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arXiv:2201.08016v1 [math.CT] 20 Jan 2022
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS
II: THE UNIVERSAL PROPERTY
FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
Abstract. We prove that the stable category associated with the category
PreOrd(C) of internal preorders in a pretopos Csatisfies a universal property.
The canonical functor from PreOrd(C) to the stable category Stab(C) uni-
versally transforms a pretorsion theory in PreOrd(C) into a classical torsion
theory in the pointed category Stab(C). This also gives a categorical insight
into the construction of the stable category first considered by Facchini and
Finocchiaro in the special case when Cis the category of sets.
Introduction
This article is meant as the sequel of [2] and it deals with the study of the uni-
versal property of the stable category Stab(C) of the category PreOrd(C) of internal
preorders in a pretopos C. It reveals the categorical feature of a natural construc-
tion due to A. Facchini and C. Finocchiaro in the category PreOrd of preordered
sets [6], that we first briefly recall.
The category PreOrd contains the full subcategories Eq and ParOrd whose objects
are equivalence relations and partially ordered sets, respectively. The pair of cate-
gories (Eq,ParOrd) has two properties making it a pretorsion theory in PreOrd [6, 7],
that is, a kind of “non-pointed torsion theory”. More explicitly, any preordered set
(A, ρ), where ρis a reflexive and transitive relation on the set A, determines an
equivalence relation (A, ∼ρ), where ∼ρ=ρ∩ρoand ρois the opposite relation of
ρ, and a partially ordered set (A/ ∼ρ, π(ρ)), where π:A→A/ ∼ρis the quotient
of Aby the equivalence relation ∼ρ, and π(ρ) is the partial order induced by ρon
the quotient A/ ∼ρ. This yields a short Z-exact sequence
(A, ∼ρ)IdA//(A, ρ)π//(A/∼ρ, π(ρ)) (SES)
where Zis the full subcategory of PreOrd whose objects are the “trivial preorders”
(B, =), with Ba set and = the equality relation on B. This subcategory Zde-
termines an ideal of trivial morphisms [5], where a morphism is called trivial if it
factors through a trivial object. The fact that the above sequence is Z-exact means
2010 Mathematics Subject Classification. Primary 06A75, 18B25, 18B35, 18B50, 18E08,
18E40.
Key words and phrases. Preorders, partial orders, equivalence relations, pretopos, stable cat-
egory, (pre)torsion theory.
The second author was partially supported by Fondazione Ing. Aldo Gini - Universit`a
di Padova, borsa di studio per l’estero bando anno 2019 and by Ministero dell’Istruzione,
dell’Universit`a e della Ricerca (Progetto di ricerca di rilevante interesse nazionale “Categories,
Algebras: Ring-Theoretical and Homological Approaches (CARTHA)”).
This work was also supported by the collaboration project Fonds d’Appui `a l’Internationalisation
“Coimbra Group” (2018-2021) funded by the Universit´e catholique de Louvain.
1
2 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
that the identity morphism IdAabove is the Z-kernel of π, and the quotient πis
the Z-cokernel of IdA, where the notions of Z-kernel and Z-cokernel are defined
by the same universal properties characterizing usual kernels and cokernels, with
the only difference that the ideal of 0-morphisms is replaced by the ideal of trivial
morphisms [7]. The Z-exact sequence (SES) has the property that the Z-kernel
belongs to Eq (the “torsion subcategory”) and the Z-cokernel belongs to ParOrd
(the “torsion-free subcategory”). Furthermore, one easily sees that any order pre-
serving morphism from an equivalence relation to a partial order is trivial. These
two properties express the fact that (Eq,ParOrd) is a pretorsion theory in PreOrd.
In their study of this pretorsion theory in PreOrd the authors of [6] introduced a
new category, the stable category Stab of preordered sets: this is a pointed category,
arising as a quotient category, with the property that the canonical functor from
PreOrd to Stab sends the trivial objects in Zto the zero object in Stab, and any
trivial morphism in PreOrd to a zero morphism in Stab.
In the first article [2] of this series we proved that, whenever Cis a coherent
category [11], it is possible to give a purely categorical construction of the stable
category Stab(C) of the category PreOrd(C) of internal preorders in C(we recall
this construction in the first section of this article). Moreover, when Cis a pretopos,
the functor Σ: PreOrd(C)→Stab(C) preserves coproducts and sends short Z-exact
sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C)
(Theorem 7.14 in [2]).
The aim of this article is to prove the universal property of the stable category
Stab(C), that relies on these two properties of the functor Σ: PreOrd(C)→Stab(C).
If we call a functor G:PreOrd(C)→Xatorsion theory functor (Definition 2.1)
when it sends the torsion and the torsion-free subcategories of the pretorsion theory
(Eq(C),ParOrd(C)) into the torsion and the torsion-free subcategory, respectively,
of a torsion theory (T,F) in the category X, the universal property can be expressed
as follows:
Theorem 2.3. The canonical functor Σ: PreOrd(C)→Stab(C) is universal among
all finite coproduct preserving torsion theory functors G:PreOrd(C)→X, where
PreOrd(C) is equipped with the pretorsion theory (Eq(C),ParOrd(C)), and Xis a
pointed category with coproducts equipped with a torsion theory (T,F). This
means that any finite coproduct preserving torsion theory functor G:PreOrd(C)→
Xfactors uniquely through Σ:
PreOrd(C)Σ//
∀G$$
■
■
■
■
■
■
■
■
■
■Stab(C)
∃!G
{{
X,
i.e. there is a unique functor Gsuch that G·Σ = G. The induced functor G
preserves finite coproducts, and it is a torsion theory functor.
This theorem reveals the nature of the stable category, namely to transform a
pretorsion theory in the sense of [7] into a “classical” torsion theory, universally.
Note that some further properties of the stable category Stab(C) can be established
when the base category Cis what we called a τ-pretopos in [2], that is a pretopos
with the additional property that the transitive closure of any relation on an object
exists. Under this assumption it is possible to show that, for any “suitable” category
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY3
X, the induced torsion theory functor Gabove preserves kernels and cokernels, hence
in particular short exact sequences (Theorem 3.12).
