb a la ice o he quo ien algeb a
E/θ
. On he ca ego ical side, we
de ine he quo ien unc o
U
:
AppEqAlg →EqAlg
and he diagonal
embedding
G
:
EqAlg →AppEqAlg
, and es ablish an adjunc ion
U⊣G
. Finally, we show ha he o ge ul unc o
V
:
AppEqAlg →
EqAlg
is opological, so ha limi s and colimi s in
AppEqAlg
a e
ob ained om hose in
EqAlg
by equipping he unde lying algeb a wi h
canonical ini ial and inal cong uences. This p o ides a uni ied ca ego ical
pe spec i e on oughness in equali y algeb as.
1. In oduc ion
The s udy o logical algeb as has long been a cen al heme in non-classical
logic, p o iding he algeb aic seman ics o a ious logical sys ems. Among
hese, equali y algeb as we e in oduced by Jenei [
1
] as a specialized
s uc u e o uzzy ype heo y. Unlike esidua ed la ices o MV-algeb as
which ely p ima ily on implica ion, equali y algeb as ake he equali y
connec i e ( uzzy equi alence) as he p imi i e ope a ion. This shi in
pe spec i e has led o a ui ul a ea o esea ch, explo ing hei opological
and algeb aic p ope ies (e.g., [2]).
Pa allel o hese de elopmen s, ough se heo y, p oposed by Pawlak [
3
],
has es ablished i sel as a powe ul ma hema ical ool o handling unce ain y
and indisce nibili y. While o iginally o mula ed in a se - heo e ic con ex ,
he heo y has been ex ensi ely applied o algeb aic s uc u es. The no ion
o “ ough algeb aic s uc u es” in es iga es how algeb aic ope a ions in e ac
wi h app oxima ion spaces. Signi ican con ibu ions ha e been made in his
Da e: No embe 23, 2025.
2020 Ma hema ics Subjec Classi ica ion. 06D20, 03G25, 18B30, 54B30.
Key wo ds and ph ases. Equali y algeb a, Rough se s, App oxima ion unc o s, Adjunc-
ion, Topological ca ego y, Closu e ope a o .
1
2 JOAQUIM REIZI HIGUCHI
di ec ion, applying oughness o g oups, ings, and a ious logical algeb as
(see, o ins ance, [4]).
Mos o he exis ing wo k on ough algeb aic s uc u es is local in na u e:
one ixes an algeb a
A
oge he wi h an equi alence o cong uence ela ion and
hen s udies, o example, whe he he lowe app oxima ion o a subalgeb a
is again a subalgeb a, o wha he p ope ies o ough ideals o ough il e s
a e. While such esul s a e undamen al, hey do no by hemsel es p o ide a
global pic u e o how oughness beha es ac oss he whole a ie y. In pa icula ,
a sys ema ic ca ego ical amewo k in which ough app oxima ions a e
ea ed no only as ope a o s on a gi en algeb a, bu as pa o unc o ial
cons uc ions be ween ca ego ies, has no been ully de eloped in he speci ic
se ing o equali y algeb as.
In his pape we ake a s ep in his di ec ion by es ablishing a ca ego ical
ounda ion o ough equali y algeb as based on cong uence-induced app oxi-
ma ion. Ou s a ing poin is he obse a ion ha , once a cong uence
θ
is
ixed on an equali y algeb a
E
, he associa ed ough app oxima ions ha e
wo complemen a y inca na ions:
•
on he algeb aic side,
θ
de e mines uppe and lowe ough app oxima-
ions on he powe se
P
(
E
), and in pa icula a closu e ope a o on
he la ice o subalgeb as o
E
oge he wi h a dual in e io ope a o
on P(E);
•
on he ca ego ical side, he same cong uence gi es ise o a quo ien
homomo phism
E→E/θ
, and quo ien s by cong uences assemble
in o a unc o
U
om a ca ego y o “ ough equali y algeb as” o he
base ca ego y o equali y algeb as.
The aim o he pape is o o ganise hese ac s in a uni o m ca ego ical
amewo k. Mo e p ecisely, we in oduce a ca ego y whose objec s a e equali y
algeb as equipped wi h a chosen cong uence ( hough o as an app oxima ion
s uc u e) and whose mo phisms a e cong uence-p ese ing homomo phisms.
