Theory and Applications of Categories, Vol. 29, No. 10, 2014, pp. 302–314.
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
Dedicated to Professor M.V. Mielke on the occasion of his seventy fifth
birthday
S.N. HOSSEINI, A.R. SHIR ALI NASAB
Abstract. In this article we give necessary and sufficient conditions for a binary
product to exist in a partial morphism category. We also give necessary and sufficient
conditions for the existence of a productive terminal in such categories.
1. Introduction and Preliminaries
Even though the category of partial morphisms is central to so many issues in mathematics, such as representability, [Herrlich, 1988]; theory of computation, [Asperti, Longo,
1991]; fibred mapping spaces, [Booth, Brown, 1978]; graph transformation, [Corradini,
Heindel, Hermann, König, 2006]; embedding CONV and MET in a quasitopos, [Lowen,
Lowen, 1988]; petri nets, [Menezes, 1998]; fuzzy graphs, [Mori, Kawahara, 1997]; and
logic, [Palmgren, Vickers, 2005], to mention a few, only little has been said about limits
in such categories. Dealing with limits is more delicate than dealing with colimits.
In [Cockett, Lack, 2007], colimits and limits in the restriction context have been considered in restriction categories (an abstract formulation of partial morphism categories),
but only the product in the partial function category (where the base category is the category of sets and functions) is given. In [Hosseini, Mielke, 2009], the universality of monos
in partial morphism categories is investigated, while in the present article, the existence
of finite products is characterized, i.e., necessary and sufficient conditions on a category
are given for the corresponding partial morphism category to have finite products. To
this end, we recall:
[Dyckhoff, Tholen, 1987], A pullback complement for the
composable pair (f, s) is a composable pair (s̄, f¯) such that
in the right diagram, the square sf = f¯s̄ is a pullback
and for every pullback square sg = ḡt and any morphism
h : V → Q with f h = g, there is a unique h̄ : Z → P with
f¯h̄ = ḡ and s̄h = h̄t. We then say the square sf = f¯s̄ is a
pullback complement square.
g
V
t
h
Q
s̄
Z
/
h̄
/P
f
p.b.c.
f¯
#
/U
s
/X
;
ḡ
Received by the editors 2012-12-12 and, in revised form, 2014-06-04.
Transmitted by Walter Tholen. Published on 2014-06-15.
2010 Mathematics Subject Classification: 18A30, 18B99.
Key words and phrases: partial morphism category, partial morphism classifier, binary product,
terminal object.
c S.N. Hosseini, A.R. Shir Ali Nasab, 2014. Permission to copy for private use granted.
302
303
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
If (s̄, f¯) is a pullback complement for (f, s) and (f, s) is a pullback complement for
(s̄, f¯), then we say the square sf = f¯s̄ is a double pullback complement square.
1.1. Lemma.
1. Pullback complements are unique up to isomorphism.
2. If (s̄, f¯) is a pullback complement for (f, s) and f is mono, then so is f¯.
Proof. Obvious.
Recalling that a partial morphism classifier in C is a morphism η : A → Ã that
represents partial morphisms into A, [Adamek, Herrlich, Strecker, 1990], we have:
1.2. Lemma.
1. λ : p → π, where p : U → W and π : V → W , is a partial
morphism classifier in C/W if and only if for any triple
(i, f, g) with i : D C, f : D → U, g : C → W such
that pf = gi, there exists a unique morphism fˆ such that
π fˆ = g and the square on the right is a pullback.
D
i
f
/
U
p
p.b.
C
fˆ
/
λ
V
///
π
///
/W
;
g
In this case λ is mono.
2. For κ : V → U , the pullback functor κ∗ : C/U → C/V preserves partial morphism
classifiers.
Proof. Obvious.
Using 1.2 we get:
1.3. Examples.
1. Let ηA : A → Ã be a partial morphism classifier and suppose the diagrams:
Ao
pr1
A×B
pr2
/
B and à o
π1
Ã × B
π2
/
B
are products. Then ηA × 1B : pr2 → π2 is a partial morphism classifier.