1. Preliminaries
In this article Cwill always be assumed to be a pretopos (see [11] for more
details). Let us recall that Cis a pretopos when
•Cis exact (in the sense of Barr [1]),
•Chas finite sums (=coproducts),
•Cis extensive [4].
This property of extensivity means that Chas pullbacks along copro jections in a
sum and the following condition holds: in any commutative diagram, where the
bottom row is the sum of Aand B
A′//
C
B′
oo
As1//A`B B,
s2
oo
the top row is a sum if and only if the two squares are pullbacks. The property
saying that the upper row of the diagram is a sum whenever the two squares are
pullbacks is usually called the “universality of sums”.
Recall that a sum of two objects Aand Bis called disjoint if the coprojections
s1:A→A`Band s2:B→A`Bare monomorphisms and their intersection
A∩Bin the pullback
A∩B//
B
s2
As1//A`B
is an initial object in the category Sub(A`B) of subobjects of A`B. For a finitely
complete category Cwith finite sums, extensivity is equivalent to the property of
having disjoint and universal finite sums. In a pretopos the supremum A∪Bof
two disjoint subobjects A→Xand B→Xis given by the coproduct A`Bof
these two objects in C(see Corollary 1.4.4 in [11]). Recall also that any pretopos
has a strict initial object, namely an initial object 0 with the property that any
morphism with codomain 0 is an isomorphism.
In this work we shall be mainly interested in the category PreOrd(C) of internal
preorders in a pretopos C, that is defined as follows.
An object (A, ρ) in PreOrd(C) is a relation hr1, r2i:ρ→A×Aon A, i.e.
a subobject of A×A, that is reflexive, i.e. it contains the “discrete relation”
h1A,1Ai:A→A×Aon A, and transitive: there is a morphism τ:ρ×Aρ→ρsuch
that r1τ=r1p1and r2τ=r2p2, where (ρ×Aρ, p1, p2) is the pullback
ρ×Aρp2//
p1
ρ
r1
ρr2//A.
4 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
A morphism (A, ρ)→(B, σ) in the category PreOrd(C) of preorders in Cis a
pair of morphisms (f, ˆ
f) in Cmaking the following diagram commute
ρ
r2
r1
ˆ
f//σ
s2
s1
Af//B,
so that f r1=s1ˆ
fand fr2=s2ˆ
f.
A preorder (A, ρ) is called an equivalence relation if there is a “symmetry”,
namely a morphism s:ρ→ρsuch that r1s=r2and r2s=r1. Equivalently,
the opposite relation ρ◦of ρis isomorphic to ρ, hence they determine the same
subobject of A×A:ρ◦=ρ. A preorder (A, ρ) is called a partial order if it
“antisymmetric”, i.e. it has the additional property that ρ∩ρ◦= ∆A, where
∆Ais the discrete equivalence relation on A. We write Eq(C) and Par(C) for the
full (replete) subcategories of PreOrd(C) whose objects are equivalence relations
and partial orders in C, respectively. We write Z=Eq(C)∩Par(C) for the full
(replete) subcategory of trivial objects in PreOrd(C) [2], whose ob jects are “discrete”
preorders, i.e. those of the form (A, ∆A). Note that the pair (Eq(C),Par(C)) is a
pretorsion theory in PreOrd(C) (see [8] for more details).
A morphism (f, ˆ
f): (A, ρ)→(B, σ ) is called a Z-trivial morphism if it factors
through a trivial object. In the following we shall often use the terms “trivial
morphism” and “trivial object” (dropping the “Z” of “Z-trivial”). As observed in
[2], the class of trivial morphisms in PreOrd(C) is an ideal of morphisms in the sense
of Ehresmann [5]. The notions of Z-kernel and of Z-cokernel are defined by the
expected universal properties obtained by replacing, in the definition of kernel and
cokernel, the ideal of zero morphisms with the ideal of trivial morphisms induced
by the subcategory Z(see [7] for more details).
Given a morphism f:A→B, where (B, σ) is an object in PreOrd(C), we denote
by f−1(σ) the inverse image of σalong f, that is the left vertical relation defined
by the following pullback:
f−1(σ)//
σ
hs1,s2i
A×Af×f//B×B
Recall then that, in any category with an initial object 0, a subobject α:A→B
of an object Bis complemented if there is another subobject αc:Ac→Bwith the
property A∩Ac= 0 and A∪Ac=B.
It was observed in [2] (Corollary 5.4) that a subobject (A, ρ)//α//(B, σ) in
PreOrd(C) is complemented in PreOrd(C) if and only if
(1) A//α//Bis a complemented subobject in C, with complement Ac//αc
//B;
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY5
(2) α−1(σ) = ρ, (αc)−1(σ) = ρcand all the commutative squares in the diagram
ρ////
r2
r1
σ
s2
s1
ρc
oooo
rc
2
rc
1
A//α//B=A`AcAc
oo
αc
oo
(i.e. the ones corresponding to the same index i∈ {1,2}) are pullbacks.
Note that, in a pretopos C, this implies that σ=ρ`ρc.
Before recalling the definition of the stable category of PreOrd(C), as an inter-
mediate step, we first define the category PaPreOrd(C) of partial morphisms in
PreOrd(C).
