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On Semi Weak Factorization Structures Volume 11, Special Issue Dedicated to Professor George A. Grätzer, July 2019, 33-56. On semi weak factorization structures A. Ilaghi-Hosseini, S.Sh. Mousavi, and S.N. Hosseini∗ Dedicated to Professor George Grätzer Abstract. In this article the notions of semi weak orthogonality and semi weak factorization structure in a category are introduced. Then the re- X lationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characteriza- tion of semi weak orthogonality, factorization of morphisms, and semi weak factorization structures by natural isomorphisms. 1 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 * Corresponding author Keywords: Quasi right (left) factorization structure, (semi weak) orthogonality, (semi weak) factorization structure. Mathematics Subject Classification[2010]: 18A32, 18A25. Received: 21 December 2017, Accepted: 22 June 2018. ISSN: Print 2345-5853, Online 2345-5861. © Shahid Beheshti University 33 34 A. Ilaghi-Hosseini, S.Sh. Mousavi, and S.N. Hosseini 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 factor- ization structures is given in Proposition 2.9. We also prove when for a given quasi right structure , there is an , with ( , ) a semi weak factorization M E E M structure. In Section 3, we present a characterization of semi weak orthog- onality, 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 and be classes of morphisms E M in . We say that is (weakly orthogonal) orthogonal to , denoted by X E M ( w ) , whenever for every commutative diagram E ⊥ M E ⊥ M u X / M e /// m Y v / Z with e and m , there is a (morphism) uniquely determined ∈ E ∈ M morphism w : Y / M with we = u and mw = v. Definition 1.2. [1] A weak factorization system in is a pair ( , ) of X E M classes of morphisms such that (i) = and = ; M E E M (ii) every morphism f has a factorization f = me with m and ∈ X ∈ M e . ∈ E Let and be two classes of morphisms in the category . Let e, e0 E M e0 X ∈ E and m, m0 be given. We denote em and e m, whenever for any f ∈ M m0 On semi weak factorization structures 35 and g making the squares e f 0 / / · @ · · @ · d0 d e m and e m g / / · · · m0 · commute, there exist diagonals rendering both triangles commutative. E Now we can define the classes and as E M M M = m em, e and m0 ; E { ∈ M | m0 ∀ ∈ E ∀ ∈ M} E e0 = e e m, e and m . M { ∈ E | ∀ 0 ∈ E ∀ ∈ M} Proposition 1.3. Let and be classes of morphisms in which are E M X closed under composition. ( , ) is a weak factorization system if and only E M if E (i) = and = , M E M E M (ii) every morphism f has a factorization f = me with m and ∈ X ∈ M e . ∈ E Proof. This follows directly from the definition. With g = ge ge is defined and e (for the class of all mor- h i { | ∈ E} E phisms, g Eis just a principal sieve, see [7]), we have h iE Definition 1.4. See [9]. Suppose that is a class of morphisms in . M X We say that has quasi right - factorizations or is a quasi right X M M f factorization structure in , whenever for all morphisms Y / X in , mf X X there exists M / X /X such that ∈ M (a) f m ; h i ⊆ h f i (b) if f m , with m /X, then m m . h i ⊆ h i ∈ M h f i ⊆ h i mf is called a quasi right part of f. The notion of a cosieve is dual to that of a sieve. A principal cosieve gen- erated by f is denoted by f . Also the notion of a quasi left -factorization i h E is dual of quasi right factorization, see [7]. 36 A. Ilaghi-Hosseini, S.Sh. Mousavi, and S.N. Hosseini 2 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 , with ( , ) a semi weak factorization structure. Some other E E M properties are investigated. Definition 2.1. Suppose that is a category and and are classes X E M of morphisms in . We say that is semi weak orthogonal to , written X E M sw , whenever E ⊥ M (SW1) for any commutative diagram E 7/ M d0 e /// m0 X m / Y d where m, m and e there exists a morphism X 0 / M making 0 ∈ M ∈ E the lower triangle commute; (SW2) for any commutative diagram e E 7/ M e /// m 0 d X / Y d where m and e, e there exists a morphism X / M making the ∈ M 0 ∈ E upper triangle commute. Proposition 2.2. Suppose that and are classes of morphisms in . E M X If w , then sw . E ⊥ M E ⊥ M Proof. The proof is straightforward. On semi weak factorization structures 37 Definition 2.3. Suppose that is a category and and are classes of X E M morphisms in . We say that has semi weak ( , )-factorizations or X X E M ( , ) is a semi weak factorization structure in , whenever E M X (SWF1) for all f : Y / X there exists m /X and e Y/ such ∈ M ∈ E that f = me; and (SWF2) sw . E ⊥ M Remark 2.4. Weak ( , )- factorizations are semi weak ( , )-factorization E M E M structures. Theorem 2.5. If has semi weak ( , )-factorizations, then has quasi X E M X right -factorizations and quasi left -factorizations. M E Proof. To show that has quasi right -factorizations, let the morphism X M f in be given. By (SWF1), there exist m and e such that X f ∈ M ∈ E f = m e. So f factors through m . Now suppose that there exist m f f ∈ M such that f m . Thus m e m and so by (SWF2), we have h i ⊆ h i h f i ⊆ h i m m . Therefore has quasi right -factorizations. Similarly h f i ⊆ h i X M X has quasi left -factorizations. E Corollary 2.6. If has semi weak ( , )-factorizations and f = me, then X E M m is a quasi right part and e is a quasi left part of f. Proof. By the fact that sw , the proof is obvious. E ⊥ M * Let have pullbacks. The partial morphism category has the same X * X objects as , with morphisms f = [(i , f)] : X / Y equivalence classes X f of pairs ( if : Df / X , f : Df / Y ) as shown in the following dia- gram f D / Y f > i f * f X where if is a universal mono, that is, its pullback along every morphism exists. Equivalence of (if , f) and (ig, g) means that there is an isomor- phism k for which if = igk and f = gk. The composition of morphisms 38 A. Ilaghi-Hosseini, S.Sh. Mousavi, and S.N. Hosseini * * f g X / Y / Z is defined by * * 1 1 g f = [(i , g)][(i , f)] = [(i f − (i ), g i− (f))], g f f ◦ g ◦ g as shown in the following diagram 1 ig− (f) g E / D / Z g ? 1 i f − (ig) p.b g * g Df / Y f = i f * f X where the commutative square is a pullback square. Now let and be classes of morphisms in and and be the E M X E0 M0 classes: = [(i , e)] dom(i ) = dom e, e and i is a universal mono E0 { e | e ∈ E e } = [(1, m)] dom 1 = dom m and m M0 { | ∈ M} We have the following proposition. Proposition 2.7. Let ( , ) be a semi weak factorization structure for E M X and be stable under pullbacks, see [2, Definition 28.13]. Then ( 0, 0) is E * E M a semi weak factorization structure for . X * * Proof. For an arbitrary morphism f = [(i , f)] : X / Y in X , since f f ∈ , we have the commutative diagram X f D / Y f ? /// e m M On semi weak factorization structures 39 where e and m . So we have ∈ E ∈ M e m D / M / Y f ? 1 p.b 1 * m D / M f e > i f * e X * f X / Y > * /// * e m M * * where m = [(1, m)] 0 and e = [(if , e)] 0.
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