On General Closure Operators and Quasi Factorization Structures

In this article the notions of (quasi weakly hereditary) general closure operator $\mb{C}$ on a category $\cx$ with respect to a class $\cm$ of morphisms, and quasi factorization structures in a category $\cx$ are introduced. It is shown that under certain conditions, if $(\ce, \cm)$ is a quasi factorization structure in $\cx$, then $\cx$ has quasi right $\cm$-factorization structure and quasi left $\ce$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $\cm$, every quasi factorization structure $(\ce, \cm)$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $\cm$, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are furnished.


Introduction and preliminaries
Closure operators have been around for almost one century in the context of categories of topological spaces and lattices. In [27], Salbany introduces a particular closure operator in the category of topological spaces. This idea was later transformed to an arbitrary category, which led to the general concept of categorical closure operators, see [8][9][10]. Weakly hereditary and idempotent closure operators play an important role, as they arise from factorization structures. In [22], quasi right factorization structures were introduced and their connection with closure operators was investigated, while quasi left factorization structures appear in [19].
There are many important structures that are not factorization structures nor even weak factorization structures; however they are quasi factorization structures, as introduced in this article. Many examples of such structures are provided and the connections between quasi right factorization structures, quasi left factorization structures, quasi factorization structures, and closure operators are investigated.
In Section 2, to develop some theory related to closure operators in the more general context of a quasi right factorization structure M on a category X , the notions of quasi mono and quasi epi are given. Then we will study some preliminary results and we will provide some examples of these notions. The strong point of these examples is to provide epimorphisms which are quasi mono and monomorphisms which are quasi epi. In Section 3, the definition of a general closure operator on a category X with respect to the class M of morphisms is introduced, some related results and several examples are also given. In Section 4, after defining quasi weakly hereditary closure operator, we prove that for a quasi idempotent closure operator we have a quasi right factorization structure and for a quasi weakly hereditary closure operator under some conditions we have a quasi left factorization structure. In Section 5, for morphism classes E and M, the notion of (E, M)-quasi factorization structure is introduced and examples of quasi factorization structures which are not weak factorization structures are furnished. It is shown that if (E, M) is a quasi factorization structure in X , then X has quasi right M-factorization structure provided that M has X -pullbacks and it has quasi left E-factorization structure provided that M ⊆ M on(X ), the class of monos, and E has X -pushouts. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure oper-ator with respect to a class M that is contained in the class of quasi monos and is closed under composition, every quasi factorization structure (E, M) yields a quasi factorization structure relative to the given closure operator. Finally it is proved that for a closure operator with respect to a class M that is contained in the class of strongly quasi monos and is a codomain, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent.
To this end we will give some basic definitions and results which will be used in the following sections.
Definition 1.1. [22] Let M be a class of morphisms in X and for every object X of X , M/X be the class of all morphisms with codomain X. We say that X has quasi right M-factorizations or M is a quasi right factorization structure in X , whenever for every morphisms Y f G G X in X , there exists M m f G G X ∈ M/X such that (a) f = m f g for some g; (b) if there exists m ∈ M/X such that f = mg for some g, then m f = mh for some h. m f is called a quasi right part of f .
With m = {mh : h ∈ X and mh is defined} denoting the sieve generated by m, see [21], (a) is equivalent to (a ) f ⊆ m f ; and (b) is equivalent to: (b ) if there exists m ∈ M/X such that f ⊆ m , then m f ⊆ m . Note that right M-factorizations, as defined in [10], are quasi right Mfactorizations.
Lemma 1.2. [22] Suppose that X has quasi right M-factorizations. Let f be a morphism in X and m f be a quasi right part of f . The class of all isomorphisms in X is denoted by Iso(X ). Proposition 1.3. Suppose M is closed under composition with isomorphisms on the left, that is, m ∈ M, α ∈ Iso(X ), and αm defined, yields that αm ∈ M. If f is a morphism in X and m f is a quasi right part of f , then αm f is a quasi right part of αf .
Proof. This follows directly from the definition.