Its objects are the same as the ones of PreOrd(C), the internal preorders (A, ρ)
in C, while a morphism (A, ρ)→(B , σ) in the category PaPreOrd(C) is a pair (α, f )
depicted as
(A′, ρ′)
zz
α
zz✉
✉
✉
✉
✉
✉
✉
✉
✉f
$$
❏
❏
❏
❏
❏
❏
❏
❏
❏
(A, ρ)(α,f)//(B, σ),
where (A′, ρ′) is an internal preorder, fis a morphism in PreOrd(C), and α: (A′, ρ′)→
(A, ρ) is a complemented subobject in PreOrd(C). Given two composable mor-
phisms (α, f ): (A, ρ)→(B, σ) and (β , g): (B, σ )→(C, τ ) in PaPreOrd(C), the
composite morphism (β , g)◦(α, f) in PaPreOrd(C) is defined by the external part
of the following diagram
(A′′, ρ′′ )
f′
&&
▼
▼
▼
▼
▼
▼
▼xx
α′
xxqqqqqq
(A′, ρ′)
yy
α
yys
s
s
s
s
sf
&&
▼
▼
▼
▼
▼
▼(B′, σ′)
xx
β
xxq
q
q
q
q
q
qg
%%
▲
▲
▲
▲
▲
▲
(A, ρ)(α,f)//(B, σ)(β,g)//(C, τ )
where the upper part is a pullback. In other words,
(β, g)◦(α, f ) = (αα′, gf ′).
The well-known properties of pullbacks guarantee that this composition is asso-
ciative. For any preorder (A, ρ), the identity on it in PaPreOrd(C) is the arrow
(A, ρ)
1
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈1
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
(A, ρ)1//(A, ρ)
As explained in [2], there is a functor I:PreOrd(C)→PaPreOrd(C) which is the
identity on objects and such that, for any f: (A, ρ)→(B, σ) in PreOrd(C), its value
6 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
I(f): (A, ρ)→(B, σ ) in PaPreOrd(C) is given by the morphism
(A, ρ)
1
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈f
$$
■
■
■
■
■
■
■
■
■
(A, ρ)I(f)//(B, σ).
To simplify the notations, from now on, we shall write Ainstead of (A, ρ) to
denote an internal preorder and A//Bfor a morphism of preorders. The fact
that the initial object 0 of PreOrd(C) is strict implies that 0 is a zero object in
PaPreOrd(C), and thus the category PaPreOrd(C) is equipped with an ideal Nof
(null) morphisms [5], where Nis the class of morphisms in PaPreOrd(C) of the form
0
~~
~~⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
❆
❆
❆
❆
❆
❆
❆
B0//C.
The stable category [2] is defined as a suitable quotient of the category PaPreOrd(C).
In the special case when Cis the category of sets this construction reduces to the
one of the stable category by Facchini and Finocchiaro in [6]. In order to define the
stable category, the following notion is needed:
Definition 1.1. Acongruence diagram in PreOrd(C) is a diagram of the form
(1.1) A1
0
c//α1
0
c
//A1
α1
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
f1
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
A0''
α2
0
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
77
α1
0
77
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
//α0//A B
A2
0
c//α2
0
c//A2
``
α2
``❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
f2
>>
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
where:
•any arrow of the form ////represents a complemented subobject in
PreOrd(C);
•the two triangles commute;
•Ai
0
c//αi
0
c
//Aiis the complement in Aiof the subobject A0//αi
0//Ai;
•f1α1
0=f2α2
0;
•each fiαi
0
cis a trivial morphism.
Two parallel morphisms (α1, f1) and (α2, f2) in PaPreOrd(C), depicted as
and
A1
}}
α1
}}⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤f1
!!
❈
❈
❈
❈
❈
❈
❈
❈A2
}}
α2
}}⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤f2
!!
❈
❈
❈
❈
❈
❈
❈
❈
A//B A //B,
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY7
are equivalent if there is a congruence diagram of the form (2.1) between them.
In this case one writes (α1, f1)∼(α2, f2). As shown in [2], the relation ∼is an
equivalence relation which is also compatible with the composition in PaPreOrd(C),
and is then a congruence (in the sense of [12]) on the category PaPreOrd(C).
Definition 1.2. [2] The quotient category Stab(C) of PaPreOrd(C) by the congru-
ence ∼defined above is called the stable category. If π:PaPreOrd(C)→Stab(C) is
the quotient functor, we also have a functor
Σ = π◦I:PreOrd(C)→Stab(C)
obtained by precomposing πwith the functor I:PreOrd(C)→PaPreOrd(C).
Remark 1.3.The definition of the stable category Stab(C) of PreOrd(C) actually
depends on the class Zof trivial objects. This fact would suggest to write “the
stable category of PreOrd(C) with respect to Z” and call it the “Z-stable category
of PreOrd(C)”. Nevertheless, we prefer to follow the notation adopted in [6] and
refer to Stab(C) as the stable category associated with PreOrd(C). It is also worth
noting that the construction we provide is based on the properties of this particular
class Zof trivial objects (such as the fact that it contains both the initial and the
terminal objects or that it is closed under coproducts) that may not hold for any
pretorsion theory in PreOrd(C). Thus, a priori, it is not possible to construct the
stable category for any pretorsion theory in PreOrd(C).
The stable category is pointed, and the zero object 0 of Stab(C) is the image
by the functor Σ of the initial object in PreOrd(C). As shown in [2], an object A
in PreOrd(C) is such that Σ(A) = 0 if and only if Ais a “discrete” object, that is
an object Aequipped with the preorder given by the discrete equivalence relation
∆Aon A. Moreover, f:A→Bin PreOrd(C) is a trivial morphism if and only if
Σ(f) = 0 in Stab(C). More generally, one has the following result, where we write
< α, f > for the image of the morphism
(A′, ρ′)
zz
α
zz✉
✉
✉
✉
✉
✉
✉
✉
✉f
$$
❏
❏
❏
❏
❏
❏
❏
❏
❏
(A, ρ) (B, σ),
by the functor π:
Lemma 1.4. For a morphism A<α,f > //Bin Stab(C) the following conditions
are equivalent:
(1) < α, f >= 0;
(2) fis a trivial morphism in PreOrd(C).
8 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
Proof. The assumption < α, f >= 0 implies that there is a congruence diagram of
the form
A1
0
c//α1
0
c
//A′
α
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
f
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
A0''
α2
0
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
77
α1
0
77
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
//α0//A B
A2
0
c//α2
0
c//0
``
``❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
>>
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
⑦
hence A0= 0 = A2
0
c(since 0 is a strict initial object). This implies that A1
0
c=A′,
α1
0
c= 1A′and fis trivial on A′.
