SUBOBJECT CLASSIFICATION AND GEOMETRIC MORPHISMS OF PARTIALLY ORDERED ACTS

UNIVERSITY OF TARTU

Faculty of Mathematics and Computer Science Institute of Pure Mathematics

Chair of Algebra

Lauri Tart

SUBOBJECT CLASSIFICATION AND GEOMETRIC MORPHISMS OF PARTIALLY ORDERED ACTS

Master thesis

Supervisor: senior researcher Valdis Laan

Tartu 2005

Table of contents

Introduction 3

1 Preliminaries 4

1.1 Posets over a pomonoid . . . 4

1.2 Monomorphisms . . . 6

1.3 Toposes . . . 7

2 Subobject classification 9 2.1 Cartesian closedness . . . 9

2.2 Subobject classifiers . . . 12

2.3 Monomorphism types . . . 14

2.4 Category of posets as a topos . . . 18

2.5 Submonomorphisms and subclassifiers . . . 19

3 Geometric morphisms 22 3.1 Pofunctors and poadjunctions . . . 22

3.2 Tensor products . . . 29

3.3 Pogeometric morphisms . . . 37

3.4 Points . . . 38

Res¨umee 44

References 45

Introduction

One way of viewing a category is to consider it as a very generic typed monoid, ie a monoid where only some elements can be multiplied (when they are of the same “type”) and whose elements form a class instead of merely a set.

Conversely a rather important example of a category is the “dot-category”

based on a monoid.

The latter view of monoids as categories also provides us with a func- torial description of acts over this monoid, these being the presheaves (or set-valued functors) on the “dot-category”. Presheaves themselves are relati- vely important tools of topos theory and give rise to a wealth of notions and results (some of which would be Grothendieck toposes, sheaves, sheafifica- tion and the logical functors arising from truth value objects). Since acts are presheaves (and rather archetypal ones at that), these notions apply to them as well, providing us with a general view of known act-specific results (for instance, that every act is a quotient of free acts, or the Hom-tensor adjunc- tion) and new ones (the sheafification technique and corresponding adjoints, or the logic on subacts).

One may take this a little farther and consider not just sets, but partially ordered sets. In the current work we work with the category of partially ordered acts or posets over a pomonoid (partially ordered monoid). These can be seen as the “dot-category” version of partial-order-valued functors.

The master thesis consists of two primary parts. In the first part we examine how close the category of ordered acts is to being a topos. It turns out that the category is complete and cocomplete, and even cartesian closed, but unfortunately is not a topos, as it lacks a subobject classifier. There are some limited subobject classifiers and generalizations thereof, but none for any of the more common kinds of monomorphisms.

Sydney Bulman-Fleming and Mojgan Mahmoudi have concurrently done a lot of the same work in their recent article [BFM]. There are things in this thesis that they did not consider, namely the regularly extremal morphisms.

Also, they study the topos-characteristic notions in much less detail, and subobject classification actually gets no explicit mention in their work.

In the second part we try to generalize the notion of geometric morphisms into one that would be useful for posets. For this we introduce the notions of pofunctors, poadjunctions and universal pococones (generalizations of coli- mits). We prove a version of the usual Hom-tensor adjunction and find some naturally occurring geometric morphisms arising from pomonoid homomor- phisms. Finally, we define the notion of a point in a poset category. In the end we find that points correspond to flat posets over pomonoids, ie posets that induce a tensor multiplication that preserves universal pocones.

1 Preliminaries

For general background in category theory we refer the reader to [Bo] or [CWM]. The basic text for topos theory used in the current work is [MLM].

For a reasonable degree of self-containment we shall review most of the basic notions in the following part of the thesis.

1.1 Posets over a pomonoid

The objects of our study are right S-posets over a pomonoid S.

Definition 1.1 A partially ordered monoid (a pomonoid) is an ordered al- gebraic structure (S,≤,·) such that the following hold:

a) ∀x, y, z ∈S (x·y)·z =x·(y·z), b) ∃1∈S :∀x∈S 1·x=x·1 =x,

c) ∀x∈S x≤x,

d) ∀x, y ∈S (x≤y∧y ≤x)⇒(x=y), e) ∀x, y, z ∈S (x≤y∧y≤z)⇒(x≤z), f) ∀x, y, z ∈S (x≤y)⇒(x·z ≤y·z), g) ∀x, y, z ∈S (x≤y)⇒(z·x≤z·y).

Definition 1.2 A partially ordered right set over a fixed pomonoid S (a right S-poset) is an ordered algebraic structure (A,≤,·s)s∈S such that the following hold:

a) ∀x∈A∀s, t ∈S (x·s)·t=x·(s·t), b) ∀x∈A x·1 =x,

c) ∀x∈A x≤x,

d) ∀x, y ∈A (x≤y∧y≤x)⇒(x=y), e) ∀x, y, z ∈A (x≤y∧y≤z)⇒(x≤z), f) ∀x, y ∈A∀s ∈S (x≤y)⇒(x·s ≤y·s), g) ∀x∈A∀s, t ∈S (s≤t)⇒(x·s≤x·t).

For better distinction between the two multiplications we write simplyst instead of s·t when s and t are elements of a pomonoid.

In the following, we also allow empty posets and empty (one-sided) ideals of pomonoids. All ideals of a pomonoid are taken as purely algebraic (one- sided) ideals, with no order restrictions (as it has been done in some studies).

Naturally, leftS-posets can be treated in the same way as rightS-posets.

Keeping this in mind we deal primarily with rightS-posets. Note that proofs for right S-posets can be carried over to left S-posets and vice versa. We refer to this operation as taking the left-(right-)sided version of the proof in question. We write A_S to emphasize that A is a right S-poset and _SA to stress that it is a left S-poset.

In the following, let S be a partially ordered monoid (pomonoid) and Pos_S the category of partially ordered right sets over this pomonoid with order-preserving act homomorphisms as morphisms. Also, let Act_S denote the usual category of right S-acts. The categories of leftS-posets andS-acts are denoted correspondingly _SPos and _SAct.

A mapping f : A_S → B_S between two S-posets is therefore an S-poset homomorphism iff the following hold:

a) ∀x∈A∀s∈S f(x)·s=f(x·s), b) ∀x, y ∈A (x≤y)⇒(f(x)≤f(y)).

The morphism sets of the category PosScan also be ordered. For this take f, g :A_S →B_S in Pos_S and define f ≤g iff f(a)≤g(a) for alla∈A. In the following we only consider this pointwise ordering for S-poset morphisms.