Notation 1.4. For each composite Z g G G X f G G Y in X we will denote by f (g) : f (Z) G G Y a chosen quasi right part of a quasi right Mfactorization of f g. Note that if m is another quasi right part of f g, by Proof. Obvious.
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 M-factorization, see [19].

Quasi mono and quasi epi
In this section, quasi monos and quasi epis as a generalization of monos and epis will be defined and some of their properties will be studied. Then we will provide some examples of these notions. The significant point of these examples is to provide epimorphisms which are quasi mono and monomorphisms which are quasi epi. Since these notions, especially "quasi mono", are used in the study of some kinds of general closure operators, some examples of quasi right M-factorization structures are given, in which the class M is contained in the class of quasi monos.
The class of all quasi monos in X is denoted by QM (X ).
(b) A morphism f is called quasi epi, whenever for each morphism a, b ∈ X if af = bf , then a = b . The class of all quasi epis in X is denoted by QE(X ).
Proposition 2.2. We have the following: (a) f is a quasi mono if and only if for all morphisms u and v with the If gf is a quasi mono, then f is a quasi mono.
(c) If f and g are quasi monos and gf is defined, then gf is a quasi mono.
is a pullback in X , then π 2 is a quasi mono.
Proof. The proof is straightforward. Proposition 2.3. We have the following: (a) If gf is a quasi epi, so is g".
(b) If f is an epi and g is a quasi epi and gf is defined, then gf is a quasi epi.
Proof. The proof is straightforward. Lemma 2.4. We have the following: (a) If f is a quasi mono and a split epi, then f is an isomorphism.
(b) If f is a quasi epi and a split mono, then f is an isomorphism.
Thus f sf = f , and so sf = 1 X , because f is a quasi mono.
Therefore there exists a morphism X t G G X such that sf t = 1 X . Hence s, and thus f , is an isomorphism. (b) Similar to (a).
Corollary 2.5. We have the following: (a) If the class of quasi monos in X is pullback stable, then quasi monos are monos.
(b) If the class of quasi epis is pushout stable, then quasi epis are epis.
Proof. (a) Suppose that a quasi mono X f G G Y is given. Consider the pullback diagram There exists a unique morphism g : X G G X × Y X in X such that π 1 g = 1 X , π 2 g = 1 X . Thus π 2 and π 1 are quasi monos and split epis, and so, by Lemma 2.4(a), they are isomorphisms. Therefore f is a mono.
Remark 2.6. In the categories, Set of sets, Top, of topological spaces and R-Mod, of left R-modules, quasi monos (quasi epis) are monos (epis).
To give examples of quasi monos which are not monos, we need the following definitions.
Recall that, [4, p.72 (ii) is essential in M , abbreviated K M , in case for every submodule L ≤ M , K ∩ L = 0 implies L = 0. It is easy to see that K is essential in M if and only if, for every nonzero m ∈ M , there is r ∈ R with rm ∈ K and rm = 0.
For a module M the socle of M (= Soc(M )) is defined as the sum of all simple (minimal) submodules of M , (see [30, p.174]).
An R-module M is called: (i) uniform if every non-zero submodule of M is essential in M (see [30, 19.9]).
(iii) cocyclic if there is an m 0 ∈ M with the property: every morphism h : Let R be a domain and M be an R-module. Then M is called torsionfree, whenever there are no nonzero x ∈ M and r ∈ R with rx = 0. Note that projective modules are torsion free (see [7,Proposition 1.1]).
Example 2.7. Let QP be the full subcategory of R-Mod whose objects are quasi-projective modules and let R be a domain.