π´1
Also with ηB : B → B̃ a partial morphism classifier and A o
A × B̃
product diagram, 1A × ηB : pr1 → π´1 is a partial morphism classifier.
π´2
/
B̃ a
2. Let C be the 2-chain category {1A : A → A, 1B : B → B, ! : A → B}. A is the initial
and B is the terminal object. Now ! : pr2 → 1B , where pr2 =! : A × B = A → B, is
a partial morphism classifier; also 1A : pr1 → 1A , where pr1 = 1A : A × B = A → A
is a partial morphism classifier. However there is no partial morphism classifier with
domain B in C.
Calling a pushout square that is preserved by pullbacks a stable pushout, and a stable
pushout that is also a pullback a stable pulation, we have:
304
S.N. HOSSEINI, A.R. SHIR ALI NASAB
1.4. Lemma.
1. Every stable pulation square in which all the morphisms are monos is a double
pullback complement.
2. If in the commutative diagram shown m, n and l are mons,
the left square is a stable pushout and the outer square is
a pullback, then the right square is a pullback.
f
A
m
/
n
B
/
g
p
C
/
E
D
l
/F
q
Proof.
1. Given monos j1 , j2 , λ1 and λ2 with the square j1 λ1 = j2 λ2
a stable pulation, let in the right diagram the outer square
be a pullback and the triangle be commutative.
/
k
Y
λ2
X
#
λ1
A
/
B
j1
s.p.l.
C
/
;P
j2
f
j1−1 (f )
/
α
Since
= λ1 k, pulling back the stable pushout square
j1 λ1 = j2 λ2 along f we get the square shown, which is a
pushout by stability.
1
/
f −1 (j2 )
f −1 (j1 )
So f −1 (j2 ) = 1 and α = f −1 (j1 ). Now j2 j2 −1 (f )
= f f −1 (j2 ) = f . Also j2 j2−1 (f )f −1 (j1 ) = f f −1 (j1 ) =
j1 j1−1 (f ) = j1 λ1 k = j2 λ2 k, implying j2−1 (f )f −1 (j1 ) = λ2 k.
Hence in the right diagram, the squares are pullbacks and
the triangles are commutative.
/
k
Y
λ2
/
X
j2−1 (f )
λ1
A
/
#
p.b.c.
B
C
/
j2
j1
;P
f
−1
Uniqueness of j2 (f ) follows from the fact that j2 is mono. So (λ2 , j2 ) is a pullback
complement for (λ1 , j1 ). Similarly (λ1 , j1 ) is a pullback complement for (λ2 , j2 ).
2. The proof in [Lack, Sobocinski, 2006] for adhesive categories works under our hypothesis too.
1.5. Lemma. If in the right diagram i1 and i2 are monos,
the inner square is a pullback and the outer square is a
stable pushout, then:
1. the outer square is a double pullback complement;
and
i−1
1 (ϕ)
2. the morphism ϕ is mono,
−1
i−1
2 (ϕ) = 1 and ϕ (i2 ) = j2 .
−1
= 1, ϕ (i1 ) = j1 ,
A
λ2
C
λ1
/
B
i1
i2
/
j1
D`
ϕ
j2
/P
305
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
Proof. First notice that because i1 and i2 are monos, so are j1 , j2 , λ1 and λ2 .
1. Obviously the pushout square is a pullback and so a stable pulation. The result
follows from 1.4.
2. In the commutative diagram on the right the left square
is a stable pushout and the outer one is a pullback. 1.4,
implies the right square is also a pullback. So we have
−1
−1
i−1
1 (ϕ) = 1 and ϕ (i1 ) = j1 . Similarly i2 (ϕ) = 1 and
ϕ−1 (i2 ) = j2 .
Now suppose for morphisms α and β, ϕα = ϕβ.
Pullback the outer square
of the above diagram along
ϕα = ϕβ to get the diagram on the right.
Since the middle square is
a stable pushout, so is the
left square. So α = β and
therefore ϕ is mono.
