Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets
Let S be a pomonoid. In this paper, Pos-S, the category of S-posets and S-poset maps, is considered. One of the main aims of this paper is to draw attention to the notion of weak factorization systems in Pos-S. We show that if S is a pogroup, or the identity element of S is the bottom (or top) eleme...
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irk-123456789-1566322019-06-19T01:26:25Z Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets Farsad, F. Madanshekaf, A. Let S be a pomonoid. In this paper, Pos-S, the category of S-posets and S-poset maps, is considered. One of the main aims of this paper is to draw attention to the notion of weak factorization systems in Pos-S. We show that if S is a pogroup, or the identity element of S is the bottom (or top) element, then (DU,SplitEpi) is a weak factorization system in Pos-S, where DU and SplitEpi are the class of du-closed embedding S-poset maps and the class of all split S-poset epimorphisms, respectively. Among other things, we use a fibrewise notion of complete posets in the category Pos-S/B under a particular case that B has trivial action. We show that every regular injective object in Pos-S/B is topological functor. Finally, we characterize them under a special case, where S is a pogroup. 2017 Article Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets / F. Farsad, A. Madanshekaf // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 235-249. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC:06F05, 18A32, 18G05, 20M30, 20M50. http://dspace.nbuv.gov.ua/handle/123456789/156632 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let S be a pomonoid. In this paper, Pos-S, the category of S-posets and S-poset maps, is considered. One of the main aims of this paper is to draw attention to the notion of weak factorization systems in Pos-S. We show that if S is a pogroup, or the identity element of S is the bottom (or top) element, then (DU,SplitEpi) is a weak factorization system in Pos-S, where DU and SplitEpi are the class of du-closed embedding S-poset maps and the class of all split S-poset epimorphisms, respectively. Among other things, we use a fibrewise notion of complete posets in the category Pos-S/B under a particular case that B has trivial action. We show that every regular injective object in Pos-S/B is topological functor. Finally, we characterize them under a special case, where S is a pogroup. |
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Farsad, F. Madanshekaf, A. |
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Farsad, F. Madanshekaf, A. Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets Algebra and Discrete Mathematics |
| author_facet |
Farsad, F. Madanshekaf, A. |
| author_sort |
Farsad, F. |
| title |
Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets |
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Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets |
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Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets |
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Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets |
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Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets |
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weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/156632 |
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Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets / F. Farsad, A. Madanshekaf // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 235-249. — Бібліогр.: 14 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT farsadf weakfactorizationsystemsandfibrewiseregularinjectivityforactionsofpomonoidsonposets AT madanshekafa weakfactorizationsystemsandfibrewiseregularinjectivityforactionsofpomonoidsonposets |
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2025-07-14T09:01:13Z |
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2025-07-14T09:01:13Z |
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1837612320605339648 |
| fulltext |
“adm-n4” 22:47 page #57
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 2, pp. 235–249
© Journal “Algebra and Discrete Mathematics”
Weak factorization systems and fibrewise regular
injectivity for actions of pomonoids on posets
Farideh Farsad and Ali Madanshekaf
Communicated by R. Wisbauer
Abstract. Let S be a pomonoid. In this paper, Pos-S,
the category of S-posets and S-poset maps, is considered. One of
the main aims of this paper is to draw attention to the notion
of weak factorization systems in Pos-S. We show that if S is a
pogroup, or the identity element of S is the bottom (or top) element,
then (DU , SplitEpi) is a weak factorization system in Pos-S, where
DU and SplitEpi are the class of du-closed embedding S-poset
maps and the class of all split S-poset epimorphisms, respectively.
Among other things, we use a fibrewise notion of complete posets
in the category Pos-S/B under a particular case that B has trivial
action. We show that every regular injective object in Pos-S/B is
topological functor. Finally, we characterize them under a special
case, where S is a pogroup.
1. Introduction
A comma category (a special case being a slice category) is a construc-
tion in category theory. It provides another way of looking at morphisms:
instead of simply relating objects of a category to one another, morphisms
become objects in their own right. This notion was introduced in 1963 by
F. W. Lawvere, although the technique did not become generally known
until many years later.
