On semi weak factorization structures

In this article the notions of semi weak orthogonality and semi weak factorization structure in a category X are introduced. Then the relationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structures by natural isomorphisms.


Introduction
The notions of (right, left) factorization structure appeared in [2], while weak factorization structures introduced in [1].In [9] and [7] the notions of quasi right, respectively, quasi left, factorization structure and some related results has been given.Since in various categories, there are important classes of morphisms that are not factorization structures nor even weak factorization structures, a weaker notion of factorization structure is deemed necessary; so the notion of semi weak factorization structure is introduced.The other main result is to look at semi weak factorization structures as certain isomorphisms in a particular quasicategory.
In the present article we first give the preliminaries in the current section, as well as a characterization of weak factorization structures in Proposition 1.3.Then in Section 2, we give the notions of semi weak orthogonality and semi weak factorization structure and its relation with quasi right, quasi left, and weak factorization structures.A characterization of semi weak factorization structures is given in Proposition 2.9.We also prove when for a given quasi right structure M, there is an E, with (E, M) a semi weak factorization structure.In Section 3, we present a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structures in terms of certain natural isomorphisms.Finally in the last section, that is, Section 4, we present several examples of semi weak factorization structures that are not weak factorization structures.Definition 1.1.See [1] and [4].Let E and M be classes of morphisms in X .We say that E is (weakly orthogonal) orthogonal to M, denoted by (E ⊥ w M) E ⊥ M, whenever for every commutative diagram Proof.This follows directly from the definition.
With g E = {ge|ge is defined and e ∈ E} (for E the class of all morphisms, g E is just a principal sieve, see [7]), we have Definition 1.4.See [9].Suppose that M is a class of morphisms in X .We say that X has quasi right M-factorizations or M is a quasi right factorization structure in X , whenever for all morphisms The notion of a cosieve is dual to that of a sieve.A principal cosieve generated by f is denoted by f .Also the notion of a quasi left E-factorization is dual of quasi right factorization, see [7].

Semi weak factorization structure
In this section, the notion of semi weak factorization structure, based on semi weak orthogonality, is introduced and its relation with quasi right, quasi left, and weak factorization structures is given.A characterization of semi weak factorization structures is given in Proposition 2.9, which is the counterpart of Proposition 1.3.We also prove when for a given quasi right structure M, there is an E, with (E, M) a semi weak factorization structure.Some other properties are investigated.Definition 2.1.Suppose that X is a category and E and M are classes of morphisms in X .We say that E is semi weak orthogonal to M, written E ⊥ sw M, whenever (SW1) for any commutative diagram where m, m ∈ M and e ∈ E there exists a morphism X d G G M making the lower triangle commute; (SW2) for any commutative diagram where m ∈ M and e, e ∈ E there exists a morphism X d G G M making the upper triangle commute.Proposition 2.2.Suppose that E and M are classes of morphisms in X .
Proof.The proof is straightforward.Definition 2.3.Suppose that X is a category and E and M are classes of morphisms in X .We say that X has semi weak (E, M)-factorizations or (E, M) is a semi weak factorization structure in X , whenever (SWF1) for all f : Y G G X there exists m ∈ M/X and e ∈ Y /E such that f = me; and (SWF2) E ⊥ sw M.
Theorem 2.5.If X has semi weak (E, M)-factorizations, then X has quasi right M-factorizations and quasi left E-factorizations.
Proof.To show that X has quasi right M-factorizations, let the morphism f in X be given.By (SWF1), there exist m f ∈ M and e ∈ E such that f = m f e.So f factors through m f .Now suppose that there exist m ∈ M such that f ⊆ m .Thus m f e ⊆ m and so by (SWF2), we have m f ⊆ m .Therefore X has quasi right M-factorizations.Similarly X has quasi left E-factorizations.
Corollary 2.6.If X has semi weak (E, M)-factorizations and f = me, then m is a quasi right part and e is a quasi left part of f .
Proof.By the fact that E ⊥ sw M, the proof is obvious.
Let X have pullbacks.The partial morphism category X has the same objects as X , with morphisms where i f is a universal mono, that is, its pullback along every morphism exists.Equivalence of (i f , f ) and (i g , g) means that there is an isomorphism k for which i f = i g k and f = gk.The composition of morphisms as shown in the following diagram where the commutative square is a pullback square.
Now let E and M be classes of morphisms in X and E and M be the classes: We have the following proposition.
Proposition 2.7.Let (E, M) be a semi weak factorization structure for X and E be stable under pullbacks, see [2,Definition 28.13].Then (E , M ) is a semi weak factorization structure for X .