Conversely, when fis a trivial morphism, it suffices to build the following con-
gruence diagram
A1A1
~~
α
~~⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
f
❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
0
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
66
66
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
////A B
0 0
``
``❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
==
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
showing that < α, f >= 0.
Note also that the “intuition” here should be that a diagram
(1.2) A′
}}
α
}}④
④
④
④
④
④
④
④f
!!
❈
❈
❈
❈
❈
❈
❈
❈
A B
“represents” a morphism < α, f > whose restriction on the (complemented) sub-
object (A′, ρ′) of (A, ρ) is f, and that is “trivial” on the complement of (A′, ρ′) in
(A, ρ), as explained in the following:
Proposition 1.5. If A<α,f >//Bis a morphism in Stab(C) the following diagram
is commutative in Stab(C), where A′c//αc
//Ais the complement of A′in Ain
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY9
PreOrd(C):
(1.3) Σ(A′)
Σ(f)
))
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
Σ(α)##
●
●
●
●
●
●
●
●
●
●
●
Σ(A)<α,f> //Σ(B)
Σ(A′c)
0
55
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
Σ(αc);;
✇
✇
✇
✇
✇
✇
✇
✇
✇
✇
Proof. In order to see that < α, f > Σ(α) = Σ(f) it suffices to consider the diagram
A′
④
④
④
④
④
④
④
④
④
④
④
④
④
④
④
④
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
❉
A′
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤!!
α
!!
❈
❈
❈
❈
❈
❈
❈
❈
❈A′
||
α
||③
③
③
③
③
③
③
③
③f
!!
❈
❈
❈
❈
❈
❈
❈
❈
❈
A′Σ(α)//A<α,f> //B
where the upper quadrangle is a pullback. On the other hand, the assumption that
A′∩A′c= 0 implies that < α, f > Σ(αc) = 0
0
}}④
④
④
④
④
④
④
④
!!
❈
❈
❈
❈
❈
❈
❈
❈
A′c
③
③
③
③
③
③
③
③
③
③
③
③
③
③
③
③!!
αc
!!
❇
❇
❇
❇
❇
❇
❇
❇
❇A′
}}
α
}}⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤f
!!
❇
❇
❇
❇
❇
❇
❇
❇
❇
A′cΣ(αc)//A<α,f> //B
since the composite 0 →A′→Bis obviously trivial. One concludes by Lemma
1.4.
The following result (Lemma 7.11 in [2]) will also be useful:
Lemma 1.6. [2] Let us consider a morphism < α, f > in Stab(C) represented by
(1.4) A′
}}
α
}}④
④
④
④
④
④
④
④f
!!
❈
❈
❈
❈
❈
❈
❈
❈
A B
and assume that for any complemented subob ject B′////Bthe induced mor-
phism f−1(B′)→B′has a Z-cokernel in PreOrd(C). Then the cokernel of < α, f >
exists in Stab(C), and
coker(< α, f >) = Σ(Z−coker(f)).
10 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
2. The universal property of Stab(C)
Definition 2.1. Let (A,T,F)pre−t. be a category Awith a given pretorsion theory
(T,F) in A. If (B,T′,F′)t. is a pointed category Bwith a given torsion theory
(T′,F′) in it, we say that a torsion theory functor is a functor G:A→Bsatisfying
the following two properties:
(1) G(A)∈ T ′for any A∈ T ,G(B)∈ F ′for any B∈ F ;
(2) if T(A)→A→F(A) is the canonical short Z-exact sequence associated
with Ain the pretorsion theory (T,F), then
0→G(T(A)) →G(A)→G(F(A)) →0
is a short exact sequence in B.
When Cis a pretopos, we write (Eq(C),ParOrd(C)) for the pretorsion theory in
PreOrd(C) where Eq(C) is the category of equivalence relations and ParOrd(C) the
category of partial orders in C.
Proposition 2.2. The functor Σ: PreOrd(C)→Stab(C) is a torsion theory functor
that preserves finite coproducts and monomorphisms.
Proof. The fact that (Eq(C),ParOrd(C)) is a pretorsion theory was observed in
[8] (for any exact category C), while the preservation of finite coproducts and
monomorphisms by the functor Σ was established in Proposition 6.2 and Proposi-
tion 6.1 in [2], respectively. It remains to prove that (Eq(C),ParOrd(C)) is a torsion
theory in the pointed category Stab(C). Consider any morphism < α, f >: (A, ρ)→
(B, σ), where ρis an equivalence relation on Aand σa partial order on B, depicted
as
(A′, ρ′)
zz
α
zz✉
✉
✉
✉
✉
✉
✉
✉
✉f′
$$
■
■
■
■
■
■
■
■
■
(A, ρ)f//(B, σ)
The fact that ρis an equivalence relation and αa complemented subobject in
PreOrd(C) implies that also ρ′=α−1(ρ) is an equivalence relation (on A′). It follows
that f: (A′, ρ′)→(B, σ) is a trivial morphism in PreOrd(C) (since (Eq(C),ParOrd(C))
is a pretorsion theory in PreOrd(C)), hence a zero morphism in Stab(C).
Next, let us prove that the canonical short Z-exact sequence in PreOrd(C)
(A, ∼ρ)i//(A, ρ)π//(A/∼ρ, π(ρ))
associated with any internal preorder (A, ρ), where ∼ρ=ρ∩ρoand iis the canonical
inclusion, becomes a short exact sequence in Stab(C).