Inside his ca ego y we iden i y:
•
aquo ien unc o
U
sending (
E, θ
) o he quo ien equali y algeb a
E/θ;
•
adiagonal embedding
G
which sends an equali y algeb a
Y
o (
Y,
∆
Y
),
whe e ∆Yis he diagonal cong uence;
•
a o ge ul unc o
V
disca ding he cong uence and emembe ing
only he unde lying equali y algeb a.
We show ha
U
and
G
o m an adjoin pai
U⊣G
, hus exp essing he
uni e sal p ope y o he quo ien
E/θ
pu ely ca ego ically. The o ge ul
unc o
V
u ns ou o be opological in he sense o Ad´amek, He lich and
S ecke , so ha limi s and colimi s in ou ough ca ego y can be cons uc ed
by equipping he co esponding limi s and colimi s in
EqAlg
wi h canonical
ini ial and inal cong uences.
Main Con ibu ions. The main con ibu ions o his pape can be sum-
ma ised as ollows:
(1)
We in oduce he ca ego y
AppEqAlg
, whose objec s a e pai s
(
E, θ
) consis ing o an equali y algeb a
E
and a cong uence ela ion
A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 3
θ
on
E
, and whose mo phisms a e cong uence-p ese ing homomo -
phisms. This ca ego y o malises he uni e se o cong uence-based
app oxima ion s uc u es o e equali y algeb as.
(2)
Fo a ixed equali y algeb a
E
and cong uence
θ
, we s udy he induced
ough app oxima ions on
P
(
E
). We show ha he uppe and lowe
app oxima ions de ine, espec i ely, a closu e ope a o and an in e io
ope a o on he comple e la ice
P
(
E
), and ha he uppe app oxima-
ion es ic s o a closu e ope a o on he subalgeb a la ice
Sub
(
E
).
Wi hin his se ing we iden i y he exac (i.e.
θ
-de inable) subalgeb as
and show ha hey o m a comple e subla ice o Sub(E).
(3) On he ca ego ical side, we de ine he quo ien unc o
U:AppEqAlg →EqAlg
and he diagonal embedding unc o
G:EqAlg →AppEqAlg
. We
p o e ha
U
is he le adjoin o
G
(
U⊣G
). This adjunc ion
p o ides a p ecise ca ego ical o mula ion o he passage om a ough
s uc u e (
E, θ
) o i s c isp quo ien
E/θ
and connec s he uni o
he adjunc ion wi h he la ice o exac subalgeb as.
(4)
We p o e ha he o ge ul unc o
V:AppEqAlg →EqAlg
is
opological. We explici ly cons uc ini ial and inal li s o s uc-
u ed sou ces and sinks, and deduce ha (co)limi s in
AppEqAlg
a e ob ained by equipping (co)limi s in
EqAlg
wi h canonical ini ial
o inal cong uences. This si ua es ough equali y algeb as wi hin
he gene al heo y o opological conc e e ca ego ies.
S uc u e o he Pape . The emainde o his pape is o ganised as
ollows. Sec ion 2 e iews he necessa y p elimina ies on equali y algeb as and
ough se heo y. Sec ion 3 is de o ed o he algeb aic s udy o cong uence-
induced ough app oxima ions on an equali y algeb a: we desc ibe uppe
and lowe app oxima ions as closu e and in e io ope a o s on
P
(
E
), analyse
he es ic ion o he uppe app oxima ion o he subalgeb a la ice, and
in oduce he no ion o exac subalgeb as, showing ha hey a e in bijec i e
co espondence wi h subalgeb as o he quo ien algeb a. Sec ion 4 de elops
he ca ego ical amewo k, de ining he ca ego y
AppEqAlg
, he unc o s
U
and
G
, and p o ing he adjunc ion
U⊣G
. Sec ion 5 in es iga es he
opological aspec s, p o ing ha he o ge ul unc o
V:AppEqAlg →
EqAlg
is opological and desc ibing ini ial and inal li s. Sec ion 6 p o ides
a conc e e example based on a h ee-elemen equali y algeb a o illus a e
he cons uc ions, and Sec ion 7 concludes he pape wi h a b ie discussion
o possible di ec ions o u u e wo k.