The pomonoid S can be made into a right poset SS over itself with its monoid multiplication and natural order. For any other rightS-poset B_S and element b ∈B, we can define a morphism b:S_S →B_S with

b(s) =b·s.

As

b(st) =b·(st) = (b·s)·t=b(s)·t

for all s, t∈S, this is an act homomorphism. Ifs ≤t inS, then b(s) =b·s ≤b·t=b(t)

and b is order-preserving as well. So this definition does give us an S-poset morphism.

Lemma 1.1 For b, c∈B, b≤c, s∈S 1) b◦s=b·s;

2) b≤c.

Proof. Obviously

(b◦s)(t) =b·(s·t) = (b·s)·t= (b·s)(t) for all t∈S. Likewise,

b(t) =b·t≤c·t =c for all t∈S.

Definition 1.3 A right T-poset A_T that is also a left S-poset _SA is called an (S, T)-biposet if

s·(a·t) = (s·a)·t for all a∈A, s∈S, t∈T.

For a categoryC, we denote the class of its objects as Ob(C).

In the same way, if A, B ∈Ob(C), then for the set of all morphisms from A to B we write MorC(A, B).

The category of all sets and functions will be denoted by Sets.

For the terminal object of a category C we write 1C or simply 1, and for any C ∈Ob(C) we will denote the unique morphism to 1 as !C :C →1.

If≤ is a partial order, then we define < as the relation <:=≤ \=.

1.2 Monomorphisms

In category theory there are several different types of monomorphisms. Let us also recall these definitions.

Definition 1.4 A morphism ι:B →C in a category C is called

• a coretraction (or a section), if it is left invertible, i.e.

(∃f :C→B)(f ◦ι= 1_B);

• a regular monomorphism, if it is an equalizer, i.e.

(∃D∈Ob(C))(∃f, g :C →D)((B, ι)≈Equ(f, g));

• a strict monomorphism, if

(∀H ∈Ob(C))(∀h:H →C)[(∀D∈Ob(C))(∀f, g:C →D) (f◦ι=g◦ι⇒f ◦h=g◦h)⇒(∃!k :H →B)(ι◦k =h)];

• a strong monomorphism, if

(∀U, V ∈Ob(C))(∀f :U →B)(∀g :V →C)(∀π:U →V) (ι◦f =g◦π∧π is epimorphism ⇒

(∃h:V →B)(f =h◦π∧g =ι◦h));

• an extremal monomorphism, if

(∀D∈Ob(C))(∀π:B →D)(∀f :D→C)

(ι=f ◦π∧π is epimorphism ⇒π is isomorphism),

• a monomorphism, if it is left cancellable, i.e.

(∀D∈Ob(C))(∀f, g:D→B)(ι◦f =ι◦g ⇒f =g).

We have the following implications (see [HS], pages 103-104, 265-266 and 110 for example):

coretraction ⇒ regular monomorphism ⇒ strict monomorphism ⇒ strong monomorphism⇒ extremal monomorphism ⇒ monomorphism

1.3 Toposes

Definition 1.5 In a finitely complete category we say that an object W is the exponential object of objects Y and X, if there exists such a morphism eval :W ×Y →X that for any other morphism α :Z ×Y → X there is a unique morphism α⁰ :Z →W such that the diagram

W ×Y _eval ^//X Z ×Y

W ×Y

α⁰×1_Y

Z ×Y

X

α

?

??

??

??

??

??

??

commutes.

The usual notation for the exponential object of Y and X is X^Y. The above definition is of course a writeout of the (alternate) definition that exponentiation is right adjoint to multiplication (ie − × Y a −^Y), with adjunction expressed in terms of a universal morphism.

Definition 1.6 In a finitely complete category the subobject classifier is a monomorphism true : 1 → Ω such that for any other monomorphism ι : B →A there is a unique morphism φ_B such that the square

A _φ Ω

B

//

B

A

ι

B ^{!B //}11

Ω

true

turns out to be a pullback. The object Ω is usually called the truth value object of this category, and it is unique up to isomorphism. Actually, the morphism true is unique up to isomorphism (in the category of morphisms) as well.

Note that monomorphisms can be replaced with subobjects in the above definition since equivalent monics are isomorphic. In the following, when dealing with partially ordered acts, if we have a subobject, we identify it with the representative monomorphism that injects the image corresponding to all the monomorphisms belonging to the equivalence class that makes up this subobject. So ι will always be an injection of a subset.

Definition 1.7 A category C is called an (elementary) topos, if (i) there exist all finite limits and colimits;

(ii) for each pair of objects E, F ∈ Ob(C) there exists their exponential object E^F;

(iii) there is a (the) subobject classifier Ω.

Definition 1.8 A category C is called a cartesian closed category, if it is finitely complete and satisfies (ii).

2 Subobject classification

Our aim in this part is to see whether the category ofS-posets is a topos and if not, whether there exist some similar constructions for replacing those in a topos.

2.1 Cartesian closedness

The following fact is also proved at the beginning of section 2.3 of [BFM]

Lemma 2.1 Monomorphisms in the category Pos_S are precisely the injective order-preserving S-act homomorphisms.

Proof.Since Pos_Sis a concrete category, every injective order-preserving S-act homomorphism is a monomorphism. To prove the converse, we only need to find a free S-poset with one generator. The S-poset S_S with its natural order is just that. Therefore no non-injective morphism can be a monomorphism and our proof is complete.

Lemma 2.2 The terminal object in PosS is the one-element poset 1with its only possible action.

Proof. For an arbitrary S-poset A, define a mapping !_A : A → 1 as

!_A(x) = ∗for allx∈A, where∗ is the only element of1. We actually cannot define any other maps from A to 1. We have !_A(x)·s= ∗ ·s =∗ =!_A(x·s) for any x ∈ A, s ∈ S and !_A(x) = ∗ ≤ ∗ =!_A(y) for any x ≤ y in A. Hence

!_A is a morphism and since we cannot get any other maps A→1, we cannot get any other morphisms either. So 1 is the terminal object in Pos_S.

The following proposition is effectively the same as Theorem 18 of [BFM].

Proposition 2.1 The category Pos_S is cartesian closed.

Proof. For the existence of limits and colimits see [Fa]. For the sake of ease of understanding later, we should mention that products are taken with componentwise order and action. Also, the equalizer of f :A→B and g :A→B is the embedding of {a∈A|f(a) =g(a)} into A.

In the following we ignore the cases when one of the S-posets is empty, since these cases can be trivially verified.