First, Proposition 7.1 in [2] implies that Σ(i) : (A, ∼ρ)→(A, ρ) is the kernel of
Σ(π): (A, ρ)→(A/∼ρ, π(ρ)). To see that Σ(π) is the cokernel of Σ(i) we shall use
Lemma 1.6. To apply this result, observe that, for any complemented subobject
(A′, ρ′) of (A, ρ), the upper horizontal morphism in the pullback
(A′, ρ′′)////
i′
(A, ρ ∩ρo)
i
(A′, ρ′)////(A, ρ)
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY11
is again a complemented subobject. This implies that ρ′′ is the restriction to A′of
the equivalence relation ρ∩ρoon A, i.e. the following square is a pullback in C:
ρ′′ //
ρ∩ρo
A′×A′//A×A
This implies that ρ′′ is an equivalence relation, and then the Z-cokernel of i′exists
(by Proposition 7.3 in [2]). The result then follows from Lemma 1.6.
Theorem 2.3. Let Cbe a pretopos. The functor Σ: PreOrd(C)→Stab(C) has
the following property: it is universal among all finite coproduct preserving torsion
theory functors G:PreOrd(C)→X, where Xhas a torsion theory (T,F) and finite
coproducts. This means that any finite coproduct preserving torsion theory functor
G:PreOrd(C)→Xfactors uniquely through Σ:
PreOrd(C)Σ//
∀G$$
❏
❏
❏
❏
❏
❏
❏
❏
❏Stab(C)
∃!G
{{
X.
Moreover, the induced functor Gpreserves finite coproducts, and is a torsion theory
functor.
Proof. Since PreOrd(C) and Stab(C) have the same objects it is clear that the
definition of the functor Gon the objects is “forced” by G:G(A) = G(A), for any
object Ain Stab(C). Let then < α, f > :A→Bbe a morphism in Stab(C) (as
in (1.4)), and recall that it is then [f , 0], the morphism induced by the universal
property of the coproduct A′`A′c=A, since the diagram 1.3 in Proposition 1.5
commutes. Again, the condition G◦Σ = Gand the fact that Ghas to preserve
binary coproducts force the definition of the functor Gon morphisms:
G(< α, f >) = [G(f),0].
The above arguments already prove the uniqueness of the functor Gwith the
above properties. We still need to check that Gis well-defined on morphisms, i.e. if
< α, f >=< α, f > in Stab(C), then G(< α, f >) = G(< α, f >) or, equivalently,
[G(f),0] = [G(f),0].
Now, the assumption < α, f >=< α, f > gives a congruence diagram
12 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
(2.1) A1//α′′
//A′
~~
α
~~⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
f
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
A0''
α′
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
77
α′
77
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
//α0//A B
A1//α′′ //A′
``
α
``❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆f
>>
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
In PreOrd(C), if we write A′′ and A′′ for the complements of A′and A′in A,
respectively, we have the decompositions
A=A′aA′′ =A0aA1aA′′
and
A=A′aA′′ =A0aA1aA′′.
Accordingly, by taking into account the distributivity law for subobjects (see Lemma
1.4.2 in [11] and recall that coproducts in PreOrd(C) are computed “component-
wise” [2, Proposition 5.3]) we get the following equalities:
A= (A0aA1aA′′)∩(A0aA1aA′′ )
= (A0∩A0)a(A0∩A1)a(A0∩A′′)a(A1∩A0)a(A1∩A1)a(A1∩A′′)a
(A′′ ∩A0)a(A′′ ∩A1)a(A′′ ∩A′′).
By taking into account the equalities A0∩A0=A0and
A0∩A1=A0∩A′′ =A1∩A0=A′′ ∩A0= 0,
we see that
A=A0a(A1∩A1)a(A1∩A′′)a(A′′ ∩A1)a(A′′ ∩A′′).
We then observe that:
•f=fon A0;
•fis trivial on A1and fis trivial on A1, hence fand fare trivial on A1∩A1;
•fis trivial on A1and <α, f > is zero in Stab(C) on A′′, hence < α, f >
and <α, f > are zero morphisms on A1∩A′′ in Stab(C);
•similarly, < α, f > and <α, f > are zero morphisms on A′′ ∩A1;
•< α, f > is zero on A′′ and < α, f > is zero on A′′, and this implies that
< α, f > and <α, f > are both zero on A′′ ∩A′′ in Stab(C).
By assumption Gis a torsion theory functor, hence it sends the trivial morphisms
in PreOrd(C) to zero morphisms in X. A zero morphism in Stab(C) is a morphism
of the form
A′
}}
β
}}④
④
④
④
④
④
④
④g
!!
❈
❈
❈
❈
❈
❈
❈
❈
A B
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY13
where gis trivial in PreOrd(C). Accordingly, in this case, G(< β, g >) = [0,0] is
the zero morphism from Ato Bin X. By assumption Gpreserves finite coproducts,
hence
G(A) = G(A0)aG(A1∩A1)aG(A1∩A′′)aG(A′′ ∩A1)aG(A′′ ∩A′′),
and from the observations above we know that [f, 0] and [f , 0] coincide on G(A0)
and are zero morphism on all the other components. It follows that [f, 0] = [f , 0],
and the definition of Gis compatible with the congruence defining the morphisms
in Stab(C).
To prove that Gis a functor consider two composable morphisms in Stab(C)
A<α,f >//B<β,g>//C,
and the following composition diagram where the upper square is a pullback
Ac
!!
αc
!!
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈A
~~
α
~~⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
f
❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
A′
}}
α
}}③
③
③
③
③
③
③
③
③
③f
!!
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈B′
}}
β
}}④
④
④
④
④
④
④
④
④
④g
❇
❇
❇
❇
❇
❇
❇
❇
❇
❇
Ahα,f i//Bhβ,gi//C
A′c
aa
aa❉
❉
❉
❉
❉
❉
❉
❉
❉0
==
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤B′c
aa
aa❈
❈
❈
❈
❈
❈
❈
❈
❈0
>>
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
and A′c,B′care the complements of A′and B′in Aand in B, respectively. We
also consider the pullback
(2.2) Ac
////A′
f
B′c////B
expressing the fact that Ac=f−1(B′c), and we observe that
A=A′aA′c=AcaAaA′c.