2. P elimina ies
In his sec ion, we e iew he basic de ini ions and p ope ies o equali y
algeb as and ough se heo y ha a e essen ial o he subsequen discussions.
Fo mo e de ails on equali y algeb as, we e e he eade o [1, 2].
2.1. Equali y Algeb as. Equali y algeb as we e in oduced by Jenei as a
s uc u e dealing wi h he equali y connec i e in uzzy logic, inspi ed by he
ac ha uzzy ype heo y elies hea ily on equali y a he han implica ion.
4 JOAQUIM REIZI HIGUCHI
De ini ion 2.1. An algeb a
E
= (
E, ∧,∼,
1) o ype (2
,
2
,
0) is called an
equali y algeb a i i sa is ies he ollowing condi ions o all x, y, z ∈E:
(E1) (E, ∧,1) is a commu a i e semila ice wi h op elemen 1.
(E2) x∼y=y∼x(Commu a i i y o ∼).
(E3) x∼x= 1.
(E4) x∼1 = x.
(E5) x≤y≤z
=
⇒x∼z≤y∼z
and
x∼z≤x∼y
, whe e
x≤y
i
and only i x∧y=x.
(E6) x∼y≤x∧z∼y∧z.
(E7) x∼y≤(x∼z)∼(y∼z).
We in e p e he ope a ion
∼
as a uzzy equali y (bi-implica ion). An
equali y algeb a is bounded i he e exis s an elemen 0
∈E
such ha 0
≤x
o all
x∈E
. In his pape , unless o he wise s a ed, we do no assume
boundedness, bu ou examples ypically in ol e bounded chains.
C ucial o ou s udy is he no ion o cong uence ela ions, as hey o m
he basis o he app oxima ion space.
De ini ion 2.2. Le
E
= (
E, ∧,∼,
1) be an equali y algeb a. An equi alence
ela ion
θ
on
E
is called a cong uence ela ion i i is compa ible wi h he
ope a ions:
(x, y)∈θand (u, )∈θ=⇒(x∧u, y ∧ )∈θand (x∼u, y ∼ )∈θ
o all x, y, u, ∈E.
The se o all cong uence ela ions on
E
is deno ed by
Con
(
E
). I is a
well-known esul in uni e sal algeb a ha
Con
(
E
) o ms a comple e la ice
unde se inclusion.
2.2. Rough Se Theo y. Rough se heo y, p oposed by Pawlak [
3
], p o-
ides a o mal ool o dealing wi h unce ain y a ising om indisce nibili y.
De ini ion 2.3. An app oxima ion space is a pai (
U, R
), whe e
U
is a non-emp y se ( he uni e se) and
R
is an equi alence ela ion on
U
.
Fo any subse
X⊆U
, he lowe app oxima ion
R
(
X
) and he uppe
app oxima ion R(X) a e de ined as:
R(X) = {x∈U|[x]R⊆X},
R(X) = {x∈U|[x]R∩X=∅},
whe e [x]Rdeno es he equi alence class o xwi h espec o R.
A subse
X
is called exac (o c isp) wi h espec o
R
i
R
(
X
) =
R
(
X
).
O he wise, i is called ough.
In ecen yea s, his heo y has been applied o a ious algeb aic s uc u es
such as g oups, ings, and logical algeb as (e.g., [
4
]). In such algeb aic
con ex s, he uni e se
U
is eplaced by an algeb aic s uc u e
A
, and
R
is ypically chosen o be a cong uence ela ion
θ
. The main in e es lies
in whe he he app oxima ions o a subalgeb a a e again subalgeb as. As
we shall see in Sec ion 3, equali y algeb as p o ide a ich se ing o such
in es iga ions.
A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 5
3. Algeb aic P ope ies o Rough App oxima ions
In his sec ion, we in es iga e he algeb aic p ope ies o ough app oxima-
ions induced by a cong uence ela ion on an equali y algeb a. While many
o he cons uc ions can be o mula ed in he gene al se ing o uni e sal
algeb as, we es ic ou sel es o equali y algeb as, since his is he ambien
a ie y o he ca ego ical amewo k de eloped la e .
Th oughou his sec ion, le
E
= (
E, ∧,∼,
1) be an equali y algeb a and
le
θ
be a cong uence ela ion on
E
. We deno e by
P
(
E
) he powe se o
E
, and by
Sub
(
E
) he se o all subalgeb as o
E
, which o ms a comple e
la ice unde se inclusion.