(a) Let P be a uniform projective module. By [30, 21.1], Soc(P ) is the intersection of essential submodules of P and since P is uniform, every non-zero submodule of P contains Soc(P ) (and hence it is minimum proper submodule of P ). Consider the following commutative diagram where P/Soc(P ) is homomorphic image of a free R-module R (Λ) for some set Λ, p is the canonical projection and qf = p. Therefore f is a non-zero map. We show that f is quasi mono in QP. To this end let the diagram, in QP be given such that f g = f h. So we have, there exists r ∈ R such that 0 = rp 0 ∈ Img. Thus rp 0 = g(t) for some t ∈ N and hence rf (p 0 ) = f (g(t)) = f (h(t)) = 0. This is a contradiction, because R (Λ) is torsion-free. So Img = 0. Thus we have two cases: (i) Img = Imh = 0, and so g = h = {0}; (ii) Img = 0 and Imh = 0. Thus equality (1) implies that Img = Imh. Since N is quasi-projective, there exist morphisms α, β : N G G N such that the diagram andh(x) = h(x). Therefore g = h and hence p is quasi mono.
(b) Let P be a cocyclic projective module and L be the intersection of all non-zero submodules of P , hence, by [30, 14.8(c)], L is a non-zero minimum submodule of P . Since every non-zero projective module contains a maximal submodule (see [4,Proposition 17.14]), L is a proper submodule of P . Consider the following commutative diagram where P/L is homomorphic image of a free R-module R (Λ) for some set Λ, p is the canonical projection and qf = p. As in (a), we can see that f is a quasi mono in QP.
Recall that, [30, p.348], for a submodule U of a left R-module M , a submodule V ≤ M is called a supplement or addition complement of U in M if V is a minimal element in the set of submodules L ≤ M with U + L = M . A module is called supplemented if every submodule has a supplement (see [30, p.349

]).
A projective left (right) module over a ring R will be called left hereditary in case every left submodule is projective (see [16]).
with pg = ph be given. Thus K + Img = K + Imh. Since K + Img has a supplement in M , by [30, 41.1(5)] we have K = K ∩(K +Img) (K +Img). Therefore Imh = K + Img and hence Img ⊆ Imh. Similarly Imh ⊆ Img, so Img = Imh. Since M is hereditary, Img is a projective submodule of M . So there exist ι 1 : Img G G N and ι 2 : Imh G G N such that gι 1 = 1 Img and hι 2 = 1 Imh . Thus g = h(ι 2 g) and h = g(ι 1 h), hence g = h . Therefore p is quasi mono. Example 2.9. Let QP be the category of quasi-projective left R-modules (R need not be a domain). Define the subcategory C of QP to have Obj(QP) as objects and for all P, Q ∈ C a morphism f : Every isomorphism in QP satisfies in the condition (C1). Note that for each morphism f : P G G Q in C (i) if Imf Q, then P P and hence P = {0}; (ii) K f P . Let p : P G G Q be a non-zero morphism in C satisfying Thus for each morphism g : L G G P , pg = 0 implies that L = {0}. We show that p is quasi mono. To this end, let the diagram in C such that pg = ph and L = 0 be given . So we have Since Img K p and Imh K p , K p K p + Img and hence, by (C2), K p K p + Img. Thus equality (4) implies that Imh = K p + Img and so Img ≤ Imh. Similarly Imh ≤ Img. Therefore Img = Imh. Since L is quasiprojective, there exist morphisms α, β : L G G L such that the diagram It is easy to see that α and β are morphisms in C and g = h . Therefore p is quasi mono in C.