/
~ Y
Y1
~
/
B
~
λ2
X
~
j2
~ A
λ2
/
β
/
P
ϕ
/
B
i1
;D
/A
 /
B
λ1

λ2
/
C
j1
j1
/P
1
1
α
/
1
B
i2
λ1
γ2
/
C
~ Y2
γ1
λ1
A

j2
/
ϕ
/
C
1
i1
D
i2

Calling a pushout square that is also a double pullback complement, a double pulation
complement, we have:
1.6. Lemma. If in the diagram on
the right, i1 and i2 are monos, the
bottom and the back faces are pullbacks and the top face is a double
pulation complement, then there is
a unique morphism ϕ : D́ → D rendering the front and right faces as
pullbacks.
Á
/
j´2
i´1
g
Ć
/
D́
i´2
A
k

C
B́
j´1
/
ϕ
j1
j2
i2
l
/
~
B
i1
D
Proof. Since i1 and i2 are monos, so are j1 , j2 , j´1 and j´2 . Now 1.1 implies i´1 and i´2 are
monos as well. Because the top face of the above cube is a pushout and i1 lj´1 = i2 k j´2 , there
exists a unique morphism ϕ : D́ → D making the front faces of the cube commutative.
306
S.N. HOSSEINI, A.R. SHIR ALI NASAB
Now transform the bottom square under
/ B̂
jˆ1
Â
pullback along ϕ : D́ → D to get the cube
jˆ2
iˆ1
on the right. Since the front faces of the
above cube are commutative and those in
/ D́
i−1
Ĉ
iˆ2
1 (ϕ)
the right cube are pullbacks, there are γ1
´
ˆ
and γ2 such that l = i−1
1 (ϕ)γ1 , i1 = i1 γ1 ,
−1
/B
ϕ
i−1
j1
k = i2 (ϕ)γ2 and i´2 = iˆ2 γ2 . Now the
A
2 (ϕ)
−1
equalities l = i−1
1 (ϕ)γ1 , k = i2 (ϕ)γ2 ,
j2
i1

~
l−1 (j1 ) = j´1 and k −1 (j2 ) = j´2 imply
/D
i2
C
γ1−1 (ĵ1 ) = j´1 and γ2−1 (jˆ2 ) = j´2 .
So we have the following diagram in which the right squares are pullbacks and the left
ones are formed to be pullbacks.
Á
j́1
B́
δ1
p.b.
/
γ1
/
Â
jˆ2
/
jˆ1 p.b.
Ĉ
B̂
iˆ1
/
Á
j́2
î2
D́
Ć
δ2
/
p.b.
/
γ2
Â
jˆ2
/
B̂
jˆ2 p.b.
Ĉ
iˆ2
î1
/ D́
j´1
j´2
Because the top face of the original cube
´
#
j
j´1 #
2
1 /
1 /
/ Ć
/ B́
is a double pullback complement, there
Á
Á
Á
Á
exist unique morphisms λ1 and λ2 such
p.b.c.
p.b.c.
j´1
j´2
i´2
i´1
that the triangles in the diagram on the
right are commutative and the squares
B̂ λ / B́ ´ / ; D́ Ĉ λ / Ć ´ / ; D́
1
2
i1
i2
are pullbacks.
iˆ1
iˆ2
We have i´1 λ1 γ1 = iˆ1 γ1 = i´1 and i´2 λ2 γ2 = iˆ2 γ2 = i´2 , yielding λ1 γ1 = 1 and λ2 γ2 = 1.
Since λ1 and λ2 are monos, they are isomorphisms. It follows that iˆ1 , is isomorphic to i´1
and iˆ2 is isomorphic to i´2 .
2. Binary Product in Partial Morphism Categories
*
By the partial morphism category C is meant the category having the same objects as C
*
and with morphisms f = [(if , f )] : A → B equivalence classes of pairs (if : Df → A, f :
Df → B) with if a universal mono (i.e., a mono whose pullback along every morphism
exists) and where equivalence of (if , f ) and (ig , g) means that there is an isomorphism
*
k for which if = ig k and f = gk. The composition of morphisms A
f
/
*
B
g
/
C is
**
defined by g f = [(if (f −1 (ig )), g(i−1
g (f )))].