2010 MSC: 06F05, 18A32, 18G05, 20M30, 20M50.
Key words and phrases: S-poset, slice category, regular injectivity, weak fac-
torization system.
“adm-n4” 22:47 page #58
236 Weak factorization systems
Injective objects with respect to a class H of morphisms have been
investigated for a long time in various categories. Recently, injective objects
in slice categories (C/B) have been investigated in detail (see [1, 6]),
especially in relationship with weak factorization systems, a concept used
in homotopy theory, in particular for model categories. More precisely,
H-injective objects in C/B, for any B in C, form the right part of a weak
factorization system that has morphisms of H as the left part (see [1, 2]).
In this paper, the notion of weak factorization system in Pos-S is
investigated. After some introductory notions in section 1, we recall
in section 2, the notion of weak factorization system and state some
related basic theorems. Also, we give the guarantee about the existence of
(Emb, Emb�) as a weak factorization system in Pos-S, where Emb is the
class of all order-embedding S-act maps. We then find that every Emb-
injective object in Pos-S/B is split epimorphism in Pos-S. In section 3,
we continue studying Emb-injectivity using a fibrewise notion of complete
posets in the category Pos-S/B in a particular case where B has the
trivial action.
For the rest of this section, we give some preliminaries which we will
need in the sequel.
Given a category C and an object B of C, one can construct the slice
category C/B (read: C over B): objects of C/B are morphisms of C with
codomain B, and morphisms in C/B from one such object f : A → B to
another g : C → B are commutative triangles in C
A
h //
f ��
C
g
~~
B
i.e, gh = f . The composition in C/B is defined from the composition
in C, in the obvious way (paste triangles side by side).
Let C be a category and H a class of its morphisms. An object I of C
is called H-injective if for each H-morphism h : U → V and morphism
u : U → I there exists a morphism s : V → I such that sh = u. That is,
the following diagram is commutative:
U
h
��
u // I
V
s
??
“adm-n4” 22:47 page #59
F. Farsad, A. Madanshekaf 237
In particular, in the slice category C/B, where B is an object of C, this
means that, an object f : X → B is H-injective if, for any commutative
square
U
u //
h
��
X
f
��
V v
// B
with h ∈ H, there exists a diagonal morphism s : V → X
U
u //
h
��
X
f
��
V
s
>>
v
// B
such that sh = u and fs = υ. The category C is said to have enough
H-injectives if for every object A of C there exists a morphism A → C
in H where C is an H-injective object in C.
Let S be a monoid with identity 1. A (right) S-act or S-set is a set A
equipped with an action µ : A×S → A, (a, s) 7→ as, such that a1 = a and
a(st) = (as)t, for all a ∈ A and s, t ∈ S. Let Act-S denote the category
of all S-acts with action-preserving maps or S-maps. Clearly S itself is an
S-act with its operation as the action. For instance, take any monoid S
and a non-empty set A. Then A becomes a right S-act by defining as = a
for all a ∈ A, s ∈ S, we call that A an S-act with trivial action (see [10]
or [11]).
Recall that a pomonoid S is a monoid with a partial order 6 which
is compatible with the monoid operation: for s, t, s′, t′ ∈ S, s 6 t, s′ 6 t′
imply ss′ 6 tt′. A (right) S-poset is a poset A which is also an S-act
whose action µ : A×S → A is order-preserving, where A×S is considered
as a poset with componentwise order. The category of all S-posets with
action preserving monotone maps is denoted by Pos-S. Clearly S itself
is an S-poset with its operation as the action. Also, if B is a non-empty
subposet of A, then B is called a sub S-poset of A if bs ∈ B, for all s ∈ S
and b ∈ B. Throughout this paper we deal with the pomonoid S and
the category Pos-S, unless otherwise stated. For more information on
S-posets see [5] or [8].