Proof. For an arbitrary morphism
where e ∈ E and m ∈ M.So we have • there exist morphisms d and d such that e = de and m = m d .Proposition 2.9.Suppose that E and M are classes of morphisms in X .(E, M) is a semi weak factorization structure in X if and only if Proof.This follows directly from the definition.Lemma 2.10.Let (E, M) be a semi weak factorization structure for the category X then: (1) For any section s, sf ∈ E implies that, f ∈ E.
(2) For any retraction r, f r ∈ M implies that f ∈ M. Proof.
(1) Suppose that s : where m ∈ M and e ∈ E. Since s is a section, there exists a morphism r : K G G C such that rs = 1 and since sf ∈ E, the following commutative diagram (2) The proof is similar to (1).It is known (see [4]) that, for a right factorization M, which is closed under composition, there is an E such that (E, M) is a factorization structure.However, for a quasi right factorization M, which is closed under composition, in general, there is no E such that (E, M) is a semi weak factorization structure.
Example 2.12.Let X be the category consisting of the following objects and morphisms only: h}, is closed under composition and Iso(X ) ⊆ M. It is easy to see that X has quasi right M-factorizations.We claim that there is not a class E of morphisms in X such that X has semi weak (E, M)-factorizations.Otherwise, if E is such a class, since f can be factored as f = 1 B f , with 1 B ∈ M, f must be in E. Similarly, g ∈ E.
On the other hand, hf = hg ⊆ g and h ∈ M. Since E ⊥ sw M, then f ⊆ g , which is a contradiction.

A characterization of semi weak factorization structure
In [9] a one to one correspondence between certain classes of quasi right factorization structures and 2-reflective subobjects of a predefined object in the category of laxed preordered valued presheaves, Lax(P rOrd X op ), has been studied.In [7] it has been shown that quasi left factorization structures correspond to subobjects of predefined objects in Lax(P rOrd X op ).It has further been shown that this correspondence is one to one when quasi left factorization structures are restricted to the so called QLF-codomains.In this section we give a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structure in a category X in terms of certain natural isomorphisms.
Denote by P rOrd the category of preordered sets and order preserving functions.Throughout this section suppose that X has pullbacks and P s : X op G G P rOrd the functor defined by y y where P s (f )( h ) = h * f and h * f is a pullback of h along f , and P c : X op G G P rOrd the functor defined by y y where In what follows X 2 is the class {(f, g)|codm(f ) = dom(g)} of all composable morphisms in X .Definition 3.1.For a class A of morphisms in X we say A satisfies the pullback condition, if for every cospan f and g, with g in A, there exists a pullback of g along f belonging to A.
The following example shows the existence of a class of morphisms that satisfies the pullback condition, which is not stable under pullbacks.
Example 3.2.(i) Let C be a category with finite products and M be the class of all pr 2 , the second factor projection.Then M satisfies the pullback condition.To prove this let f : (ii) Let K be a field, R be a finite dimensional K-algebra and C be a full subcategory of RMod, of left R-modules, whose objects are finitely generated R-modules.Let M be the class of all morphisms in C which factors through a non isomorphism pr 2 .Then M satisfies the pullback condition.To prove this let f : The morphism pr 2 is not an isomorphism, because otherwise, since every module is finite dimensional as a K-vector spaces, dim K N = 0 and so N = 0. Therefore pr 2 is an isomorphism, and this is a contradiction.First, we characterize semi weak orthogonality in two steps.
Step 1: Suppose E and M are classes of morphisms in X which satisfy the pullback condition.Define the equivalence relation ∼ s on X 2 as (f, g) ∼ s (f , g ) if and only if gf = g f and g = g .
Define the functor R s : Also the family r s : R s G G P s defined, for each X, by (r s ) X ([(e, m)] s ) = me can be easily verified to be a natural transformation.
Lemma 3.3.Suppose that E and M are classes of morphisms in X that satisfy the pullback condition, M is closed under composition and for all morphisms f in X there exist morphisms m f ∈ M and e f ∈ E such that f = m f e f .Then for all m, m ∈ M and for all e ∈ E, me ⊆ m implies m ⊆ m if and only if r s is a natural isomorphism.
Proof.To prove the necessity, the family α : P s G G R s defined, for each X, by α X ( f ) = [(e f , m f )] s is a natural transformation.It is easy to see that αr s = 1 and r s α = 1 so r s is a natural isomorphism.To prove the sufficiency, suppose that β : Step 2: let E and M be classes of morphisms in X such that M is closed under composition, for all morphisms f in X there exist morphisms m f ∈ M and e f ∈ E such that f = m f e f and we have (a) for all g ∈ E, e g = g ; (b) f ⊆ g implies e f ⊆ e g .By a similar fashion, define the equivalence relation ∼ c on X 2 as Next we give a characterization of the factorizations of morphisms.To this end, assume E and M are classes of morphisms in X that satisfy the pullback condition, E is closed under composition, and Iso(X ) ⊆ E, where Iso(X ) is the class of isomorphisms in X .Suppose that f ∈ X /X is given.
y y where We denote the equivalence class of (f, g) by [(f, g)] E .
Define the functor where the relation is defined by , where f * h and g * are obtained by the pullback diagrams .G G P s E defined, for each X, by Since E and M satisfy the pullback condition, R (E,M) : X op G G P rOrd defined by E is a subfunctor of P 2 E .So we have the natural transformation X ([(e, m)] E ) = me E .Proposition 3.6.Suppose E and M are classes of morphisms in X which satisfy the pullback condition, E is closed under composition, and Iso(X ) ⊆ E. Then for all morphisms f ∈ X there exist morphisms m ∈ M and e ∈ E such that f = me if and only if * E,M is a natural isomorphism.
Since E is closed under composition, e ∈ E. Thus f = me.For the proof of the converse, suppose for all f ∈ X there exist morphisms e f ∈ E and Finally, the following theorem gives a characterization of semi weak factorization structures, under certain conditions.Theorem 3.7.Suppose E and M are classes of morphisms in X that satisfy the pullback condition, are closed under composition, and Iso(X ) ⊆ E. Then (E, M) is a semi weak factorization structure in X if and only if r s , r c , and * E,M are natural isomorphisms.