Calling a category weakly adhesive if monos of the category are universal and pushouts
of monos along monos exist and are stable, we have:
2.1. Proposition. Suppose A
o
*
π1 =[(i1 ,π1 )]
*
A×B
*
π2 =[(i2 ,π2 )]
/
*
B is a product in C . Then:
307
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
π1 λ1
π 2 λ2
/ B is a product diagram in C,
1. A o
P
where λ1 and λ2 are obtained by the pullback
square given on the right;
λ1
P /
/
D 1
p.b.
λ2
D2 /
i2
/
i1
*
A×B
2. the above square is a double pullback complement;
3. assuming C is weakly adhesive, the above square is a pushout; and
4. with λn , n = 1, 2, as in part (1) and prn = πn λn , λn : prn → πn is a partial
morphism classifier (in the appropriate slice category).
Proof.
1. If A o
f
g
C
/
*
B is a diagram in C, then there exists a unique morphism h=
* *
*
**
[(ih , h)] : C → A × B such that π 1 h= f and π2 h= g. This implies ih h−1 (i1 ) =
−1
−1
ih h−1 (i2 ) = 1, π1 i−1
1 (h) = f and π2 i2 (h) = g. Since ih is mono, ih = h (i1 ) =
−1
h−1 (i2 ) = 1 (up to isomorphism). Then i2 i−1
2 (h) = h = i1 i1 (h), and so the above
−1
pullback yields a morphism γ : C → P such that λ1 γ = i1 (h) and λ2 γ = i−1
2 (h).
−1
Therefore π1 λ1 γ = π1 i−1
(h)
=
f
and
π
λ
γ
=
π
i
(h)
=
g.
To
show
uniqueness,
2 2
2 2
1
suppose θ : C → P is a morphism such that π1 λ1 θ = f and π2 λ2 θ = g. Then
*
*
*
*
π 1 i1 λ1 θ = f =π1 i1 λ1 γ, π 2 i1 λ1 θ = g =π2 i1 λ1 γ, implying i1 λ1 θ = i1 λ1 γ and so
θ = γ.
2. Consider the diagram on the right in
which both squares are pullbacks and
the triangle commutes.
k
Y
l−1 (i1 )
X
/
λ1
A×B
λ2
/
'
D1
D2
i2
/A
7
i1
*
×B
l
pr1 =π1 λ1
*
pr2 =π2 λ2
/ B is a product in C. We have π1 l
By part 1, A o
A×B
*
−1
−1
= [(l−1 (i1 ), pr1 k)] and π2 l = [(l−1 (i2 ), π2 i−1
2 (l))]. Let α = i2 (l). Since i1 (l)
= λ1 k, we get l−1 (i1 ) = l−1 (i1 λ1 ) = l−1 (i2 λ2 ) = l−1 (i2 )α−1 (λ2 ). This in turn
*
*
*
−1
−1
yields π1 [(l−1 (i2 ), ll−1 (i2 ))] = [(l−1 (i1 ), π1 i−1
1 (l))] =π1 l and π2 [(l (i2 ), ll (i2 ))]
*
−1
−1
= [(l−1 (i2 ), π2 i−1
2 (l))] =π2 l. Uniqueness gives, l = [(l (i2 ), ll (i2 ))]. It follows
−1
−1
−1
that l (i2 ) = 1 and l = ll (i2 ) = i2 i2 (l). We have i2 λ2 k= i1 λ1 k = i1 i−1
1 (l)
−1
−1
−1
−1
−1
−1
= ll (i1 ) = i2 i2 (l)l (i1 ). i2 mono yields λ2 k = i2 (l)l (i1 ). So i2 (l) is the
required morphism. Uniqueness follows from the fact that i2 is mono. Hence (λ2 , i2 )
is a pullback complement for (λ1 , i1 ). Similarly (λ1 , i1 ) is a pullback complement for
(λ2 , i2 ).