Take arbitrary objects X, Y ∈ Ob(Pos_S), both of these are of course S-posets. As the exponential object X^Y take the morphism set

X^Y := MorPos_S(SS ×Y, X),

i.e. the set of all order-preserving S-act homomorphisms from the product S-posetS_S×Xto the posetY. Here,S_S is just the pomonoidS considered as a poset over itself. The order on X^Y is the pointwise one. Define the S-action exactly in the same way as it is done in the case of S-acts (see [MLM] page 62): for f : S_S ×Y → X and s ∈ S, define f ·s : S_S ×Y → X with the equation

(f·s)(t, y) = f(st, y) for all t∈S, y ∈Y. We have

(f·s)((t, y)·u) = (f·s)(tu, y·u) = f(stu, y·u) = (f(st, y))·u= ((f·s)(t, y))·u for all t, u∈S, y ∈Y and if (t₁, y₁)≤(t₂, y₂), then

(f ·s)(t₁, y₁) =f(st₁, y₁)≤f(st₂, y₂) = (f·s)(t₂, y₂)

since f itself preserves the order and so does multiplication in S. So indeed f ·s∈Mor_PosS(S_S×Y, X).

Moreover, (f·1)(s, y) = f(1s, y) = f(s, y) for alls∈S, y ∈Y, sof·1 = f.

Similarly,

((f·s)·t)(u, y) = (f·s)(tu, y) = f(s(tu), y) =f((st)u, y) = (f ·(st))(u, y) for all s, t, u ∈ S, y ∈ Y and therefore (f ·s)·t = f ·(st). For the order, if f ≤g, then for any s, t∈S, y ∈Y

(f·s)(t, y) =f(st, y)≤g(st, y) = (g·s)(t, y), that is, f·s ≤g·s. In the same line of thought, if s1 ≤s2, then

(f ·s₁)(t, y) = f(s₁t, y)≤f(s₂t, y) = (f ·s₂)(t, y) for allt∈S, y ∈Y sincef is order-preserving ands₁t ≤s₂t. This concludes our verification that X^Y with the S-action defined above is an S-poset.

Take the evaluation morphism also precisely the same as in the case of Act_S: for eval:X^Y ×Y →X, have

eval(f, y) =f(1, y) for allf ∈X^Y, y ∈Y.

Then

eval((f, y)·s) = eval(f ·s, y·s) = (f ·s)(1, y·s)

= f(s·1, y·s) = (f(1, y))·s= (eval(f, y))·s

for all f ∈X^Y, y ∈Y, s∈S. If (f₁, y₁)≤(f₂, y₂), then

eval(f₁, y₁) = f₁(1, y₁)≤f₁(1, y₂)≤f₂(1, y₂) =eval(f₂, y₂).

Thus eval is indeed a morphism in Pos_S.

X^Y ×Y _eval ^//X Z ×Y

X^Y ×Y

α⁰×1Y

Z ×Y

X

α

?

??

??

??

??

??

??

To show that this morphism setX^Y is indeed the exponential object, we have to show that for any object Z ∈Ob(Pos_S) and any S-poset morphism α : Z × Y → X there is a unique morphism α⁰ : Z → X^Y such that eval◦(α⁰ ×1_Y) = α. For that we define α⁰ as follows: for any z ∈ Z, t ∈ S, y ∈Y

α⁰(z)(t, y) = α(z·t, y).

Then we have

α⁰(z)((t, y)·s) = α(z·(ts), y·s) = (α(z·t, y))·s= (α⁰(z)(t, y))·s for all s, t∈S, z ∈Z, y∈Y. Likewise, if (t1, y1)≤(t2, y2), then

α⁰(z)(t₁, y₁) = α(z·t₁, y₁)≤α(z·t₂, y₂) = α⁰(z)(t₂, y₂).

Therefore α⁰(z) is in X^Y = Mor_PosS(S_S×Y, X).

Additionally, if z ∈Z and s∈S, then

α⁰(z·s)((t, y)) = α((z·s)·t, y) =α(z·(st), y) = α⁰(z)(st, y) = (α⁰(z)·s)(t, y).

So α⁰(z·s) = α⁰(z)·s. Also, for z₁ ≤z₂

α⁰(z₁)(t, y) =α(z₁ ·t, y)≤α(z₂·t, y) = α⁰(z₂)(t, y)

for allt∈S, y ∈Y asz₁·t≤z₂·t andαis order-preserving. This shows that α⁰ is a morphism of Pos_S.

Consider the equation

eval◦(α⁰×1_Y) =α.

This is equivalent to

(eval◦(α⁰×1_Y))(z, y) = α(z, y)

holding for arbitrary z ∈Z and y ∈Y. Some calculation yields α(z·s, y) = (eval◦(α⁰×1_Y))(z·s, y) = eval(α⁰(z·s), y)

= (α⁰(z)·s)(1, y) = α⁰(z)(s1, y) =α⁰(z)(s, y).

This shows that the α⁰ defined earlier does make this equation true, and is also the only Pos_S morphism to do so. Proof completed.

2.2 Subobject classifiers

In [BFM] the nonexistence of a subobject classifier and the fact that Pos_S is not a topos is shown through the fact that not all monomorphisms are regular (Remark 19, point (2) of [BFM]). Here, we will see why exactly the subobject classifier cannot exist and we will also seek to remedy that in some manner.

Proposition 2.2 For any pomonoid S, the category Pos_S does not have a subobject classifier. Furthermore, it does not have one for even only the em- beddings, ie order-reflecting monomorphisms.

Proof. TakeA={a < b < c} and x·s =x for all s ∈S, x∈A. Then A turns out to be an S-poset. We have the subobject B ={a < c} ofA, which is the equivalence class of the monomorphism ι : B → A, with ι = 1_A|_B. If we had a subobject classifier true : 1 → Ω, where Ω is the truth value object, we would have to have the unique map φ_B :A→Ω which makes the diagram

A _φ Ω

B

//

B

A

ι

B ^{!B //}11

Ω

true

into a pullback. Obviously true fixes one element true(∗) in Ω. If the afore- mentioned diagram is a pullback, thenB ∼={(x,∗)∈A×1|φ_B(x) = true(∗)}

(for more details see [BFM] page 5).

Soφ_B(a) =φ_B(c) =true(∗). Sincea < b < c and φ_B is order-preserving, we have φ_B(b) =true(∗). But then {(x,∗) ∈ A×1|φ_B(x) = true(∗)} = A.

We have thus found out that regardless of the nature of Ω, B cannot even have a characteristic map φ_B making the above diagram into a pullback, not to mention its uniqueness. Therefore, no object is fit to be the truth value object. Since ι was also an embedding, we do not have a subobject classifier for embeddings instead of monomorphisms either.