Examples
In this section we give several examples of semi weak factorization structures which are not weak factorization structures.
Example 4.1.Consider the category Set of sets.Let E be the class of monomorphisms and M be defined by (i) each pr 2 , the second factor projection, is in M; (iii) the class M is generated by (i) and (ii).
For an arbitrary function f : X G G Y we have f = pr 2 1, f .Since M is in the class of epimorphisms, E ⊥ w M. Since Iso X M, this factorization system is not weak.
Example 4.2.Consider the category RMod such that R is a left semisimple ring.Let E and M be as in Example 4.1.For an arbitrary R-module homomorphism f : X G G Y we have f = pr 2 1, f .Since every module is injective and projective, E ⊥ sw M. Since Iso X M, this factorization system is not weak.
Example 4.3.Let K be a field and R be a finite dimensional K-algebra, left injective R-module, and semihereditary ring, see [12,Definition 39.1].Let P be a full subcategory of RMod whose objects are finitely generated projective R-modules.Let E be the class of monomorphisms in P and M be the class of those morphisms in P which factor through a non isomorphism pr 2 .For each momorphism f : M G G N in P we have where F 1 and F 2 are finitely generated free R-modules, r 1 s 1 = 1 M , r 2 s 2 = 1 N , α = s 1 , s 2 f , and β = r 2 pr 2 .Thus f = βα, α ∈ E and β ∈ M.
Since every object in P is also injective, E ⊥ sw M. Since Iso X M, this factorization system is not weak.
Example 4.4.Consider the category T op of topological spaces.Let M be the class of initial maps and E be the class of all continuous maps with identity as the underlying map.Suppose that f : X G G Y in T op is given.We have where τ f is the induced topology by f on X.Now suppose that there is m, m ∈ M and e ∈ E such that me ⊆ m .So there is a morphism λ such that me = m λ.It is easy to see that λ = λ in the diagram (X, τ X ) makes the triangles commute and is in T op.So m ⊆ m .The proof of the second part is similar.Hence E ⊥ sw M. Since Iso X E, this factorization system is not weak.
where j is a trivial cofibration and p is a trivial fibration.Let M be the class of trivial fibrations.So by [5, and M any collection of retractions constitute a semi weak factorization system.Since Iso X E, this factorization system is not weak.
Example 4.9.Consider the category Set of sets.Define the classes E and M as For an arbitrary function f : X G G Y we have f = pr 2 1, f .Since M is in the class of epimorphisms, E ⊥ w M. Since Iso X M, this factorization system is not weak.
For an arbitrary morphism f : G G G H , we have f = (f )e.Suppose that f, g ∈ M and e ∈ E are given such that f e ⊆ g , thus f e = gh.So we have H where d = h and is the operation of G. Now suppose that f ∈ M and e ∈ E are given such that f e ⊆ e .Thus f e = ke.So we have H where d = 1 G×G .Therefore, (E, M) constitutes a semi weak factorization structure for Ab.Since Iso X M, this factorization system is not weak.
Example 4.11.Let X be a pointed category, see [2].Fix a non terminal object B ∈ X .Define the classes E and M as where pr 1 is the first factor projection.Every morphism f : X G G Y in X can be factored as f = pr 1 f.To show (E, M) constitutes a semi weak factorization structure for X , suppose pr 1 , pr 1 ∈ M and f ∈ E are given such that pr 1 f ⊆ pr 1 .Thus pr 1 f = pr 1 u.So we have commutes.Since pr 1 f = vg we have f = vg.Also pr 1 dg = vg = f and pr 2 dg = vg = f , hence dg = f.Since Iso X M, this factorization system is not weak.
Example 4.12.In the category Grp, let E and M be the following classes of morphisms: For an arbitrary morphism f : G G G H, we have f = pr 2 f.Suppose that g ∈ E and pr 2 , pr 2 ∈ M are given such that pr 2 g ⊆ pr 2 , thus pr 2 g = pr 2 u.So the map d = e, pr 2 is a diagonal for the following diagram.
So the condition (SW1) holds.Now let f, g ∈ E and pr 2 ∈ M such that G f G G K, g : G G G H and pr 2 f ⊆ g be given.Thus pr 2 f = vg.So the map d = pr 1 , v is a diagonal for the diagram