2. Weak factorization system
The concept of weak factorization systems plays an important role in
the theory of model categories. Formally, this notion generalizes factor-
“adm-n4” 22:47 page #60
238 Weak factorization systems
ization systems by weakening the unique diagonalization property to the
diagonalization property without uniqueness. However, the basic examples
of weak factorization systems are fundamentally different from the basic
examples of factorization systems.
Here, we introduce from [1] the notion which we deal with in the
paper.
Notation. We denote by � the relation diagonalization property on the
class of all morphisms of a category C: given morphisms l : A → B and
r : C → D then
l�r
means that in every commutative square
A //
l
��
C
r
��
B
d
>>
// D
there exists a diagonal d : B → C rendering both triangles commutative.
In this case, l is also said to have the left lifting property with respect to
r (and r to have the right lifting property with respect to l).
Let H be a class of morphisms. We denote by
H� = {r| r has the right lifting property with respect to each l ∈ H}
and
�H = {l| l has the left lifting property with respect to each r ∈ H}.
Let D be an object in C and HD be the class of those morphisms in C/D
whose underlying morphism in C lies in H. Now, r : C → D ∈ H� if and
only if r is an HD-injective object in C/D. Dually, all morphisms in �H
are characterized by a projectivity condition in HD.
Recall from [1] that a weak factorization system in a category is a pair
(L, R) of morphism classes such that:
(1) every morphism has a factorization as an L-morphism followed by an
R-morphism,
(2) R = L� and L = �R.
Remark 2.1. If we replace “�” by “⊥” where “⊥” is defined via the
unique diagonalization property (i.e., by insisting that there exists precisely
“adm-n4” 22:47 page #61
F. Farsad, A. Madanshekaf 239
one diagonal), we arrive at the familiar notion of a factorization system
in a category. Factorization systems are weak factorization systems. For
instance, let E be the class of all S-poset epimorphisms. Then, by Theo-
rem 1 of [5] one can easily seen that (E , Emb) in Pos-S is a factorization
system.
Now, consider a functor G : A → X. Recall from [1] that a source
(A
fi→ Ai)i∈I in A is called G-initial provided that for each source (B
gi→
Ai)i∈I in A and each X-morphism h : GB → GA with Ggi = Gfi ◦ h
for each i ∈ I, there exists a unique A-morphism h̄ : B → A in A with
Gh̄ = h and gi = fi ◦ h̄ for each i ∈ I.
Also, a source (A
f̄i→ Ai)i∈I lifts a G-structured source (X
fi→ GAi)i∈I
provided that Gf̄i = fi for each i ∈ I.
Definition 2.2. (cf. [1]) A functor G : A → X in the category Cat (of
all categories and functors) is topological if every G-structured source
(X
fi→ GAi)i∈I has a unique G-initial lift (A
f̄i→ Ai)i∈I .
Example 2.3. (1) In the category Set (of all sets and functions between
them) pair (Mono, Epi) is a weak factorization system. But (Epi, Mono)
is a factorization system in this category, where Mono is the class of all
one-to-one maps and Epi is the class of all surjective maps.
(2) The pair (Full, Top) is a weak factorization system in the category
Cat, where Full is the class of those morphisms in Cat that are full and
Top is the class of those morphisms in Cat that are topological.
(3) In the category Pos of all posets with monotone maps, the pair
(Emb, Top) is a weak factorization system, where Emb is the class of all
order-embeddings; that is, maps f : A → B for which f(a) 6 f(a′) if
and only if a 6 a′, for all a, a′ ∈ A and Top is the class of all topological
monotone maps. For more details of the proof see [1].
We record the following two results from [1], that will be used later on.
Proposition 2.4. Let C be a category and H a class of morphisms closed
under retracts in arrow-category C→. Then the following conditions are
equivalent:
(1) (H, H�) is a weak factorization system;
(2) for all objects B of C, the slice category C/B has enough HB-
injectives.
Proposition 2.5. Let C be a category. Then (L, R) is a weak factoriza-
tion system if and only if
“adm-n4” 22:47 page #62
240 Weak factorization systems
(1) Any morphism h ∈ C has a factorization h = gf with f ∈ L and
g ∈ R.