In Xwe have to check that [G(g),0][G(f),0] and [G(gf),0] coincide on
G(A) = G(Ac)aG(A)aG(A′c).
On G(A) we have G(g)G(f) in both cases, hence
G(< β, g >)·G(< α, f >) = G(< β , g >< α, f >).
Now, on Acthe morphism ffactors through B′c(see diagram 2.2), hence
G(< β, g >)·G(< α, f >)
14 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
is the zero morphism on G(Ac). But G(< β , g >< α, f >) is also the zero morphism
on G(Ac), hence these two morphisms are equal on G(Ac). Finally, on A′cwe have
< α, f >= 0, hence again
G(< β, g >)·G(< α, f >) = 0 = G(< β , gi>< α, f >)
on G(A′c), completing this part of the proof.
One clearly has that G(1A) = G(1A) = 1G(A), since Gis a functor. To see that
Gpreserves finite coproducts one has to observe that Gpreserves finite coproducts
and these are calculated in Stab(C) as in PreOrd(C) (see Corollary 6.3 in [2]). In
order to check that Gis a torsion theory functor, since Gand Gcoincide on ob jects,
it will suffice to prove that Gpreserves the canonical short exact sequences in the
torsion theory. This follows from Proposition 2.2, since the canonical short exact
sequence in the torsion theory in Stab(C) is the image by Σ of the canonical short
Z-exact sequence in the pretorsion theory in PreOrd(C) and, by assumption, G
preserves this kind of sequences.
3. The case of τ-pretoposes
The aim of this section is to prove that, when Cis a τ-pretopos (in the sense
of Definition 3.7), all the short exact sequences in Stab(C) are images (up to iso-
morphism) by the functor Σ : PreOrd(C)→Stab(C) of a short Z-exact sequence in
PreOrd(C).
Proposition 3.1. The stable category Stab(C) has disjoint binary coproducts.
Proof. Consider any commutative diagram in Stab(C)
C<α,f > //
<β,g>
A
Σ(sA)
BΣ(sB)//A`B.
where sAand sBdenote the copro jections of the coproduct in PreOrd(C). This
means that there is a congruence diagram
A1
0
c////A1
α
⑧
⑧
⑧
⑧
⑧
⑧
⑧
⑧
⑧
⑧
⑧
f//A
sA
!!
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
A0''
α1
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
77
α0
77
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
////A A `B
A2
0
c////A2
__
β
__❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
g//B
sB
==
④
④
④
④
④
④
④
④
④
④
④
④
In the category PreOrd(C) the equality sAfα0=sBgα1induces a unique morphism
A0→A×A`BBto the pullback A×A`BBof sAand sB. Since A×A`BB
is the initial object 0 in PreOrd(C) and 0 is strict, it follows that A0= 0. This
implies that A1
0
c=A1and A2
0
c=A2, and the morphisms sAfand sBgare both
trivial. Since Σ(sA) and Σ(sB) are monomorphisms in Stab(C) (by Proposition
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY15
2.2), it follows that < α, f >= 0 and < β , g >= 0 . From the fact that 0 is a zero
object in Stab(C) it follows that the square
0//
A
Σ(sA)
BΣ(sB)//A`B.
is a pullback in Stab(C), as desired.
Definition 3.2. Let Xbe a category with binary coproducts. One says that binary
coproducts in Xare pre-universal if, given any morphism f:C→A`B, there
exists a commutative diagram of the form
(3.1) A′sA′//
fA
C
f
B′
sB′
oo
fB
AsA//A`B B
sB
oo
where the top row of the diagram is a sum (i.e. C=A′`B′and s′
Aand s′
Bare
the coprojections).
Since in Stab(C) binary coproducts (exist and) are computed as in PreOrd(C) [2,
Corollary 6.3], in the sequel we shall often write AsA//A`B B
sB
oofor the
coprojections of the coproduct of Aand Bboth in Stab(C) and in PreOrd(C).
Proposition 3.3. The stable category Stab(C) has pre-universal binary coprod-
ucts.
Proof. Let us consider any morphism < α, f > :C→A`B, and then the diagram
C′
Σ(f)
((
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
Σ(α)
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
C<α,f > //A`B
C′c
0
66
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
Σ(γ)
>>
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
in Stab(C) and
A′′ sA′′ //
f′′
A
C′
f
B′′
sB′′
oo
f′′
B
AsA//A`B B
sB
oo
in PreOrd(C), respectively, where in the second one A′′ and B′′ are the inverse
images along f, and then C′=A′′ `B′′ . Note that, by Proposition 1.5, in the
16 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
stable category we have the equality < α, f > Σ(γ) = 0. There is then the following
factorisation Σ(fA) in Stab(C):
A′′
Σ(f′′
A)
))
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
❚
σA′′
$$
■
■
■
■
■
■
■
■
■
■
■
■
■
A′′ `C′cΣ(fA)//A
C′c
0
55
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
σC′c
::
✉
✉
✉
✉
✉
✉
✉
✉
✉
✉
✉
✉
✉
where σA′′ and σC′care the coproduct coprojections. One then sets A′=A′′ `C′c
and gets the diagram
(3.2) A′sA′//
Σ(fA)
C
<α,f >
B′′
Σ(α)sB′′
oo
Σ(f′′
B)
AsA//A`B B,
sB
oo
whose commutativity can be checked as follows. We have
C=C′aC′c=A′′ aB′′ aC′c=A′aB′′.
By Proposition 1.5, we know that
< α, f > sA′σA′′ =< α, f > Σ(α)sA′′ = Σ(f)sA′′
and
< α, f > sA′σC′c=< α, f > Σ(γ) = 0.
Since we also have that
sAΣ(fA)σA′′ =sAΣ(f′′
A) = Σ(f)sA′′ .
and
sAΣ(fA)σC′c=sA0 = 0,
we conclude that < α, f > sA′=sAΣ(fA), as desired. On the other hand, the
following equalities show that the right-hand side of diagram 3.2 commutes:
< α, f > Σ(α)sB′′ = Σ(f)sB′′ =sBΣ(f′′
B).