3.1. Rough app oxima ions on subse s and subalgeb as. We i s
ex end he de ini ion o ough app oxima ions o a bi a y subse s o
E
; he
case o subalgeb as will hen be ob ained by es ic ion.
De ini ion 3.1. Fo any subse
X⊆E
, he uppe app oxima ion
θ
(
X
)
and he lowe app oxima ion
θ
(
X
) o
X
wi h espec o
θ
a e de ined by:
θ(X) = {x∈E|[x]θ∩X=∅},
θ(X) = {x∈E|[x]θ⊆X}.
I
A
is a subalgeb a o
E
(deno ed
A≤ E
), we w i e
θ
(
A
) and
θ
(
A
) o he
app oxima ions o he unde lying se A.
These ope a o s sa is y he usual Pawlak- ype inequali ies a he le el o
he powe se .
Lemma 3.2. Fo all X, Y ⊆E, he ollowing hold:
(i) θ(X)⊆X⊆θ(X).
(ii) I X⊆Y, hen θ(X)⊆θ(Y)and θ(X)⊆θ(Y).
(iii) θ(θ(X)) = θ(X)and θ(θ(X)) = θ(X).
P oo .
(i) I
x∈θ
(
X
), hen by de ini ion [
x
]
θ⊆X
, and in pa icula
x∈X
.
Con e sely, i
x∈X
, hen [
x
]
θ∩X
=
∅
(because
x∈
[
x
]
θ∩X
), hence
x∈θ(X).
(ii) Mono onici y is immedia e: i
X⊆Y
and [
x
]
θ⊆X
, hen [
x
]
θ⊆Y
, so
x∈θ(Y); simila ly, i [x]θ∩X=∅, hen [x]θ∩Y=∅.
(iii) Fo idempo ence o
θ
, ex ensi i y (i) gi es
θ
(
X
)
⊆θ
(
θ
(
X
)). Fo he
con e se inclusion, le
x∈θ
(
θ
(
X
)). Then he e exis s
y∈θ
(
X
) such ha
(
x, y
)
∈θ
. By de ini ion o
θ
(
X
), he e exis s
z∈X
wi h (
y, z
)
∈θ
. By
ansi i i y o θwe ob ain (x, z)∈θ, so [x]θ∩X=∅and hence x∈θ(X).
The dual a gumen gi es idempo ence o
θ
: i
x∈θ
(
X
), hen o any
y∈
[
x
]
θ
we ha e [
y
]
θ
= [
x
]
θ⊆X
, so
y∈θ
(
X
) and hus [
x
]
θ⊆θ
(
X
), i.e.
x∈θ(θ(X)). Toge he wi h (i) his yields θ(θ(X)) = θ(X). □
Theo em 3.3 (Uppe app oxima ion as closu e).The ope a o
θ
:
P
(
E
)
→
P
(
E
)is a closu e ope a o on he comple e la ice (
P
(
E
)
,⊆
). Tha is, o
all X, Y ⊆E:
(a) X⊆θ(X) (ex ensi i y);
(b) X⊆Y=⇒θ(X)⊆θ(Y) (mono onici y);
(c) θ(θ(X)) = θ(X) (idempo ence).

6 JOAQUIM REIZI HIGUCHI
In pa icula , whene e
A∈Sub
(
E
), he se
θ
(
A
)is a subalgeb a o
E
, and
he es ic ion o
θ
o
Sub
(
E
)de ines a closu e ope a o on he subalgeb a
la ice Sub(E).
P oo .
The h ee axioms o a closu e ope a o on
P
(
E
) a e exac ly Lemma 3.2
(i)–(iii). I emains o show ha i
A
is a subalgeb a o
E
, hen
θ
(
A
) is again
a subalgeb a.
Le
A≤ E
and le
x, y ∈θ
(
A
). By de ini ion, he e exis
a, b ∈A
such
ha (
x, a
)
∈θ
and (
y, b
)
∈θ
. Since
θ
is a cong uence, i is compa ible wi h
he ope a ions ∧and ∼, so
(x∧y, a ∧b)∈θand (x∼y, a ∼b)∈θ.