3. Let D1 /
j1
/
Po
j2
o
D2 be a pushout of D1 o
*
λ1
o
λ
A×B / 2
/
D2 . Then there
exists a unique morphism ϕ : P −→ A × B such that i1 = ϕj1 and i2 = ϕj2 . 1.5
308
S.N. HOSSEINI, A.R. SHIR ALI NASAB
−1
−1
−1
implies ϕ is mono and i−1
1 (ϕ) = 1, ϕ (i1 ) = j1 , i2 (ϕ) = 1 and ϕ (i2 ) = j2 . Now
*
*
*
*
π1 [(ϕ, ϕ)] =π1 and π2 [(ϕ, ϕ)] =π2 implying that ϕ is an isomorphism.
4. We show that λ2 : pr2 → π2 is a partial morphism classifier. Given (i, f, g) with
i : D C, f : D −→ A × B and g : C −→ B such that gi = pr2 f , we have partial
morphisms [(i, pr1 f )] : C → A and [(1, g)] : C → B. Then there exists a unique
*
**
**
morphism k = [(ik , k)] such that π1 k = [(i, pr1 f )] and π2 k = g. This implies ik = 1,
−1
−1
k −1 (i1 ) = i, π1 i−1
1 (k) = pr1 f , k (i2 ) = 1 and π2 i2 (k) = g.
Consider the diagram
β
/
/C
X
on the right, in which
the right squares are
α
pullbacks and the left
A × B / λ / D2 /
squares are formed by
2
taking pullbacks.
Since the square i1 λ1 = i2 λ2 is a pullback, so is the square on the right. Then
X = D, β́ = 1, β = i. These and the
previous equations imply:
/
1
i2
/
β́
X/
C
α
k
*
/
A×B A×B /λ
β́
X /
β
/
1
C
1
/
D/
/
i
D1 /
i1
/
C
k
*
A×B
D
/
i
C
−1
pr1 α = π1 λ1 α = π1 i−1
1 (k)β́ = π1 i1 (k) = pr1 f
pr2 α = π2 λ2 α = π2 i−1
2 (k)β = gβ = gi = pr2 f
It follows that α = f . So we have the diagram on the right, in which the square
is a pullback and π2 i−1
2 (k) = g.
D
i
C
f
/
A × B
p.b. λ2
i−1
2 (k)
/
pr2
D2
#
π2
/B
To show uniqueness, suppose there exists a morphism h such that in the above
diagram when i−1
2 (k) is replaced by h, the square is a pullback and π2 h = g. One
*
**
*
**
can easily verify that π1 i2 h = [(i, pr1 f )] =π1 k and π2 i2 h = g =π2 k . So i2 h = k.
−1
Because k −1 (i2 ) = 1, we have i2 h = k = i2 i−1
2 (k) and so h = i2 (k). By 1.2, we are
done. Similarly λ1 : pr1 → π1 is a partial morphism classifier.
309
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
2.2. Proposition. Suppose C is weakly adhesive.
If
pr1
pr2
o
/
A
A×B
B is a product in C, λn : prn → πn is
a partial morphism classifier for n = 1, 2, where π1 : BA → A,
π2 : AB → B and there exist morphisms i1 and i2 such that
the square on the right is a double pulation complement, then
A
o
*
π1 =[(i1 ,π1 )]
*
π2 =[(i2 ,π2 )]
P
/
λ1
A×B
λ2
/
BA
d.p.l.c.
AB
i1
/P
i2
*
B is a product diagram in C
Proof. Because by 1.2, λ1 , λ2 are monos, (λ1 , i1 ) is a pullback complement for (λ2 , i2 )
and (λ2 , i2 ) is a pullback complement for (λ1 , i1 ), by 1.1, i1 and i2 are monos.
*
*
To show A
let Df o
o
o π1
P
π2
Df ∩ Dg /
/
/
*
*
B is a product in C , given A
Dg be a pullback of Df /
f i−1
f (ig )
if
/
ig
Co
*
f =[(if ,f )]
o
o
g =[(ig ,g)]
C
/B
,
Dg . Then we have mor-
gi−1
g (if )
/ B , and so a unique morphism l exists such that
phisms A o
Df ∩ Dg
−1
pr1 l = f i−1
f (ig ) and pr2 l = gig (if ).
Also we have unique morphisms f˜, g̃ such that π1 f˜ = f , π2 g̃ = g and the below squares
are pullbacks.
/
l
Df ∩ Dg
p.b.