(2) For all f ∈ L and g ∈ R, f has the left lifting property with respect
to g.
(3) If f : A → B and f ′ : X → Y are such that there exist morphisms
α : B → Y and β : A → X then
(a) If αf ∈ L and if α is a split monomorphism then f ∈ L.
(b) If f ′β ∈ R and if β is split epimorphism then f ′ ∈ R.
If f : A → B and g : A → C are morphisms in a category C such that
there exist morphisms α : C → B and β : B → C with βα = 1C , αg = f
and βf = g then we say that g is a retract of f . In categorical terms, g
is a retract of f in the coslice category A/C.
Notice that in 3(a) above, f is a retract of αf and that all retracts
can be written in this way. So this result is simply saying that L is
closed under retracts. Similarly 3(b) is equivalent to R being closed under
retracts.
Recently, Bailey and Renshaw in [2], provide a number of examples of
weak factorization systems for S-acts such as the following theorem. But
first we need a definition. Following [2] an S-act (S-poset) monomorphism
f : X → Y is unitary if y ∈ im(f) whenever ys ∈ im(f) and s ∈ S.
Notice that in the case of S-acts, it is clear this is equivalent to saying
that there exists an S-act Z such that Y ∼= X∪̇Z (the disjoint union of
X and Z) or in other words, im(f) is a direct summand of Y, while in
the other case this is not true (see Remark 3.3 below).
Theorem 2.6. Let S be a monoid and let U be the class of all unitary
S-monomorphisms and SplitEpi be the class of all split S-epimorphisms.
Then (U , SplitEpi) is a weak factorization system in Act-S.
3. Weak factorization systems via down and up-closed
embeddings
Now, consider Emb as the class of all embedding S-poset maps. We
try to provide a weak factorization system for Pos-S with Emb as the left
part. In this section, we consider down and up-closed embeddings, briefly
du-closed embeddings, as a subclass of Emb and find a weak factorization
system in Pos-S with some conditions on pomonoid S.
Let C be a category with binary coproducts, Sum the class of all
coproduct injections, and SplitEpi the class of all split epimorphisms
in C. The following is a particular case of [13, Theorem 2.7].
“adm-n4” 22:47 page #63
F. Farsad, A. Madanshekaf 241
Proposition 3.1. If Sum is stable under pullback in C, then
(Sum, SplitEpi) is a weak factorization system in C.
Proposition 3.2. Let S be an arbitrary pomonoid. Then the class of all
unitary monomorphisms in Pos-S is stable under pullback in Pos-S.
Proof. Let
Z
f ′
//
g′
��
Y ′
g
��
X
f
// Y
be a pullback in Pos-S, where f : X → Y is unitary and g : Y ′ → Y is
an S-poset map. Suppose that y′ ∈ Y ′ is such that y′s ∈ im(f ′) for some
s ∈ S. Then there is an element z ∈ Z with f ′(z) = y′s, and we have:
f(g′(z)) = (fg′)(z) = (gf ′)(z) = g(f ′(z)) = g(y′s) = g(y′)s.
Thus g(y′)s ∈ imf and since f is assumed to be unitary, one concludes
that g(y′) ∈ imf . Thus g(y′) = f(x) for some x ∈ X and since Z is a
pullback of f and g, it follows that there is a unique element z0 ∈ Z
such that f ′(z0) = y′ and g′(z0) = x. In particular, y′ ∈ imf ′. Thus f ′ is
unitary.
Remark 3.3. Note that the assertion ‘any unitary monomorphism in
Pos-S is just coproduct injection’ is not true, in general. For instance,
consider the trivial case S = {1}, then we have Pos-S=Pos and so any
injective monotone map trivially is unitary. Next, let A = {0, 1} be a
poset with two elements 0 and 1 and with 0 6 1 and let {0} be a discrete
poset with only one element 0. Then the map {0} → A that takes 0 to 0
is monotone and injective. But A = {0 6 1} is not a coproduct of {0} and
{1}, since {0}∪̇{1} = {0, 1} with discrete ordering, while A = {0 6 1}
has a nontrivial ordering on it.