Because Ais a subalgeb a, a∧b∈Aand a∼b∈A. Hence
[x∧y]θ∩A=∅and [x∼y]θ∩A=∅,
which shows
x∧y∈θ
(
A
) and
x∼y∈θ
(
A
). Mo eo e , 1
∈A
and (1
,
1)
∈θ
,
so 1
∈θ
(
A
). The e o e
θ
(
A
) is closed unde
∧
,
∼
and con ains 1, i.e.
θ
(
A
) is
a subalgeb a.
Since
θ
is ex ensi e, mono one and idempo en on
P
(
E
), and maps
Sub
(
E
)
in o i sel , i s es ic ion
θ↾Sub(E):Sub(E)→Sub(E)
is a closu e ope a o on he comple e la ice Sub(E). □
Theo em 3.4 (Lowe app oxima ion as in e io ).The ope a o
θ
:
P
(
E
)
→
P(E)is an in e io ope a o on (P(E),⊆). Tha is, o all X, Y ⊆E:
(a) θ(X)⊆X(con ac i i y);
(b) X⊆Y=⇒θ(X)⊆θ(Y) (mono onici y);
(c) θ(θ(X)) = θ(X) (idempo ence).
P oo .
All h ee p ope ies ha e al eady been e i ied in Lemma 3.2. Con-
ac i i y is Lemma 3.2(i), mono onici y is Lemma 3.2(ii), and idempo ence
is Lemma 3.2(iii). □
Rema k 3.5. In con as o he uppe app oxima ion, he lowe app ox-
ima ion
θ
does no in gene al map
Sub
(
E
) in o i sel : o a subalgeb a
A≤ E
, he se
θ
(
A
) is always a
θ
-sa u a ed subse o
A
, bu i need no be
a subalgeb a unless addi ional hypo heses a e imposed. We he e o e wo k
wi h
θ
as a closu e ope a o on
Sub
(
E
), while
θ
will be used a he le el o
P(E) and o cha ac e izing exac (i.e. θ-sa u a ed) subalgeb as.
3.2. Exac subalgeb as and he quo ien algeb a. We now isola e hose
subalgeb as ha a e “c isp” wi h espec o he gi en cong uence, i.e. unions
o θ-classes.
De ini ion 3.6. A subalgeb a
S∈Sub
(
E
) is called exac (o
θ
-de inable)
i θ(S) = S. We deno e he amily o all exac subalgeb as by
Tθ(E) = {S∈Sub(E)|θ(S) = S}.
I
S
is exac , hen i is
θ
-sa u a ed by de ini ion: whene e
x∈S
and
(
x, y
)
∈θ
, we ha e [
y
]
θ
= [
x
]
θ⊆θ
(
S
) =
S
, so
y∈S
. Con e sely, any
θ
-sa u a ed subalgeb a
S
sa is ies
θ
(
S
) =
S
, since o e e y
x∈θ
(
S
) we ha e
[x]θ∩S=∅, hence [x]θ⊆Sby sa u a ion and hus x∈S.
A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 7
Lemma 3.7. I S∈ Tθ(E), hen θ(S) = S=θ(S).
P oo .
We al eady obse ed ha
θ
(
S
) =
S
by de ini ion. Fo he lowe
app oxima ion, ecall ha
S
is a union o
θ
-classes. I
x∈S
, hen [
x
]
θ⊆S
,
hence
x∈θ
(
S
). Thus
S⊆θ
(
S
). On he o he hand,
θ
(
S
)
⊆S
holds o
e e y subse Sby con ac i i y, so θ(S) = S.□
The nex heo em shows ha exac subalgeb as can be iden i ied wi h
subalgeb as o he quo ien algeb a E/θ in a la ice- heo e ic way.
Theo em 3.8. Le
E/θ
be he quo ien equali y algeb a. The comple e la ice
o exac subalgeb as
Tθ
(
E
)is isomo phic o he comple e la ice o subalgeb as
o he quo ien algeb a, Sub(E/θ).
P oo .
Le
q
:
E → E/θ
be he canonical quo ien homomo phism de ined by
q(x)=[x]θ o all x∈E.