Remark 3.4.Let us observe that the choice of the objects A′and B′in Definition
3.2 is by no means unique. Indeed, in the proof of Proposition 3.3, we could as
well have chosen A′=A′′ and B′=B′′ `C′c(with reference to diagram (3.2)).
So the pre-universality of binary coproducts could be rephrased as the existence
of three objects A′′, B ′′, C′′ , respectively mapped by fin A, B, 0, and such that
C=A′′ `B′′ `C′′.
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY17
Lemma 3.5. Let Xbe a category with a zero object and binary coproducts which
are disjoint and pre-universal. Assume that
Kk//Af//B
and
Nn//Cg//D
are composable morphisms such that k= ker(f) and n= ker(g). Then the mor-
phism k`n:K`N→A`Cis the kernel of f`g:A`C→B`D.
Proof. First of all the composite (f`g)(k`n) is clearly the zero morphism:
(fag)(kan) = fk agn = 0 a0 = 0.
Next consider any arrow h:E→A`Csuch that (f`g)h= 0. By the pre-
universality of binary coproducts we can form the commutative diagram
E1
s1//
h1
E
h
E2
s2
oo
h2
AsA//A`C C
sC
oo
where E=E1`E2. We have the equality
sBfh1= (fag)sAh1= (fag)hs1= 0,
where sB:B→B`Dis a monomorphism, hence there is a unique morphism m1
such that km1=h1. Similarly, one can prove that there is a unique m2such that
nm2=h2. The universal property of the coproduct E=E1`E2gives a unique
morphism m=m1`m2:E→K`Nsuch that
(kan)ms1= (kan)sKm1=sAkm1=sAh1=hs1.
Symmetrically, one has that (k`n)ms2=hs2, yielding the equality (k`n)m=
h. To check the uniqueness of the factorization consider then another morphism
r:E→K`Nsuch that (k`n)r=h. Again by pre-universality we have a
commutative diagram
˜
E1
˜s1//
r1
E
r
˜
E2
˜s2
oo
r2
KsK//K`N N
sN
oo
18 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
where E=˜
E1`˜
E2. Consider the following commutative diagram, where E1=
ˆ
E11 `ˆ
E12 and E2=ˆ
E21 `ˆ
E22,
ˆ
E11
ˆs11 //
s11
E1
s1
ˆ
E12
ˆs12
oo
s12
˜
E1˜s1
//E˜
E2
˜s2
oo
ˆ
E21 ˆs21
//
s21
OO
E2
s2
OO
ˆ
E22
ˆs22
oo
s22
OO
that exists by the pre-universality of binary coproducts. By assumption the restric-
tions to ˆ
E11 of mand rare equal when composed with k`n
(kan)rs1ˆs11 = (kan)ms1ˆs11
and these composites both factor through sAk
(kan)ms1ˆs11 =hs1ˆs11 =sAh1ˆs11 =sAkm1ˆs11
and sAkis a monomorphism (as a composite of monomorphisms). This means that
mand rinduce a unique morphism m1ˆs11 =r1ˆs11 :ˆ
E11 →K. Similarly, mand
ralso induce a unique morphism m2ˆs22 =r2ˆs22 :ˆ
E22 →N. On ˆ
E12 we get the
following equalities
sAh1ˆs12 =hs1ˆs12
= (kan)ms1ˆs12
= (kan)rs1ˆs12
= (kan)r˜s2s12
= (kan)sNr2s12
=sCnr2s12
showing that the induced morphisms ˆ
E12 →Aand ˆ
E12 →Care the zero mor-
phisms, since the coproduct A`Cis disjoint. But kand nare both monomor-
phisms, hence both the morphisms ˆ
E12 →Kand ˆ
E12 →Nare zero as well.
Similarly, the morphisms ˆ
E21 →Kand ˆ
E21 →Nare also zero. By compos-
ing with sK:K→K`Nand sN:N→K`Nwe obtain that the restrictions
ˆ
E11 →K`Nof mand rare equal. Similarly, the restrictions ˆ
E12 →K`Nof
mand rare both zero, hence the restriction of mand rto E1are equal. In a
similar way one checks that the restrictions of mand rto E2are equal, and, finally,
m=r.
Proposition 3.6. The category Stab(C) has kernels. If < α, f > :A→Bis a
morphism in Stab(C) as in diagram 1.2, its kernel is given by
Σ(k`1A′c) : K`A′c//A′`A′c
where kis the Z-kernel of fin PreOrd(C) and A′cis the complement of A′in A.
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY19
Proof. In PreOrd(C) we have the Z-kernels
Kk//A′f//B
and
A′cA′cτ//1.
By Proposition 7.1 in [2] we know that Σ: PreOrd(C)→Stab(C) sends these Z-
kernels to the kernels
Kk//A′f//B
and
A′cA′c//0
in Stab(C), respectively. By Lemma 3.5 we know that k`1A′cis then the kernel of
f`0: A′`A′c→B`0 = B. From Proposition 1.5 it follows that in the following
diagram in Stab(C)
K`A′ck`1A′c//A<α,f> //B
the morphism k`1A′cis the kernel of < α, f >, as desired.
We now recall the following definition that had a role in proving some of the
results in [2]:
Definition 3.7. Aτ-pretopos is a pretopos Cwith the property that the transitive
closure of any relation on an object in Cexists in C.
Any σ-pretopos is in particular a τ-pretopos [11]. As observed in [2], any ele-
mentary topos is a τ-pretopos (Proposition 7.7).
Proposition 3.8. When Cis a τ-pretopos, two composable morphisms in Stab(C)
A<α,f >//B<β,g>//C
form a short exact sequence if and only if, up to isomorphism, they are the image
by Σ of a short exact sequence in PreOrd(C).