Definition 3.4. A possibly empty sub S-poset A of an S-poset B is said
to be down-closed (up-closed) in B if for each a ∈ A and b ∈ B with b 6 a
(a 6 b) we have b ∈ A. By a du-closed embedding, we mean an embedding
S-poset map f : A → B such that imf is both down-closed and up-closed
sub S-poset of B.
Proposition 3.5. Let S be an arbitrary pomonoid. Then the class D
(resp. U) of all down-closed (resp. up-closed) embeddings is stable under
pullback in Pos-S.
“adm-n4” 22:47 page #64
242 Weak factorization systems
Proof. Let
Z
f ′
//
g′
��
Y ′
g
��
X
f
// Y
be a pullback in Pos-S, where f ∈ D (resp. f ∈ U) and g : Y ′ → Y
is an S-poset map. Suppose that y′ ∈ Y ′ is such that y′ 6 f ′(x) (resp.
y′ > f ′(x)) for some x ∈ imf ′. Then g(y′) 6 gf ′(x) = fg′(x) (resp.
g(y′) > gf ′(x) = fg′(x)), and since f is assumed to be down-closed (resp.
up-closed), we conclude that g(y′) ∈ imf. Thus there is an element x′ ∈ X
with f(x′) = g(y′). But since Z is a pullback of f and g, it follows that
there is a unique element z ∈ Z such that f ′(z) = y′ and g′(z) = x. In
particular, y′ ∈ imf ′. Thus f ′ is down-closed (resp. up-closed).
A corollary follows immediately:
Corollary 3.6. Let S be an arbitrary pomonoid. Then the class DU of
all du-closed embeddings is stable under pullback in Pos-S.
Now, we prove the following crucial lemma in the category Pos-S.
Lemma 3.7. Let f : A → B be a du-closed S-poset embedding. Then f
is a unitary if and only if imf is a direct summand of B.
Proof. Let f be a unitary monomorphism. First we show that imf and
B\imf are two S-posets. These are S-acts because of the unitary property
of f . Now, consider a, b ∈ B with a 6 b. If a ∈ imf , since imf is an up-
closed subset in B so b ∈ imf . This gives a 6 b in imf . Also, if a /∈ imf
then we have b /∈ imf ; in fact b ∈ imf implies that a ∈ imf as imf is
down-closed, which is a contradiction. So a 6 b in B \ imf . Hence if a 6 b
in B, then a 6 b in imf ∪̇(B \ imf). Now if a 6 b in imf ∪̇(B \ imf), then
by definition of order on the coproduct, we have a 6 b in imf or a 6 b in
B \ imf . Since the order on imf and B \ imf is inherent from B, then
a 6 b in B. Consequently we get
a 6 b in B ⇐⇒ a 6 b in imf ∪̇(B \ imf).
Hence B = imf ∪̇(B \ imf). The converse is trivially true by definition of
unitary monomorphism.
Let DUU denote the class of all unitary du-closed embedding S-poset
maps.
“adm-n4” 22:47 page #65
F. Farsad, A. Madanshekaf 243
Proposition 3.8. Let S be an arbitrary pomonoid. Then DUU = Sum.
Combining Propositions 3.2, 3.8 and Corollary 3.6 gives
Proposition 3.9. Let S be an arbitrary pomonoid. Then Sum is stable
under pullback in Pos-S.
It then follows from Propositions 3.1 and 3.9 that
Theorem 3.10. Let S be a pomonoid. Then (Sum, SplitEpi) is a weak
factorization system in Pos-S.
Next we state two useful results.
Lemma 3.11. Let S be a pomonoid whose identity element e is the
bottom element (resp. the top element) and f : X → Y be an S-poset map.
If imf is a down-closed (resp. up-closed) subset of Y , then f is unitary.