S ep 1: De ini ion and basic p ope ies o he maps. De ine
Ψ : Sub(E/θ)→ Tθ(E),Ψ(T) = q−1(T) = {x∈E|[x]θ∈T}
o each subalgeb a
T≤ E/θ
. Since
q
is a homomo phism o algeb as, he
in e se image o a subalgeb a is a subalgeb a. Mo eo e , i
x∈q−1
(
T
) and
(
x, y
)
∈θ
, hen
q
(
x
) =
q
(
y
), so
q
(
y
)
∈T
and hence
y∈q−1
(
T
). Thus
q−1
(
T
)
is θ-sa u a ed, so Ψ(T)∈ Tθ(E).
Con e sely, de ine
Φ : Tθ(E)→Sub(E/θ),Φ(S) = q(S) = {[x]θ|x∈S}
o each exac subalgeb a
S∈ Tθ
(
E
). As
q
is a su jec i e homomo phism
and Sis a subalgeb a, q(S) is a subalgeb a o E/θ.
S ep 2: Φand Ψa e in e se o each o he . Le T∈Sub(E/θ). Then
Φ(Ψ(T)) = q(q−1(T)) = T,
since qis su jec i e. Thus Φ ◦Ψ = idSub(E/θ).
Now le S∈ Tθ(E). Then
Ψ(Φ(S)) = q−1(q(S)).
Clea ly
S⊆q−1
(
q
(
S
)) o e e y subse
S⊆E
. To show he e e se inclusion,
le
x∈q−1
(
q
(
S
)). Then
q
(
x
)
∈q
(
S
), so he e exis s
s∈S
wi h
q
(
x
) =
q
(
s
),
i.e. (
x, s
)
∈θ
. Since
S
is
θ
-sa u a ed, we ha e
x∈S
. Hence
q−1
(
q
(
S
))
⊆S
,
and he e o e Ψ(Φ(S)) = S. Thus Ψ ◦Φ = idTθ(E).
S ep 3: P ese a ion o a bi a y mee s and joins. I emains o check ha
Ψ is a comple e la ice isomo phism. Le (
Ti
)
i∈I
be a amily o subalgeb as
o E/θ. Then
Ψ
i∈I
Ti=q−1
i∈I
Ti=
i∈I
q−1(Ti) =
i∈I
Ψ(Ti),
so Ψ p ese es a bi a y mee s (in e sec ions).
Fo joins, ecall ha he join in he la ice o subalgeb as is he subalgeb a
gene a ed by he union:
_
i∈I
Ti=⟨[
i∈I
Ti⟩.
8 JOAQUIM REIZI HIGUCHI
Using s anda d ac s om uni e sal algeb a abou in e se images o gene a ed
subalgeb as unde su jec i e homomo phisms, we ob ain
Ψ_
i∈I
Ti=q−1[
i∈I
Ti=q−1[
i∈I
Ti
=[
i∈I
q−1(Ti)=_
i∈I
q−1(Ti) = _
i∈I
Ψ(Ti).
Thus Ψ p ese es a bi a y joins as well. Since Φ is he in e se map o Ψ,
i ollows ha bo h Ψ and Φ a e comple e la ice isomo phisms be ween
Sub(E/θ) and Tθ(E). □
This heo em shows ha he “exac ” (i.e.
θ
-de inable) pa o he ough
s uc u e on
E
is comple ely cap u ed by he o dina y algeb a o he quo ien
E/θ
. In ca ego ical e ms,
Tθ
(
E
) can be iewed as he ull subla ice o
Sub
(
E
) consis ing o
θ
-closed subobjec s, and Theo em 3.8 iden i ies his
subla ice wi h he subobjec la ice o he co esponding quo ien in he
ca ego y o equali y algeb as.
4. Ca ego ical F amewo k: The Adjunc ion
In his sec ion, we li he local algeb aic p ope ies o Sec ion 3 o a global
ca ego ical se ing. We in oduce he ca ego y o app oxima ion equali y
algeb as, deno ed by
AppEqAlg
, in which objec s a e equali y algeb as
equipped wi h a chosen cong uence ( he “app oxima ion s uc u e”). The
passage om a ough objec (
E, θ
) o i s “c isp” quo ien
E/θ
hen becomes
a unc o
U:AppEqAlg −→ EqAlg,
and ou main esul is ha his unc o is le adjoin o a canonical embedding
G:EqAlg −→ AppEqAlg.