Df
Let Df /
jf
/
Ṕ o
o
λ1
/
f˜
jg
A × B
pr1
BA
"
π1
/
A
Dg
λ2
/
g̃
o
pr2
AB
Df ∩ Dg /
/
π2
/
"
B
Dg .
/ Dg
Df ∩ Dg
jg
Df
{
*

Ṕ
g̃
A×B
*
/
jf
l
f˜
BA
*
A × B
p.b.
Dg be a pushout of Df o
There exists a unique morphism
k : Ṕ −→ C such that ig = kjg
and if = kjf . By 1.5, k is mono
and the pushout square is a double pullback complement. Also
1.6, implies the existence of a
unique morphism h : Ṕ −→ P
such that the front faces in the
right diagram are pullbacks.
/
l
Df ∩ Dg
z
λ1
i1
*
One easily verifies that π1 [(k, h)] = f and π2 [(k, h)] = g .
/
h
λ2
/
~
P
i2
AB
310
S.N. HOSSEINI, A.R. SHIR ALI NASAB
To show uniqueness, suppose
there exists a morphism ψ such
**
*
**
*
that π1 ψ= f and π2 ψ= g . Then
iψ ψ −1 (i1 ) = if , π1 i−1
1 (ψ) = f ,
iψ ψ −1 (i2 ) = ig and π2 i−1
2 (ψ) =
g. Form the cube on the right,
where all the squares are pullbacks.
Df
}
/
m
} X
*
i−1
1 (ψ)
/
BA
{
λ1
λ2
ψ −1 (i1 )
}
Dψ
ψ −1 (i
}
Dg
i1
ψ
−1
i
o Df ∩ Dg g/
Df o
cube a pushout.
(if )
/
/ { AB
i−1
2 (ψ)
2)
One can see X = Df ∩ Dg and that the pair of morphisms Df o
i−1
f (ig )
A{ × B
/
i2
P
{
o
X/
/
Dg equals
Dg . Pushout stability now renders the left square in the above
i−1
So Dψ = Ṕ , ψ −1 (i1 ) =
jf
i−1
j
g (if )
f (ig )
/
/
/
/ Dg / g
/
/
D
Ṕ
D
∩
D
D
∩
D
−1
f
f
g
f
g
jf and ψ (i2 ) = jg and
therefore we have the
m
m
pb
ψ
pb
pb
pb
i−1
i−1
1 (ψ)
2 (ψ)
pullback squares on the
/P
/ BA /
/ AB /
A×B / λ
A×B / λ
i1
i2
right.
1
2
−1
−1
−1
It follows that pr1 m = π1 λ1 m = π1 i1 (ψ)if (ig ) = f if (ig ) = pr1 l. Similarly
pr2 m = pr2 l. Therefore m = l.
/
/
Ṕ
ψ
P
we get
Now we have the dial=m /
l=m /
A × B
A × B
Df ∩ Dg
Df ∩ Dg
gram on the right, in
pr
pr2
1
i−1
pb
pb
λ1
λ2
i−1
g (if )
f (ig )
which the squares are
"
"
pullbacks, π1 i−1
/A
/B
/ BA
/ AB
1 (ψ) = f
D
D
f
g
π
π
−1
−1
1
2
i1 (ψ)
i2 (ψ)
and π2 i−1
(ψ)
=
g.
2
−1
−1
˜ −1
˜
It follows that i−1
1 (ψ) = f , i2 (ψ) = g̃. Now ψjf = ψψ (i1 ) = i1 i1 (ψ) = i1 f and
−1
similarly ψjg = i2 g̃. The uniqueness of h yields h = ψ. Now we have if = iψ ψ (i1 ) = iψ jf
*
and ig = iψ ψ −1 (i2 ) = iψ jg , and uniqueness of k yields iψ = k. Hence ψ= [(k, h)].
2.3. Theorem. Suppose C is weakly adhesive.
A
o
*
π1
pr1
*
A×B
*
π2
/
A product
*
B exists in C if and only if a product
pr2
/ B , along with morphisms π1 , π2 and monos
Ao
A×B
λ1 , λ2 , i1 and i2 exist in C such that λn : prn → πn is a partial morphism classifier and the square on the right is a double
*
pulation complement. In this case, πn = [(in , πn )].