K
So the condition (SW2) holds.Therefore, (E, M) is a semi weak factorization system.Since Iso X M, this factorization system is not weak.

X
with e ∈ E and m ∈ M, there is a (morphism) uniquely determined morphism w : Y G G M with we = u and mw = v.Definition 1.2.[1] A weak factorization system in X is a pair (E, M) of classes of morphisms such that (i) M = E and E = M; (ii) every morphism f ∈ X has a factorization f = me with m ∈ M and e ∈ E. Let E and M be two classes of morphisms in the category X .Let e, e ∈ E and m, m ∈ M be given.We denote e e m and e m m, whenever for any f and g making the squares • exist diagonals rendering both triangles commutative.Now we can define the classes E M and E M as E M = {m ∈ M | e m m, ∀e ∈ E and ∀m ∈ M}; E M = {e ∈ E | e e m, ∀e ∈ E and ∀m ∈ M}.Proposition 1.3.Let E and M be classes of morphisms in X which are closed under composition.(E, M) is a weak factorization system if and only if e)] | dom(i e ) = dom e, e ∈ E and i e is a universal mono } M = {[(1, m)] | dom 1 = dom m and m ∈ M}

D
m = [(1, m)] ∈ M and e = [(i f , e)] ∈ E .Suppose that e = [(i e , e)] ∈ E and m , m ∈ M are given such that m e ⊆ m .Thus m e = m α , hence [(i α , mα)] = [(i e , m e)].Since (E, M) is a semi weak factorization structure for X , we have the diagram d.Now d = [(1, d)] gives the diagonal for the diagram A So the condition (SW1) holds.Now suppose that e , e ∈ E and m ∈ M are given such that m e ⊆ e .Thus m e = γ e .So, [(i e • e −1

Let
E and M be two classes of morphisms in the category X , with e, e ∈ E and m, m ∈ M. We write e e ¡ m and e¡ m m , respectively, whenever in the unbroken commutative diagrams •

Remark 2 . 8 .
(i) e e ¡ m if and only if, whenever me ⊆ e , then e ⊆ e ; (ii) e¡ m m if and only if, whenever me ⊆ m , then m ⊆ m .Now we can define the classes E ¡ M and E ¡ M as follows: E ¡ M = {e ∈ E | e e ¡ m, ∀e ∈ E and ∀m ∈ M}; E ¡ M = {m ∈ M | e¡ m m , ∀e ∈ E and ∀m ∈ M}.