Theorem 2.1 The category Pos_S has the subobject classifier for subobjects that are downwards closed.

The truth value object is

Ωd ={X|Xis a right ideal ofSandXis downwards closed}

with true(∗) = S. The order on Ω_d is that of reverse inclusion, i.e.

X ≤Y ⇔Y ⊆X.

The S-action is

X·s={t∈S|st∈X}.

Proof.Since the S-action is taken directly from the topos ofS-acts (see [MLM] page 35), we only have to see that X·s is downwards closed. Take t ∈ X ·s and t⁰ ≤ t. Since S is a pomonoid, st⁰ ≤ st. As X is downwards closed, st⁰ ∈X and consequently t⁰ ∈X·s.

We have to verify that Ω_dis anS-poset. As it is done in the case ofS-acts, it can be shown that Ω_d is an S-act. If Y ⊆X and s∈S, then

Y ·s={t∈S|st∈Y} ⊆ {t ∈S|st∈X}=X·s.

Also, if s≤s⁰, thenst≤s⁰t for any t ∈S and

X·s⁰ ={t ∈S|s⁰t∈X} ⊆ {t∈S|st∈X}=X·s

sinceX is downwards closed. This concludes the proof that Ω_d is anS-poset.

Take a monomorphism ι : B → A where B is downwards closed and define

φ_B(x) = {s∈S|x·s∈B}

for any x∈A. Take x·s ∈B and t≤s. Then x·t≤x·s and consequently x·t ∈ B. So φ_B is a well-defined homomorphism of S-acts. If x ≤ y, then x·s≤y·sand we haveφB(y) ={s∈S|y·s∈B} ⊆ {s∈S|x·s∈B}=φB(x) as B is downwards closed. This shows that φ_B is also order-preserving and hence a morphism in the category Pos_S.

A _φ Ω_d

B

//

B

A

ι

B ^{!B //}11

Ω_d

true

Obviouslyφ_B◦ι=true◦!_B asB is a subact. Also{(a,∗)∈A×1|φ_B(a) = true(∗) = S} = B, because if a /∈ B, then 1 ∈ {s/ ∈ S|a · s ∈ B}, or equivalently S 6= {s ∈ S|a ·s ∈ B} = φ_B(a). So we have that B is the pullback of φ_B and true.

Let ψ_B :A → Ω_d be a morphism such that B is the pullback of ψ_B and true. Then B ∼= {(a,∗) ∈ A×1|ψ_B(a) = true(∗) = S}. If x·s ∈ B, then {t ∈ S|st ∈ ψ_B(x)} = ψ_B(x)·s = ψ_B(x·s) = S 3 1 and so s·1 ∈ ψ_B(x).

Hence {s ∈S|x·s∈ B} ⊆ψ_B(x). If x·s /∈B (then ψ_B(x·s)6= S), we get S 6= ψ_B(x·s) = ψ_B(x)·s = {t ∈ S|st ∈ ψ_B(x)} and so 1 ∈ {t/ ∈ S|st ∈ ψ_B(x)}. So s /∈ ψ_B(x) and ψ_B(x) = {s ∈ S|x·s ∈ B} = φ_B(x). We have verified, that φ_B is the only morphism that gives us the desired pullback.

This completes our proof.

By dualizing the order onS, using Theorem 2.1 and dualizing back toS, we get

Corollary 2.1 The category PosS has the subobject classifier for subobjects that are upwards closed.

The truth value object is

Ω_u ={X|Xis a right ideal ofSandXis upwards closed}

with true(∗) = S. The order on Ω_u is that of inclusion, i.e.

X ≤Y ⇔X ⊆Y.

The S-action is

X·s={t∈S|st∈X}.

Proof.

2.3 Monomorphism types

Now that we have seen that monomorphisms and embeddings fail to give us a subobject classifier, we shall examine whether any other types of mono- morphism might do better.

In the following we first examine which notions of monomorphisms are different in the category of S-posets over a pomonoid S. Recall that an em- bedding is an injective homomorphism of S-acts that both preserves and reflects the order.

Proposition 3 of [BFM] shows a part of the following proof (epimorp- hisms are surjective homomorphisms) and Theorem 7 of [BFM] describes the extremal monomorphisms.

Lemma 2.3 In Pos_S extremal monomorphisms are embeddings.

Proof. Take an extremal morphism ι: B → C and suppose it is not an embedding. Then we have x, y ∈ B such that ι(x) ≤ ι(y), but x 6≤ y. We can thus define π : B → D and f : D → C with D = Im(ι), π(z) = ι(z) for all z ∈ B and f inserting D = Im(ι) into C. As π is a surjective S-act homomorphism, it must also be an S-act epimorphism.

B _π ^//D _f ^//C

ι

&&

To see that, takeg◦π=f◦π, withf andg being morphisms in Pos_S. We get f =g inS-Act which means they coincide as mappings and consequently as Pos_S homomorphisms.

It is trivial to see that π preserves the order. So π is an epimorphism of S-posets. Obviouslyι =f◦π. Butπ is not left invertible, because if we had i◦π = 1_B, then x= (i◦π)(x) =i(ι(x))≤i(ι(y)) = (i◦π)(y) = y, as imust preserve the order. Therefore we have a factorization ι = f ◦π, with π an epimorphism but not an isomorphism. This cannot happen as ι is extremal and therefore our assumption that ιwas not an embedding does not hold.

Lemma 2.4 In Pos_S, every embedding is a regular monomorphism.

Proof. Letι:B →C be an embedding and B⁰ = Imι. Take D=Cq(C\B⁰).

For a clearer notation, have

D=C₁qC₂ qB⁰ with C₁ =C₂ =C\B⁰.

We need an order on D and for that define x ≤ y in D iff one of the following holds

• x∈B⁰, y ∈C_i, i= 1,2 and x≤y in C;

• x∈C_i, i= 1,2, y ∈B⁰ and x≤y in C;

• x, y ∈B⁰ and x≤y inC;

• x, y ∈C_i, i= 1,2 and x≤y inC;

• x∈C_i, y ∈C_j, i6=j and ∃z ∈B⁰ such thatx≤z ≤y in C.

Essentially this is the transitive closure of the amalgamated coproduct of C with itself over B⁰ in the sense of Act_S. Obviously we get a reflexive relation. By taking transitive closure we ensure transitivity, since the only failures to remain transitive have to compare elements from different copies of C. Antisymmetry within the two copies of C carries over, and if we take x ∈C_i, y ∈C_j, i 6=j from different copies, then x ≤y and y ≤x holds only if there are u, v ∈ B⁰ such that x ≤u ≤y ≤ v ≤ x, whence x =u∈ B⁰, an impossibility. So we have a partial order on D.