Proof. Suppose e is the bottom element of S. We show that if y ∈ imf
whenever ys ∈ imf and s ∈ S. In fact, by hypothesis we have e 6 s and
we get y 6 ys, for every y ∈ Y and s ∈ S. Now, as imf is down-closed
and ys ∈ imf then have y ∈ imf . The other case is similar.
By Proposition 3.8 and Lemma 3.11, we deduce
Proposition 3.12. Let S be a pomonoid such that its identity element
is either the bottom or top element. Then DU = DUU , and hence Sum =
DU .
Proposition 3.13. If S is a pogroup, then any morphism in Pos-S is
unitary.
Proof. Consider any morphism f : X → Y in Pos-S and suppose that
y ∈ Y is such that ys ∈ imf for some s ∈ S. Then ys = f(x) for some
x ∈ X and we have:
y = (ys)s−1 = f(x)s−1 = f(xs−1)
proving that y ∈ imf . Thus f is unitary.
Proposition 3.14. If S is a pogroup, then DU = DUU , and hence
Sum = DU .
Proof. By propositions 3.8 and 3.13, it is obvious.
“adm-n4” 22:47 page #66
244 Weak factorization systems
Next, combining Propositions 3.12 and 3.14 with Theorem 3.10, we
get the following result.
Theorem 3.15. Suppose that
(i) S is a pogroup, or
(ii) the identity element of S is the bottom element, or
(iii) the identity element of S is the top element,
Then (DU , SplitEpi) is a weak factorization system in Pos-S.
Recall that each poset can be embedded (via an order-embedding)
into a complete poset, called the Dedekind-MacNeille completion. In fact,
given a poset P , its MacNeille completion is the poset P̄ consisting of all
subsets A of P for which LU(A) = A, where
U(A) = {x ∈ P : x > a, ∀a ∈ A}
and
LU(A) = {y ∈ P : y 6 x, ∀x ∈ U(A)},
and the embedding ↓ (−) : P → P̄ is given by
a 7→↓ a = {x ∈ P : x 6 a}
for every a ∈ P (see [3]).
Notice that in the category Pos-S/B, regular monomorphisms corre-
spond to regular monomorphisms in Pos-S and these are exactly order-
embeddings (in Pos-S) (see [5, 9]). We state the following theorem which
gives us enough Emb-injectivity property in Pos-S/B. For details of the
proof see [9].
Theorem 3.16. For an arbitrary S-poset B, the category Pos-S/B has
enough regular injectives. More precisely, each object f : A → B in
Pos-S/B can be regularly embedded into a regular injective object πĀ(S)
B :
Ā(S) × B → B in Pos-S/B in which Ā(S) is the set of all monotone
maps from S into Ā, with pointwise order and the action is given by
(fs)(t) = f(st) for s, t ∈ S and f ∈ Ā(S) and πĀ(S)
B : Ā(S) × B → B is
the second projection.
It is easy to show that the class Emb is closed under retracts in
Pos-S/B. So by Proposition 2.4 and the theorem above, we can say
that (Emb, Emb�) is a weak factorization system for Pos-S. This implies
that Emb is saturated, that is, Emb is closed under pushouts, transfinite
compositions and retracts (see [2]).
“adm-n4” 22:47 page #67
F. Farsad, A. Madanshekaf 245
Up to now, we can not succeed to determine if there is a class R such
that (Emb, R) is a weak factorization system in Pos-S. However we do
have:
Proposition 3.17. Let S be a pomonoid. Suppose (Emb, R) is a weak
factorization system in Pos-S. Then R ⊆ SplitEpi.
Proof. Since Sum ⊂ Emb, it follows that Emb� ⊂ Sum�. But since
(Sum, SplitEpi) is a weak factorization system in Pos-S by Theorem 3.10,
one has that Sum� = SplitEpi. Thus
R = Emb� ⊆ SplitEpi.
4. Fibrewise regular injectivity of S-poset maps
In the previous section, we deduced that every Emb-injective object
in Pos-S/B is a split epimorphism in Pos-S (see Proposition 3.17). In
this section, we are going to characterize them using a fibrewise notion of
complete posets. (A poset is said to be complete if each of its subsets has
an infimum and a supremum.)