Proof. Follows from 2.1 and 2.2.
A×B
λ2
λ1
/
d.p.l.c.
AB
i2
/
BA
i1
*
A×B
311
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
2.4. Examples.
1. Let C be an adhesive category [Lack Sobocinski, 2004]. Then by 1.4, pushout of
pr1
pr2
/B
monos along monos are double pullback complements. So if A o
A×B
is a product in C and λn : prn → πn is a partial morphism classifier, then with
*
π1 =
BA
[(i1 , π1 )] and
i1
/
i2
Po
*
π2 =
*
[(i2 , π2 )], A
o π1
*
P
AB is a pushout of BA o
*
π2
/
λ1
A×B
B is a product diagram in C , where
λ2
/
AB .
2. Let C be an adhesive category in which the partial morphism classifiers ηA : A → Ã
pr1
and ηB : B → B̃ and the products A o
and à o
πÃ
Ã × B
πB
/
iA
/
Po
pr2
/
πA
B, Ao
A × B̃
πB̃
/
B̃
B exist. By 1.3, 1 × ηB : pr1 → πA and ηA × 1 : pr2 → πB
are partial morphism classifiers. So A o
where A × B̃
A×B
iB
*
π1 =[(iA ,πA )]
*
P
π2 =[(iB ,πB )]
/B
*
is a product in C ,
ηA ×1
1×ηB
Ã × B is a pushout of A × B̃ o
A×B
/ Ã × B
.
3. Let C be a topos. Because topoi are adhesive, [Lack, Sobocinski, 2006], and all
*
partial maps are representable, [Johnstone, 1977], by (2), C has all binary products.
*
In this case for A, B ∈ C, A × B = Ã × B ∪ A × B̃, where ηA : A → Ã is a partial
morphism classifier and ”∪” is the union taken as subobjects of Ã × B̃.
4. Let C be the category corresponding to a distributive prelattice (i.e., a preordered class with binary
meets and joins, in which meet is distributive over
join). One can easily verify that C is weakly adhesive, but it is not generally adhesive as the diagram
on the right, in which A 6= B, suggests. Also it is
not hard to see λ : p → π, where p : A → C and
π : B → C, is a partial morphism classifier if and
only if π is an isomorphism and the only morphism
to A ∧ B is the identity morphism (or equivalently,
in the preordered class, B ≤ C, C ≤ B and A ∧ B
*
is minimal). So A × B exists if and only if A ∧ B
is minimal. In this case, since by 1.4, the square on
the right is a double pulation complement, we get
*
A × B = A ∨ B.
/
B
/
B
A
A

/
~
B
A

/
B
/
A∧B
B
d.p.l.c.
/
~
B
A
A∨B
5. Let C be a category in which all monos are isomorphisms and not all pullbacks exist
(e.g, the category generated by parallel isomorphisms f, g and a morphism h such
that hf = hg), so that C is not adhesive. It can be easily verified that C is weakly
312
S.N. HOSSEINI, A.R. SHIR ALI NASAB
*
adhesive. Also for A, B ∈ C, the existence of A × B is equivalent to that of A × B.
*
*
In fact A × B = A × B. But this is also the case because C is isomorphic to C.
6. Let C be a weakly adhesive category that is not adhesive and D be any weakly
adhesive category. Then C × D is weakly adhesive but not adhesive. The product
*
(A, A0 ) × (B, B 0 ) exists if and only if the pairs A, B and A0 , B 0 satisfy the conditions
of 2.3 in C and D, respectively.
Calling an object productive, if its product with every object exists, we have:
2.5. Lemma. Let C be a category with a terminal 1 and T be a productive object. Then
*
T is a terminal object in C if and only if T → 1 is a partial morphism classifier in C.
Proof.
*
Assume T is a terminal object in C . Let
/ T be a partial morphism.
[(iu , u)] : A
Since T is productive, the right square is a
pullback.
*
A×T
pr1
pr2
/
p.b.