In pa icula , he uni e sal p ope y o he quo ien
E/θ
is exp essed pu ely
ca ego ically as an adjunc ion
U⊣G
. Combined wi h Theo em 3.8, his
shows ha he exac (i.e.
θ
-de inable) subalgeb as o (
E, θ
) a e con olled by
he uni o his adjunc ion.
4.1. The Ca ego y
AppEqAlg
.We i s spell ou he ca ego y o equali y
algeb as endowed wi h a dis inguished cong uence ela ion.
De ini ion 4.1. The ca ego y
AppEqAlg
o app oxima ion equali y algeb as
is de ined as ollows:
(1)
Objec s: Pai s (
E, θ
), whe e
E
is an equali y algeb a and
θ
is a
cong uence ela ion on E.
(2) Mo phisms: A mo phism
: (E1, θ1)−→ (E2, θ2)
is a homomo phism o equali y algeb as :E1→E2such ha
(x, y)∈θ1=⇒( (x), (y)) ∈θ2 o all x, y ∈E1.
We call such mo phisms ela ion-p ese ing homomo phisms.
(3) Composi ion is gi en by he usual composi ion o unc ions.
(4) Iden i ies a e he iden i y homomo phisms idE:E→E.
A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 9
I is s aigh o wa d o e i y ha
AppEqAlg
is indeed a ca ego y:
he iden i y on (
E, θ
) is ela ion-p ese ing, and he composi e o ela ion-
p ese ing homomo phisms is again ela ion-p ese ing.
4.2. App oxima ion (quo ien ) unc o s. We now de ine he basic unc-
o s ela ing
AppEqAlg
o he ambien ca ego y
EqAlg
o equali y algeb as
and homomo phisms.
De ini ion 4.2 (Quo ien unc o ).We de ine a unc o
U:AppEqAlg −→ EqAlg
by:
•On objec s: Fo any (E, θ)∈AppEqAlg,
U(E, θ) = E/θ.
Tha is,
U
sends an equali y algeb a oge he wi h a cong uence o
he co esponding quo ien equali y algeb a.
•On mo phisms: Fo a mo phism
: (E1, θ1)−→ (E2, θ2),
we de ine U( ): E1/θ1→E2/θ2by
U( )([x]θ1)=[ (x)]θ2, x ∈E1.
The map
U
(
) is well-de ined: i [
x
]
θ1
= [
y
]
θ1
, hen (
x, y
)
∈θ1
, and since
is ela ion-p ese ing, (
(
x
)
,
(
y
))
∈θ2
, so [
(
x
)]
θ2
= [
(
y
)]
θ2
. One easily
checks ha
U
(
id(E,θ)
) =
idE/θ
and
U
(
g◦
) =
U
(
g
)
◦U
(
), so
U
is a unc o .
Nex we embed
EqAlg
in o
AppEqAlg
by a aching he ines possible
cong uence, namely he diagonal.
De ini ion 4.3 (Diagonal unc o ).We de ine a unc o
G:EqAlg −→ AppEqAlg
by:
•On objec s: Fo an equali y algeb a Y∈EqAlg,
G(Y) = (Y, ∆Y),
whe e ∆Y={(y, y)|y∈Y}is he diagonal (equali y) ela ion.
•
On mo phisms: Fo a homomo phism
g:Y1→Y2
in
EqAlg
, we
se
G(g) = g: (Y1,∆Y1)−→ (Y2,∆Y2).
The map
g
is au oma ically ela ion-p ese ing: i (
y, y
)
∈
∆
Y1
, hen
(g(y), g(y)) ∈∆Y2. Thus Gis indeed a unc o G:EqAlg →AppEqAlg.
4.3. The adjunc ion heo em. We now show ha he quo ien unc o
U
is le adjoin o he diagonal unc o G.
Theo em 4.4. The unc o
U:AppEqAlg →EqAlg
is le adjoin o he
unc o G:EqAlg →AppEqAlg. Tha is, he e is a na u al bijec ion
Φ(E,θ),Y : HomEqAlg(U(E, θ), Y )∼
=HomAppEqAlg((E, θ), G(Y))
o all (
E, θ
)
∈AppEqAlg
and
Y∈EqAlg
, na u al in bo h a iables.
Equi alen ly,
U⊣G.