Theorem 2 . 11 .
Let M be a class of morphisms in X which is closed under composition and is a quasi right factorization structure.If the class E = {e|∃f : f = me with m a right part of f } is closed under composition and for all m 1 , m 2 in M and e 1 , e 2 in E, the equality m 1 e 1 = m 2 e 2 implies that m 1 = m 2 and e 1 = e 2 , then (E, M) is a semi weak factorization structure for X .Proof.Since X has quasi right M-factorizations, for all morphisms f in X there exists m f ∈ M such that f = m f e 1 .By definition of E we have e 1 ∈ E. To show (SW1), suppose that m and m in M and e in E are given such that me ⊆ m .Thus me = m g, and so me = m m g e 2 .Since M is closed under composition and e and e 2 are in E, m = m m g .Therefore m ⊆ m .The Condition (SW2) is proved similarly.Hence E ⊥ sw M and so X has semi weak (E, M)-factorizations.
s is defined as [(e, m)] s s [(e , m )] s if and only if me ⊆ m e and m ⊆ m ; and R s (h)([(e, m)] s ) = [(e * , m * h )] s , where m * h and e * are obtained by the following pullback diagrams • e G G p.b.
s is the inverse of r s and m, m ∈ M and e ∈ E are given such that me ⊆ m .Thus there exists a morphism k such that me = m k and so r s ([(e, m)] s ) = r s ([(e k , m m k )] s ), where k = m k e k .Therefore (e, m) ∼ s (e k , m m k ) and hence m = m m k ⊆ m .

Lemma 3 . 4 .
c is defined as [(e, m)] c c [(e , m )] c if and only if me ⊆ m e and e ⊆ e ; and R c (h)([(e, m)] c ) = [(e eh , m m eh )] c , where eh = m eh e eh such that m eh ∈ M and e eh ∈ E. It is easy to see that R c is a lax functor.Also the family r c : R c G G P c defined, for each X, by (r c ) X ([(e, m)] c ) = me can be easily verified to be a natural transformation.Suppose that E and M are classes of morphisms in X which are closed under composition.Also suppose that for all morphisms f in X there exist morphisms m f ∈ M and e f ∈ E such that f = m f e f and we have (a) for all g ∈ E, e g = g ; and (b) f ⊆ g implies e f ⊆ e g .Then for all e, e ∈ E and for all m ∈ M, me ⊆ e , implies e ⊆ e if and only if r c is a natural isomorphism.Proof.The proof is similar to Lemma 3.3.Proposition 3.5.Let E and M be classes of morphisms in X that satisfy the pullback condition are closed under compositions.Also for all morphisms f in X there exist morphisms m f ∈ M and e f ∈ E such that f = m f e f and we have (a) for all g ∈ E, e g = g ; and (b) f ⊆ g implies e f ⊆ e g .Then E ⊥ sw M if and only if r s and r c are natural isomorphisms.Proof.It follows from Lemmas 3.3 and 3.4.
Thus the family * E : P 2 E

Lemma 4 . 5 .
If (E, M) and (E, M ) are weak factorization systems in X , then M = M Proof.Let m ∈ M be given.Thus, there exist e ∈ E and m ∈ M such that m = me and hence there exists a morphism d such that de = 1 and m d = m.So d is a retraction and m d ∈ M. Therefore [1, Observation 1.3 (2b )] implies that m ∈ M and hence M ⊆ M. Similarly, we have M⊆ M .Example 4.6.Let C be a closed model category whose objects are cofibrantfibrant.The pair (E, M) of morphisms in C, where E is the class of cofibrations and M is the class of weak equivalences which are retractions form a semi weak factorization structure.Because, by [5, Definition 7.1.3]every morphism f in C has a factorization f = pj, where j is a cofibration and p is a trivial fibration and by [5, Proposition 7.6.11(2)] p is a retraction.It is easy to see that M = E ¡ M .To prove E = E ∈ E, w ∈ M, and v is an arbitrary.By [5, Proposition 7.2.6]

Example 4 . 10 .
Consider the category Ab of abelian groups.Define the classes E and M as E = { G e G G G × G | e(x) = (x, e), where e = 1, e } and h is a zero morphism.Now suppose that f, g ∈ E and pr 1 ∈ M are given such that pr 1 f ⊆ g .Thus pr 1 f = vg.So we have X is a map and f = e, f , where e is the zero map}M = { A × B pr 2 G G B | pr 2is the second factor projection}.

Remark 4 .
13.Note that examples 4.1 to 4.4 satisfy the conditions of Theorem 3.7.
M ) is a weak factorization system, Lemma 4.6 implies that M = M .However this is not the case, for instance in T op let E = {0} × I I × {0} (with the topology induced by R 2 ), B = I × {0} and p : E G G B the projection on the first factor.Then p is not a fibration, see [10, Exercises 2.2.9].Therefore p / ∈ M but p ∈ M.
Definition 7.1.3],we have E ⊥ w M .Thus, there exists a morphism C d G G W such that p d = v and d i = ji.r G G B such that rj = 1 B .Put d := rd , so di = i.Therefore E =