The action onC can clearly be transferred to D, asB⁰ is a subact and so both partial actions (of S on C) are confined to respective copies of C.

Now take from D two elements d ≤ d⁰ and s ∈ S. Then ds ≤ d⁰s. The only place where that might not hold (as C is an S-poset) is when d ∈ C_i and d⁰ ∈ C_j, i 6= j. Then d ≤ d⁰⁰ ≤ d⁰, with d⁰⁰ ∈ B⁰. But in that case ds ≤d⁰⁰s ≤d⁰s, and consequently ds≤d⁰s.

Likewise, for d ∈ D and s, s⁰ ∈ S, with s ≤ s⁰, we get ds ≤ ds⁰ as this holds within both copies of C without problems. SoD is indeed an S-poset.

B ^{ι //}C

f1 //

f2

//DD

H

hsssss99 ss ss

kOO

Define f1, f2 : C ^// D, with fi(x) = x if x ∈ B⁰ and fi(x) = gi(x) if x /∈B⁰, whereg_i :C →C_iqB⁰ are the isomorphisms. Triviallyf₁◦ι =f₂◦ι.

Since f_i are both essentially identities, they preserve the order and the S- action onD(which was unchanged within the separate copies ofC). So they are both morphisms in Pos_S.

Moreover, ifh:H →C is an order-preservingS-act homomorphism and f1◦h=f2◦h then clearly Imh⊆B⁰.Letx∈H.Then there exists a unique b ∈B such thatι(b) =h(x). Define

k(x) =b.

Because ι is an embedding, k is order-preserving, and obviously also a ho- momorphism. We have

(ι◦k)(x) =ι(b) =h(x)

for all x ∈ H and hence ι◦k = h. Since ι is a monomorphism, this means that we have shown (B, ι) to be the equalizer of f₁ and f₂. Therefore, ι is a regular monomorphism.

So far, we have not managed to get anything different from embeddings and usual monomorphisms (injective homomorphisms). One might try to get something different and generalize the extremal monomorphisms as follows.

Definition 2.1 A morphism ι :B →C in a category C is a regularly extre- mal morphism, if

(∀D∈Ob(C))(∀π :B →D)(∀f :D→C)

(ι=f◦π∧π is regular epimorphism ⇒π is isomorphism).

The following result shows that this definition is much weaker and is actually not even a generalization, but more of an overgeneralization.

Lemma 2.5 Monomorphisms are always regularly extremal morphisms, but regularly extremal morphisms do not have to be monomorphisms.

Proof. First, let us see that a regularly extremal morphism does not have to be a monomorphism. Take a category with four objects A, B, C and D. Let f, g : A → B, m : B → C, n : C → D and their composites be the only non-trivial morphisms in this category, with the added restrictions m ◦ f = m ◦ g and n ◦ f = n ◦ g. The morphism m is obviously not a monomorphism. The only possible factorizations of m are either m◦1_B or 1C◦m. 1Bis a regular epimorphism, but also an isomorphism. The morphism m itself is not an isomorphism, not even a coretraction. Therefore it can only be the coequalizer of f andg. But there are no morphisms from C toD and thus m is not a regular epimorphism and therefore is a regularly extremal morphism.

A B

A B

f //

g //

mjjjj44 jj j

nTTTT**

TT

T C

D C D

Now, let ι : B → C be a monomorphism in an arbitrary category and let ι = f ◦π, where π : B → D is a regular epimorphism. Then π must be the coequalizer of a single morphism a (with a copy of this morphism to coequalize with), since whatever π coequalizes,f◦π=ι makes equal andm is a monomorphism.

X B

X ^{a //}B ^{π //}

f

ι

?

??

??

??

??

?? D

C D

C

From this we establish thatπ has a left inverse, since 1_B◦a= 1_B◦a and thus there must be a unique b : D → B such that b◦π = 1_B. In the same line of reasoning, π◦a=π◦a and so there is a uniquec:D→D such that c◦π=π. But (π◦b)◦π =π= 1_D◦π, therefore c=π◦b = 1_D and π is an isomorphism. This proves that ι is regularly extremal.

It turns out that in the case of Pos_S, the converse holds as well.

Lemma 2.6 InPos_S all monomorphisms are precisely the regularly extremal morphisms.

Proof. We have seen that all monomorphisms are regularly extremal.

Suppose ι:B →C is not a monomorphism, then

∅ 6={(x, y)∈B²|ι(x) = ι(y), x6=y}={(x_i, y_i)∈B²|i∈I}

for some index set I. We can take D = Im(ι), with π : B → D defined as π(z) = ι(z) for all z ∈ B. Define f : D → C with f(z) = z for all z ∈ D.

Both π and f are S-poset morphisms and clearly f◦π =ι.

I ×S I ×S ^{a //}

b // B _π ^//D _f ^//C

ι

&&

Take a new poset, denotedI×S, such that it consists of all (i, s), where i∈I, s ∈S and (i, s)·t = (i, st) for allt∈S. WithS-induced order (that is, the only order relations are i·s ≤i·s⁰ with s ≤s⁰) and the obvious action, it is evidently an S-poset. The epimorphism π is the coequalizer of the pair a, b : I×S → B of S-poset morphisms with a(i, s) = x_i ·s, b(i, s) = y_i·s.

To see this, have g◦a = g◦b for some morphism g :B → G, which means g(x_i) = g(y_i) for each i ∈ I. Then we define a new morphism k : D → G as follows. Take z ∈ D, in which case there is at least one u∈ B such that π(u) =z. Define

k(z) = g(u).

For any other suchu⁰ thatπ(u⁰) = zwe haveι(u⁰) = ι(u) =z, sog(u) =g(u⁰) as well and the map is well-defined. It is also easy to see that k preserves both the order and theS-action. Of coursek◦π=g. Asπis an epimorphism, k must be unique. Therefore ι is not regularly extremal because π is not an isomorphism, but is a regular epimorphism.

2.4 Category of posets as a topos

By Lemmas 2.3, 2.4 and 2.6 we obtain the following:

Proposition 2.3 InPosS the notions of regular, strict, strong and extremal monomorphism coincide, being precisely all the embeddings ofPos_S. Regularly extremal morphisms are the same as monomorphisms.