We recall [14] that in the category Pos of partially ordered sets and
monotone maps, an Emb-injective monotone map can be characterized
as follows:
Theorem 4.1. A monotone map f : X → B is Emb-injective in Pos/B
if and only if it satisfies the following conditions:
(I) f−1(b) is a complete poset, for every b ∈ B;
(II) f is a fibration (that is, for every x ∈ X and b ∈ B with f(x) 6 b,
{x′ ∈ f−1(b) | x 6 x′} has a minimum element) and a cofibration
(=dual of fibration).
By [6, Theorem 1.2] we have:
Theorem 4.2. Let S be a pomonoid. Then f : X → B is a regular
injective object in Pos-S/B if and only if the following two conditions
hold:
(1) 〈1X , f〉 : f → πX
B is a section in Pos-S/B where πX
B : X × B → B
is the second projection;
(2) the object S(f) of sections of f is a regular injective object in Pos-S.
Remark 4.3 ([9]). For a pomonoid S, the category Pos-S is cartesian
closed (see [5]). Indeed, given two S-posets A and B, the exponential
“adm-n4” 22:47 page #68
246 Weak factorization systems
BA is given by BA = HomPos-S(S × A, B), the set of all S-poset maps
from the product S-poset S × A to B. (Note that the action on S × A
operates on both components.) This set is an S-poset, with pointwise
order and the action is given by (f · s)(t, a) = f(st, a) for f ∈ BA, a ∈ A
and s, t ∈ S (see [5, 11]). Now, given f : X → B in Pos-S/B we have
S(f) = {h ∈ HomPos-S(S × B, X) | fh = πS
B}.
Moreover, if B has the trivial action, then we get the following embedding
induced by fibres of f :
m : S(f)
∏
b∈B
f−1(b) given by m(h) = (h(s, b))s∈S,b∈B
Now, we supply a partial answer to the characterization of regular
injectivity in the category Pos-S/B in a special case, when the S-poset
B has the trivial action.
First recall the following result from [9].
Proposition 4.4. Let S be a pomonoid. If f : X → B is a regular
injective object in the category Pos-S/B, then:
(1) 〈1X , f〉 : f → πX
B is a section in Pos-S/B.
(2) for every b ∈ B, the sub S-poset f−1(b) of X is a regular injective
object in Pos-S, so it is a complete poset.
In this section, we are going to give a new characterization of injective
objects in Pos-S/B that removes condition (1) of the above proposition.
Proposition 4.5. Let S be a pomonoid and f : X → B be a S-poset
map. If 〈1X , f〉 : f → πX
B is a section in Pos-S/B then for every x ∈ X
and b ∈ B with f(x) 6 b, the poset {x′ ∈ f−1(b) | x 6 x′} has a minimum
element.
Proof. Let r : X × B → X be a retraction of 〈1X , f〉 over B. For every
x ∈ X and b ∈ B with f(x) 6 b, let r(x, b) = xb. Then we have
x = r(x, f(x)) 6 r(x, b) = xb,
so xb ∈ {x′ ∈ f−1(b) | x 6 x′}. Also, take x′ in f−1(b) with x 6 x′ then
xb = r(x, b) 6 r(x′, b) = x′.
This means that xb is the minimum of {x′ ∈ f−1(b) | x 6 x′}.
“adm-n4” 22:47 page #69
F. Farsad, A. Madanshekaf 247
Corollary 4.6. Let S be a pomonoid and f : X → B be a regular injective
S-poset map. Then
(i) for every b ∈ B, the sub S-poset f−1(b) of X is regular injective
object in Pos-S, so it is a complete poset;
(ii) for x ∈ X and b ∈ B with f(x) 6 b, {x′ ∈ f−1(b) | x 6 x′} has a
minimum element xb (also we have the dual of this fact).
Proof. Applying Proposition 4.4 and the above proposition we get the
result.