Proof. One implication follows from Theorem 7.14 in [2] (for which the assumption
that Cis a τ-pretopos is needed). Conversely, consider a short exact sequence
(3.3) A<α,f >//B<β,g>//C
in Stab(C). By Proposition 3.6 we know that the kernel of < β, g > is a morphism
of type Σ(n):
< α, f >= ker < β, g >= Σ(n).
More precisely, going back to the construction of the kernel Ker(< β, g >) as in
Proposition 3.6 we have the diagram
K//k//
||
s1
||②
②
②
②
②
②
②
②
②
②
②
②
②
②
②B′
~~
β
~~⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
g
!!
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈
❈B′c
vv
βc
vv♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠
K`B′c
k`1B′c//B C
20 FRANCIS BORCEUX, FEDERICO CAMPANINI, AND MARINO GRAN
and we set n=k`1B′c. When Cis a τ-pretopos the functor Σ: PrOrd(C)→
Stab(C) sends Z-cokernels to cokernels (Corollary 7.13 in [2]): this implies that
< β, g > =coker(< α, f >)
=coker(Σ(n))
= Σ(Z−coker(n)).
The construction of the cokernel coker(< α, k >) as in Lemma 7.11 of [2] shows
that the Z-cokernel q:B→Qof nis such that
Σ(q) = coker(<1K`B′c, k a1B′c>) = coker(Σ(n)).
Since the sequence (3.3) is exact, < β, g > is isomorphic to Σ(q), and the sequence
(3.3) is isomorphic to the exact sequence
Σ(K`B′c)Σ(n)//Σ(B)Σ(q)//Σ(Q),
as desired.
Corollary 3.9. When Cis a τ-pretopos, under the assumptions of Theorem 2.3,
the functor Gpreserves the short exact sequences whenever the functor Gsends
short Z-exact sequences to short exact sequences.
Proof. This follows immediately from Proposition 3.8.
Corollary 3.10. When Cis a τ-pretopos, under the assumptions of Theorem 2.3,
the functor Gpreserves cokernels whenever the functor Gsends Z-cokernels to
cokernels.
Proof. Let < α, f >:A→Bbe a morphism in Stab(C) and q:B→Cits cokernel.
As we have noticed in the proof of Proposition 3.8, this cokernel is the image by
Σ of the Z-cokernel of f, i.e. q= Σ(Z-coker(f)). In the category Xwe have that
G(q) = G(q) = coker(G(f)) = coker(G(f)).Consider then the following diagram
G(A′)
G(α)
ww♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥
G(f)
##
●
●
●
●
●
●
●
●
●
●
●
●X
G(A) = G(A′)`G(A′c)[G(f),0] //G(B)G(q)//
x
<<
①
①
①
①
①
①
①
①
①
①
①
①
①G(Q)
∃!y
OO
G(A′c)
G(αc)
ggPPPPPPPPPPPPPPPPP
0
;;
✇
✇
✇
✇
✇
✇
✇
✇
✇
✇
✇
✇
One sees that G(q)G(f) = 0, since G(q) = coker(G(f)) and, obviously, G(q)0 = 0,
hence G(q)[G(f),0] = 0. Then, if x[G(f),0] = 0 we get
xG(f) = x[G(f),0]G(α) = 0G(α) = 0,
yielding a unique factorisation yof xthrough G(q) = coker(G(f)). It follows that
G(q) = coker([G(f),0]), and this implies that
G(q) = G(q) = coker([G(f),0]) = coker(G(< α, f >)).
THE STABLE CATEGORY OF PREORDERS IN A PRETOPOS II: THE UNIVERSAL PROPERTY21
Proposition 3.11. If in Theorem 2.3 we also assume that
•Cis a τ-pretopos,
•Xhas finite coproducts that are disjoint and pre-universal,
•G:PreOrd(C)→Xsends Z-kernels to kernels,
then the functor G:Stab(C)→Xpreserves kernels.
Proof. Consider a morphism < α, f >:A→Bin Stab(C) and its kernel k`1A′c
in Stab(C) as in Proposition 3.6
Kk//
s1
A′
{{
α
{{✇
✇✇
✇
✇
✇
✇
✇✇
✇
✇
✇
f
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆A′c
s2
rr❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡
α′
~~⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
⑥
K`A′c
k`1A′c//A′`A′c
<α,f > //B,
where kis the Z-kernel of f. From the assumptions it follows that
G(k) = G(k) = ker(G(f)) = ker(G(f)),
and one also has
1G(A′c)=ker(0),where 0: G(A′c)→0.
From Lemma 3.5 and Theorem 2.3 we get the equalities
G(ka1A′c) = G(k)a1G(A′c)=ker(G(f)a0) = ker(G(< α, f >)),
hence Gpreserves the kernel of < α, f >.
Theorem 3.12. Let Cbe a τ-pretopos. The functor Σ : PreOrd(C)→Stab(C) is
universal among all finite coproduct preserving torsion theory functors G:PreOrd(C)
→X, where Xhas a torsion theory (T,F), and it has binary coproducts that are
disjoint and pre-universal. Consider any finite coproduct preserving torsion theory
functor G:PreOrd(C)→Xthat sends Z-kernels and Z-cokernels (then in particu-
lar short Z-exact sequences) to kernels, cokernels (and short exact sequences). The
functor Gthen factors uniquely through Σ
PreOrd(C)Σ//
∀G$$
❏
❏
❏
❏
❏
❏
❏
❏
❏Stab(C)
∃!G
{{
X.
and the induced functor Gpreserves finite coproducts, is a torsion theory functor
that preserves kernels and cokernels (then in particular short exact sequences).
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Universit´
e catholique de Louvain, Institut de Recherche en Math´
ematique et Physique,
1348 Louvain-la-Neuve, Belgium
Email address:francis.borceux@uclouvain.be
Universit`
a degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”,
35121 Padova, Italy
Email address:federico.campanini@unipd.it
Universit´
e catholique de Louvain, Institut de Recherche en Math´
ematique et Physique,
1348 Louvain-la-Neuve, Belgium
Email address:marino.gran@uclouvain.be