As we can see, most notions of monomorphism lead us to embeddings, with (actual) monomorphisms (the same as regularly extremal morphisms) and coretractions being separate. In Pos_S we cannot classify neither embed- dings nor monomorphisms. Therefore we cannot use any of those notions to get Pos_S to have a partial (i.e. valid only for a certain class of monomorp- hisms) subobject classifier.

In the end, Pos_S is not a topos, but it is rather close. It has arbitrary limits and colimits, exponentials and a restricted subobject classifier. Our hope was to get a result like that of usual S-acts, which form a Grothendieck topos (see [MLM], third chapter). This was motivated by S-posets being very similar to S-acts in the functorial sense, the former as 2-functors to the category of posets and the latter as functors to the category of sets.

Unfortunately, we can not get a Grothendieck topos out of it (since these have to be toposes), and one can only hope that there is a suitable 2-categorical notion of Grothendieck topology that might be used on S-posets as some sort of 2-sheaves.

2.5 Submonomorphisms and subclassifiers

In the study of S-posets, in some situations it has proved to be useful to substitute the equalities of composites with inequalities of composites in the definitions of (co)limits. Subpullbacks, subequalizers etc thus derived have been introduced in [BFM] and [BFL]. Such an approach motivates our defi- nition of submonomorphisms in the same way. From [BFM] we obtain that subpullbacks of f :A→C andg :B →C can be canonically constructed as

{(a, b)|a∈A, b∈B, f(a)≤g(b)}

and subpullbacks of g and f as

{(b, a)|a∈A, b ∈B, g(b)≤f(a)}.

Definition 2.2 We call anS-poset morphismm:A_S →B_S asubmonomor- phism iff for all S-poset morphisms f, g :C_S →A_S whenever m◦f ≤m◦g, f ≤g as well.

Lemma 2.7 Submonomorphisms in PosS are precisely embeddings.

Proof.Ifm is an embedding andm◦f ≤m◦g forf, g :C_S →A_S, then m(f(c))≤m(g(c)) for allc∈C. Therefore f(c)≤g(c) and f ≤g.

If m is a submonomorphism and m(x) ≤ m(y), then we can consider morphisms x:S_S →A_S and y:S_S →A_S. Then

(m◦x)(s) =m(x·s) =m(x)·s≤m(y)·s =m(y·s) = (m◦y)(s)

for all s∈ S. Thus m◦x≤m◦y, whence x≤y. In particular, x=x·1 = x(1)≤y(1) =y·1 =y. So x≤y and m reflects order. Every submonomor- phism is a monomorphism, since equality implies both inequalities and vice versa. Therefore m is indeed an embedding.

While we could define subregular, substrict, substrong etc morphisms, by [BFM] subequalizers are embeddings, subcoequalizers are surjections and we would not get any new classes of morphisms.

Instead, we define subobject subclassifiers as follows. Note that since ter- minal objects are not defined with equalities of composites, subterminal ob- jects are the same as terminal objects.

Definition 2.3 In a category with finite sublimits thesubobject subclassifier (subobject supclassifier) is a monomorphism true: 1→ Ω such that for any other monomorphism ι : B → A there is a unique morphism φB such that the square

A _φ Ω

B

//

B

A

ι

B ^{!B //}11

Ω

true

turns out to be a subpullback of φB and true (oftrue and φB).

These objects are also unique up to isomorphism, which is proved in the same way as in the case of subobject classifiers.

Once again, it suffices to consider only injections of subsets.

In the case of Pos_S we have the following result.

Theorem 2.2 The category Pos_S has the subobject subclassifier for subob- jects that are downwards closed and the subobject supclassifier for subobjects that are upwards closed. Moreover, any other subobjects can not be subclas- sified or supclassified.

Proof. Take precisely the same Ω_d, true and φ_B for downwards closed subobjects as we did in Theorem 2.1. Take a monomorphism ι : B → A.

Since true(∗) =S,

φ_B(a)≤true(∗) ⇔ S ⊆φ_B(a)⇔S=φ_B(a)

⇔ ∀s∈S a·s ∈B ⇔a∈B

for every a ∈A. Then

{(a,∗)∈A×1|φ_B(a)≤true(∗)}=B×1∼=B.

Therefore we do obtain a subpullback.

LetψB :A→Ωdbe a morphism such thatB is the subpullback ofψB and true. ThenB ∼={(a,∗)∈A×1|ψ_B(a)≤true(∗) =S}. We are trying to prove φ_B =ψ_B. If x·s ∈ B, x∈ A, s ∈S, then {t ∈ S|st∈ψ_B(x)} =ψ_B(x)·s= ψB(x·s)⊇S 31 and sos·1∈ψB(x). Hence {s∈S|x·s∈B} ⊆ψB(x). If x·s /∈B (then ψ_B(x·s)6=S because otherwise ψ_B(x·s)⊇S and x·s∈B from the subpullback), we getS 6=ψ_B(x·s) =ψ_B(x)·s={t∈S|st∈ψ_B(x)}

and so 1 ∈ {t/ ∈ S|st ∈ ψB(x)}. So s /∈ ψB(x) and ψB(x) = {s ∈ S|x·s ∈ B} = φ_B(x). We have verified, that φ_B is the only morphism that gives us the desired subpullback.

Upwards closed subobjects are supclassified by using Ωu, true and φB

implied by Corollary 2.1.

Suppose a subobject ι : B → A can be subclassified and there is an a ∈A such that a≤b for someb ∈B, but a /∈B. ThenφB(a)⊇φB(b) =S, so a ∈B, a contradiction. Similarly only upwards closed subobjects can be supclassified.

3 Geometric morphisms

As the previous part demonstrated, Pos_S is not a topos and therefore the usual notion of a geometric morphism (for more details, see [MLM], chap- ter VII) does not apply there. In the following we show that by adding a few order-related restrictions, we can obtain similar morphisms (pogeomet- ric morphisms) between S-poset categories.

3.1 Pofunctors and poadjunctions

Definition 3.1 We say that a functor F : PosS → PosT is a pofunctor if for any pair of S-poset morphisms f₁, f₂ : B_S → B_S⁰ with f₁ ≤ f₂ also F(f₁)≤F(f₂).

Definition 3.2 We say that two adjoint pofunctors L : Pos_S → Pos_T and R : Pos_T → Pos_S, La R, form a poadjunction (L is left poadjoint to R), if the corresponding binatural isomorphism

α : Mor_PosT(L(−),−)→Mor_PosS(−, R(−))

also preserves and reflects the pointwise order of S- and T-poset morphisms.