Now, we consider the category Pos-S as a sub category of Cat. On
the other words, every S-poset is a category as a poset and all action-
preserving monotone maps are functors. Further, by a topological S-poset
functor we mean an S-poset map which is topological as a functor. So we
get the following result.
Theorem 4.7. Every regular injective object in Pos-S/B is a topological
S-poset functor.
Proof. First by Corollary 4.6 and Theorem 4.1, one concludes that every
regular injective object in Pos-S/B is a regular injective object in Pos/B.
Then, part (3) of Example 2.3, implies that in Pos, (Emb)� = Top, and
we get the result.
Our next goal is to prove the converse of the above fact in a special
case where S is a pogroup. But first we record a well known result.
Theorem 4.8. Let A
F //
B
U
oo with F ⊣ U be an adjunction. Then for
any B ∈ B, one has an adjunction FB ⊣ UB, where FB : A/U(B) → B/B,
and UB : B/B → A/U(B), are such that
• UB(f : X → B) = (U(f) : U(X) → U(B)), and
• FB(g : Y → U(B)) = (F (Y )
F (g)
→ FU(B)
ǫB→ B)
In light of [5, Theorem 12], the following result is a particular case of
Theorem 4.8, but for the convenient of the reader we give a proof here.
Theorem 4.9. The functor GB : Pos/B → Pos-S/B which assigns
every object in Pos/B (i.e., a monotone map in Pos) to itself (which
equips with the trivial action) has a left adjoint.
“adm-n4” 22:47 page #70
248 Weak factorization systems
Proof. Define the functor HB : Pos-S/B → Pos/B given by HB(h) =
h̄ : A/θA → B with h̄([a]) = h(a) for every object h : A → B in Pos-
S/B, where the poset A/θA was introduced in [5, Theorem 12] and h̄
is a monotone map. If g : A → C is an S-poset map over B, then
HB(g) : A/θA → C/θC defined by HB(g)([a]) = [g(a)], is a well-defined
monotone map over B. The unit of this adjunction
A
ηf
//
f ��
A/θA
GBHB(f)
||
B
for an object f : A → B in Pos-S/B, is the canonical S-poset map over
B, i.e, ηf = π. It is a universal arrow to GB because for a given S-poset
map h such that the diagram
A
h //
f ��
P
GB(l)=l��
B
commutes, where l : P → B is a monotone map, we have a unique S-poset
map h̄ as in the following diagram
A/θA
h̄ //
HB(f)
""
P
l��
B
(4.1)
given by h̄([a]) = h(a). By a similar proof as in [5, Theorem 12] one
can prove that h̄ is a well-defined S-poset map. The diagram (4.1) is
commutative, since for every [a] ∈ A/θA we have:
l(h̄[a]) = l(h(a)) = f(a) = f̄ [a] = HB(f)[a].
Theorem 4.10. Let S be a pogroup. Then all topological S-poset functors
are regular injective as S-poset maps with trivial action.
Proof. By an analogue proof as in [7, Theorem 4.6], we can show that
the functor HB : Pos-S/B → Pos/B preserves order-embeddings, these
are the regular monomorphisms in two categories Pos-S/B and Pos-S.
Therefore, by [9, Lemma 3.2] and the adjunction as mentioned in Proposi-
tion 4.9, the functor GB : Pos/B → Pos-S/B preserves regular injective
objects. Since, part (3) of Example 2.3 implies that (Emb)� = Top so we
get the result.
“adm-n4” 22:47 page #71
F. Farsad, A. Madanshekaf 249
Acknowledgments
The authors are very grateful to the anonymous referee for reading
the paper at least twice and giving very helpful suggestions.
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Contact information
F. Farsad,
A. Madanshekaf
Department of Mathematics, Faculty of Mathe-
matics, Statistics and Computer Science,
Semnan University, P. O. Box 35131-19111,
Semnan, Iran
E-Mail(s): faridehfarsad@yahoo.com,
amadanshekaf@semnan.ac.ir
Received by the editors: 21.04.2015.
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