T
/
A
1
*
Now since T is a terminal in C , T → 1 is mono in C , and so mono in C, [Hosseini,
Mielke, 2009].
So pr1 is a universal mono,
putting
u
/
T
[(pr1 , pr2 )] in C .
It now follows that
iu
[(iu , u)] = [(pr1 , pr2 )] and so the square on
/1
A
the right is a pullback.
Conversely assume T → 1 is a partial morphism classifier in C. Given an object A,
we have a pullback square as shown in the first square above and so a partial morphism
/ T . Uniqueness follows from the fact that any partial morphism from
[(pr1 , pr2 )] : A
*
A to T can be represented by T
/
D
1.
2.6. Examples.
1. Let C be a category with binary products. If C has a strict initial object 0 and a
terminal object 1, then 0 → 1 is a partial morphism classifier and therefore 0 is a
*
*
*
terminal object in C . Also 0 is an initial object in C . Hence C has a zero object.
2. Let C be a topos. Then C has a strict initial object [Mac Lane, Moerdijk, 1992]. So
*
C has a zero object.
FINITE PRODUCTS IN PARTIAL MORPHISM CATEGORIES
313
References
J. Adamek, H. Herrlich, G.E. Strecker: Abstract and Concrete Categories, John Wiley &
Sons, Inc., 1990, http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf.
A. Asperti, G. Longo: Categories Types and Structures, M.I.T. Press, 1991.
P.I. Booth, R. Brown: Spaces of partial maps, fibred mapping spaces and the compactopen topology, Gen. Topol. Appl. 8, pp 181-195, 1978.
J.R.B. Cockett, S. Lack: Restriction categories III: colimits, partial limits, and extensivity,
Mathematical Structures in Computer Science, 17, 775-817, 2007
A. Corradini, T. Heindel, F. Hermann, B. König: Sesqui-pushout rewriting, In Third International Conference on Graph Transformations (ICGT 06), volume 4178 of Lecture
Notes in Computer Science, pages 30-45, Springer, 2006.
R. Dyckhoff, W. Tholen: Exponentiable morphisms, partial products and pullback Complements, J. Pure Appl. Algebra 49, 103-116, 1987.
H. Herrlich: On the representability of partial morphisms in Top and in related constructs.
Springer Lect. Notes Math. 1348, pp 143-153, 1988.
S.N. Hosseini, M.V. Mielke: Universal monos in partial morphism categories, Appl Categor
Struct, 17:435-444, 2009, DOI 10.1007/s10485-007-9123-2.
P.T. Johnstone: Topos Theory, Academic Press, London, New York, San Francisco, 1977.
S. Lack, P. Sobocinski: Adhesive categories, In Foundations of Software Science and Computation Structures (FoSSaCS 04), volume 2987 of Lect. Notes Comput. Sc., pages
273-288. Springer, 2004.
S. Lack, P. Sobocinski: Toposes are adhesive, In International Conference on Graph Transformation (ICGT 06), volume 4178 of Lect. Notes, Comput. Sc., pages 184-198.
Springer, 2006.
E. Lowen, R. Lowen: A Quasitopos containing CONV and MET as full subcategories,
Internat. J. Math. & Math. Sci., Vol 11 No 3, pp 41-438, 1988.
S. Mac Lane, I. Moerdijk: Sheaves in Geometry and Logic, A First Introduction to Topos
Theory, Springer-Verlag, New York, 1992.
P.B. Menezes: Diagonal Compositionality of Partial Petri Nets, Electronic Notes in Theoretical Computer Science 14, 1998.
M. Mori, Y. Kawahara: Rewriting Fuzzy Graphs, Department of Informatics, Kyushu
University 33, Fukuoka 812-81, Japan, 1997.
314
S.N. HOSSEINI, A.R. SHIR ALI NASAB
E. Palmgren, S.J. Vickers: Partial Horn Logic and Cartesian Categories, U.U.D.M. Report
36, ISSN 1101-3591, 2005.
Mathematics Department
Shahid Bahonar University of Kerman
Kerman, Iran
Email: nhoseini@uk.ac.ir
ashirali@math.uk.ac.ir
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