This means that for all S-posets B_S, T-posets C_T and f₁, f₂ :L(B_S) →C_T, f₁ ≤ f₂ we have α_B,C(f₁) ≤ α_B,C(f₂). Also, for all g₁, g₂ : B_S → R(C_T), g1 ≤g2 it must hold that α_B,C⁻¹ (g1)≤α⁻¹_B,C(g2).

Definition 3.3 LetF :I →Pos_Sbe a functor on an index categoryI, where the objects form a poset (Ob(I),≤). Let C:<→Ob(I)×Mor_PosS×Mor_PosS be such a mapping that for each i < j, where i, j ∈Ob(I), there is an object C(i < j)₁ ∈Ob(I) and a pair of morphismsC(i < j)₂ :F(C(i < j)₁)→F(i) and C(i < j)₃ : F(C(i < j)₁) → F(j) in Pos_S. We say that a cocone (C,(f_i)i∈Ob(I)) on a diagram (F(i))i∈Ob(I) of shape F is apococone of F with respect to mapping Cif

f_i◦C(i < j)₂ ≤f_j◦C(i < j)₃ for all i < j, i, j∈Ob(I).

F(i) _f C

i

//

F(C(i < j)₁)

F(i)

C(i<j)2

F(C(i < j)₁) ^{C(i<j)3 //}FF(j(j))

C

fj

If we have another functorG: Pos_S →Pos_T, then we write G◦Cinstead of the longer (1×G×G)◦C to denote the corresponding mapping for the functor G◦F, ie (G◦C)(i < j) = (C(i < j)₁, G(C(i < j)₂), G(C(i < j)₃)).

A universal pococone of a functor F :I →Pos_S with respect to mapping Cis a pococone (U,(u_i)_i∈Ob(I)) with the property that for every other pococo- ne (V,(v_i)i∈Ob(I)) of the same functor and with respect to the same mapping there is a unique morphismf :U →V such thatf◦u_i =v_i for alli∈Ob(I).

Note that universal pococones of the same functor with respect to the same mapping C are unique up to isomorphism, this is proved in the usual way. Universal pococones of trivial orders are ordinary colimits.

Lemma 3.1 If (U,(u_i)i∈Ob(I)) is the universal pococone of F : I → Pos_S with respect to U, (V,(v_i)_i∈Ob(I)) is the universal pococone of G : I → Pos_S with respect to V, functors F and G are naturally isomorphic with natural isomorphisms α_i : F(i) → G(i) for all i ∈ Ob(I), U(i < j)₁ = V(i < j)₁ and

V(i < j)₂◦α_U(i<j)1 =α_i◦U(i < j)₂, V(i < j)3◦αU(i<j)1 =αj ◦U(i < j)3

for all i < j in Ob(I), then there is a unique morphism α_U : U → V such that

α_U◦u_i =v_i◦α_i for all i∈Ob(I), which is also an isomorphism.

Proof. Due to naturality of α,

G(k)◦αi =αj ◦F(k) for all morphisms k:i→j in I.

F(j) _αj ^//G(j) F(i)

F(j)

F(k)

F(i) ^{αi //}G(i)G(i)

G(j)

G(k)

This implies

v_j ◦α_j◦F(k) =v_j◦G(k)◦α_i =v_i◦α_i for k :i→j. Therefore

(V,(v_i◦α_i)i∈Ob(I))

is a cocone of F. Also,

v_i◦α_i◦U(i < j)₂ = v_i◦V(i < j)₂◦α_U(i<j)1

≤ vj◦V(i < j)3 ◦α_U(i<j)1

= v_j◦α_j◦U(i < j)₃.

So (V,(vi◦αi)i∈Ob(I)) is a cocone ofF with respect to U. Therefore we have a unique α_U :U →V such that

α_U ◦u_i =v_i◦α_i.

G(V(i < j)₁) ^{V(i<j)3 //}G(j) G(V(i < j)₁)

G(i)

V(i<j)2

G(i) _vi ^//V

G(j)

V

vj

F(U(i < j)₁) ^{U(i<j)3 //}F(j) F(U(i < j)₁)

F(i)

U(i<j)2

F(i) _ui ^//U

F(j)

U

uj

F(i)

F(j)

F(k)

??





























G(i)

G(j)

G(k)

22

F(U(i < j)₁) G(V(i < j)₁)

α_U(i<j)1

__???

?????????

??????

???????

F(j)

G(j)

αj

JJ

F(i) G(i)

αi

ttjjjjjjjjjjjjjjjjj U

V

αU

?

? ?

Reversing the roles of U and V in the previous argument, we obtain that there is a unique α_V :V →U such that

α_V ◦v_i =u_i◦α_i⁻¹. Therefore

α_V ◦α_U◦u_i =α_V ◦v_i◦α_i =u_i◦α⁻¹_i ◦α_i =u_i. Since also

1_U ◦u_i =u_i,

the uniqueness property of universal cocone U gives usα_V ◦α_U = 1_U. Addi- tionally

α_U◦α_V ◦v_i =α_U◦u_i◦α⁻¹_i =v_i ◦α_i◦α⁻¹_i =v_i

and the universal property of V provides α_U ◦ α_V = 1_V. Thus α_U is the isomorphism we wanted.

Proposition 3.1 Left poadjoints preserve universal pococones. More preci- sely, if (C,(f_i)i∈Ob(I)) is the universal pococone of F : I → Pos_S with res- pect to mapping C and the functor L : Pos_S → Pos_T is a left poadjoint, then (L(C),(L(f_i))i∈Ob(I)) is the universal pococone of L◦F with respect to mapping L◦C.

Proof. Let L: Pos_S →Pos_T be left poadjoint to R : Pos_T → Pos_S with the corresponding natural isomorphisms

α_A,B : Mor_PosT(L(A), B)→Mor_PosS(A, R(B)),

A ∈ Ob(Pos_S), B ∈ Ob(Pos_T). Let I be an index category the object set of which is also ordered and consider a functorF :I →Pos_S. Let (C,(f_i)i∈Ob(I)) be the universal pococone of F with respect toC, which means

f_i◦C(i < j)₂ ≤f_j◦C(i < j)₃

for all i < j, i, j ∈ Ob(I). We are trying to prove that (L(C),(L(f_i))i∈Ob(I)) is a universal pococone of L◦F with regard to L◦C.

The functorial image of a cocone is a cocone. Since L is a pofunctor, we have

L(fi)◦L(C(i < j)2)≤L(fj)◦L(C(i < j)3)

for all i < j, i, j ∈ Ob(I). Therefore (L(C),(L(f_i))i∈Ob(I)) is a pococone of L◦F with respect toL◦C.

Now consider a pococone (D,(g_i)_i∈Ob(I)) of L◦F with respect to L◦C, implying

g_i◦L(C(i < j)₂)≤g_j◦L(C(i < j)₃).

L(F(i)) ^{L(fi) //}L(C) L(F(C(i < j)₁))

L(F(i))

L(C(i<j)2)

L(F(C(i < j)₁)) L(C(i<j)3) //L(FL(F(j))(j))

L(C)

L(fj)

g[i[[[[[[[[[[[[[[[[[--D

[[ [[ [ [[ [[ [[ [[ [[ [

D

gj

-- ---- ---- ---- --

s

""

FF FF

Due to adjointness we have morphisms

h_i =α_F(i),D(g_i) :F(i)→R(D) for all i∈Ob(I).

L(F(i)) ^{gi //}D

F(i) ^{hi //}R(D)

_

α_F(i),D

For a morphism k :i→j the naturality of the isomorphisms yields h_i = α_F(i),D(g_i) = α_F(i),D(g_j ◦L(F(k)))

= (α_F(i),D◦Mor_PosT(L(F(k)), D))(g_j)

= (Mor_PosS(F(k), R(D))◦α_F(j),D)(g_j)

= Mor_PosS(F(k), R(D))(h_j) = h_j◦F(k).

Additionally, the poadjunction implies

αF(C(i<j)1),D(g_i◦L(C(i < j)₂))≤α_F(C(i<j)1),D(g_j◦L(C(i < j)₃)) for i < j, i, j ∈Ob(I), whence

h_i◦C(i < j)₂ = α_F(i),D(g_i)◦C(i < j)₂

= [Mor_PosS(C(i < j)₂, R(D))◦α_F(i),D](g_i)

= [α_F(C(i<j)1),D◦Mor_PosT(L(C(i < j)₂), D)](g_i)

= α_F(C(i<j)1),D(gi◦L(C(i < j)2))

≤ α_F(C(i<j)1),D(g_j ◦L(C(i < j)₃))

= [α_F(C(i<j)1),D◦Mor_PosT(L(C(i < j)₃), D)](g_j)

= [Mor_PosS(C(i < j)₃, R(D))◦α_F(j),D](g_j)

= α_F(j),D(g_j)◦C(i < j)₃ =h_j ◦C(i < j)₃. This makes (R(D),(h_i)i∈Ob(I)) into a pococone on F with respect to C.

F(i) ^{fi //}C

F(C(i < j)₁)

F(i)

C(i<j)2

F(C(i < j)₁) ^{C(i<j)3 //}FF(j(j))

C

fj

hZiZZZZZZZZZZZZZZ--R(D)

ZZ ZZ ZZ ZZ ZZ ZZ ZZ

R(D)

hj

-- ---- ---- ---- --

r

""

FF FF

From this we get the unique factorizationh_i =r◦f_i with r:C →R(D), for all i∈ Ob(I). By adjointness we have s =α⁻¹_C,D(r) :L(C) → D, and by naturality of α_C,D⁻¹ also

s◦L(f_i) = Mor_PosT(L(f_i), D)(α⁻¹_C,D(r)) = α⁻¹_F(i),D(Mor_PosS(f_i, R(D))(r))

= α_F⁻¹_(i),D(r◦fi) = α⁻¹_F(i),D(hi) = gi.

Mor_PosS(F(i), R(D)) Mor_PosT(L(F(i)), D)

α⁻¹_F(i),D

//

Mor_PosS(C, R(D))

Mor_PosS(F(i), R(D))

MorPosS(fi,R(D))

Mor_PosS(C, R(D)) Mor_PosT(L(C), D)

α⁻¹_C,D

//Mor_PosT(L(C), D)

Mor_PosT(L(F(i)), D)

MorPosT(L(fi),D)

For any others⁰ =α⁻¹_C,D(r⁰) :L(C)→Dsuch thats⁰◦L(f_i) =g_i for every i∈Ob(I), the naturality ofα_C,D gives

r⁰◦f_i = α_C,D(s⁰)◦f_i = Mor_PosS(f_i, R(D)(α_C,D(s⁰))

= αF(i),D(MorPosT(L(fi), D)(s⁰))

= α_F(i),D(s⁰◦L(fi)) =α_F(i),D(gi) =hi. As (C,(f_i)_i∈Ob(I)) is universal, r=r⁰ and s is unique.

Lemma 3.2 Every S-poset is a universal pococone of free S-posets S_S. Proof. Let BS be an S-poset. Construct a new category El(B), where Ob(El(B)) = B and if b, c∈ B, then Mor_El(B)(b, c) = {s ∈ S|c·s = b} and the composition is the multiplication of S. Note that B is a partial order.

Define a functor F : El(B)→PosS by F(b) =S_S and

F(s) =s

for all b, c∈B, s ∈Mor_El(B)(b, c). From Lemma 1.1 F(ss⁰) = ss⁰ =s◦s⁰ =F(s)◦F(s⁰), F(1) =1= 1_S and we do obtain a functor.

We have the S-poset morphisms f_b :F(b)→B_S defined as f_b =b

for all b∈B.

Ifb < b⁰, b, b⁰ ∈B, we take

B(b < b⁰)₁ =b and

B(b < b⁰)₂ =B(b < b⁰)₃ =1= 1_S :S_S →S_S.

We want to show that (B_S,(f_b)b∈B) is the universal pococone of F with respect to the order of B and mapping B.

First of all, take b, c∈B, c·s=b. Then by Lemma 1.1 f_c ◦F(s) =c◦s=c·s=b=f_b for any s :b →cin El(B) and we have a cocone.

Secondly, take b < c, b, c∈B. Then

f_b◦B(b < c)₂ =b◦1=b≤c=b◦1=f_c◦B(b < c)₃ and (B_S,(f_b)b∈B) turns out to be a pococone.

Take another pococone (C_S,(g_b)b∈B) with respect to B, which means g_b =g_b◦B(b < c)₂ ≤g_c◦B(b < c)₃ =g_c

when b < c, b, c ∈B.

F(b) =S_S ^{fb //}B_S F(b) =S_S

F(b) =S_S

1S

F(b) =S_S ^{1S //}FF(c) =(c) = SS_SS

B_S

fc

C_S

gbZZZZZZZZZZZZZZZ-- ZZ

ZZ ZZ ZZ ZZ ZZ ZZ

gc

//

////

////

////

///

α

''P

PP PP PP PP P

To prove the universal property, we define α:B_S →C_S with α(b) = g_b(1).

If b < c, then

α(b) = g_b(1) ≤g_c(1) =α(c)