Recall that a semiperfect ring is one for which every finitely generated module has a projective cover, (see [25, p.179]). Semiperfect rings T , whose indecomposable, projective left and right modules have simple and essential socles, are called QF -2 rings (see [30, p.557]). Example 2.10. Let R be a QF -2 ring and P be a projective cover of a simple module. Since every simple module is trivially quasi-projective and indecomposable, P is indecomposable, (see [4,Exercises 17,p.203]). Thus P has a simple and essential socle and hence each non-zero submodule of P contains Soc(P ). Furthermore Soc(P ) is a fully invariant submodule of P , ( [28, p.16]), and so P/Soc(P ) is quasi-projective, (see [26,Lemma 4.2]). Now consider the canonical projection p : P G G G G P/Soc(P ) . Let K ≤ P , p(K) P/Soc(P ) and K + L = P for some L ≤ P . Thus p(K)+p(L) = P/Soc(P ) and hence p(L) = P/Soc(P ). Therefore p −1 p(L) = P . Let x ∈ p −1 p(L), so there exists l ∈ L such that p(x) = p(l). Thus x − l ∈ K p = Soc(P ). Since Soc(P ) ≤ L, x − l ∈ L. This implies that x ∈ L and so p −1 p(L) = L. Therefore L = P and hence K P . Since p is an epimorphism and K p is simple and essential, it fulfills the conditions (C1) and (C2) of 2.9 and hence p is quasi mono in C. Let Then R is a commutative local ring and R has a simple essential socle J(R) 2 = Z 2 m as R-module. In particular, R is uniform. Note that Soc(R) ⊆ J(R). Thus R has no non-zero semisimple direct summand, hence every simple R-submodule of R is superfluous, (see [23,Lemma 2.4]). Therefore Soc(R) R. As 2.10 the canonical projection p : R G G G G R/Soc(R) fulfills the conditions (C1) and (C2) of 2.9 and hence p is quasi mono in C.
( Example 2.12. Let X and X 0 be two sets with X 0 X and C be a subcategory of Set, such that X 0 ∈ C and X / ∈ C. Now define the subcategory D of Set to have obj(C) ∪ {X} as objects and for all A, B ∈ D, To prove this, suppose the diagram It is easy to see that f = gh. Thus f ∈ g . Similarly g ∈ f and hence f = g .
In the following examples we give a quasi right M-factorization structure in which each m in M is a quasi mono. where M/K is the homomorphic image of a free R-module R (Λ) for some set Λ, p is the canonical projection and q = pf . Since pf is an epi and p is a superfluous epi, f is epi, (see [4,Corollary 5.15.]). As we have shown in where pf = nu and so n is epi. Since M is projective, there exists a morphism d such that nd = p. Thus the factorization q = pf is a quasi right M-factorization and p is a quasi mono.
Example 2.14. Recall that a ring R is completely hereditary if its class of quasi-projective modules is closed under taking submodules, (see [14]). Let C be as in Example 2.9 and Re i /Je i and so N R. So the above diagram is in C. Also for each i, since Je i is superfluous and maximal submodule of Re i , p i satisfies the condition (C2) and hence is a quasi mono. Since R/J ∼ = Re 1 /Je 1 ⊕ · · · Re n /Je n and R/J is a projective R-module, for each i, Re i /Je i is a projective R-module and hence p i is split epi. Thus there exists s i : Re i and hence s i is a morphism in C. Therefore for each i, p i ∈ M. Similar to 2.13, the factorization q = p i f i is a quasi right M-factorization.
In the following two examples we give a class of quasi epis which are not epics.
Example 2.15. Let C be the category of subrings of real numbers R. Then is given in C such that f j = gj. Thus for each m ∈ Z, f (m) = g(m) and thus for each a ∈ Q, f (a) = g(a). On the other hand f ( In the first case f = g. In the second case define Example 2. 16. Let X and X 0 be in Set with X 0 X and C be a subcategory of Set, such that X − X 0 ∈ C and X / ∈ C. Now define the subcategory D of Set to have obj(C) ∪ {X} as objects and

General closure operator
Consider a category X and a fixed class M of morphisms (not necessarily monomorphisms) in X which we think of as generalized subobjects. For every object X of X , let M/X be the class of all morphisms in M with codomain X. The relation given by m n ⇔ m ⊆ n is reflexive and transitive, hence M/X is a preordered class. Note that m n means that there exists a morphism j such that m = nj. Since M is an arbitrary class of morphisms, j is not uniquely determined. Also m n and n m do not imply that j is an isomorphism. If m n and n m, then we say m and n are -equal and we write m ∼ n. Of course, ∼ is an equivalence relation, and M/X modulo ∼ is a partially ordered class. In fact, we shall use the notations and ∼ for elements of M/X rather than for their ∼-equivalence classes. So, with m denoting the ∼-class of m, we have m n ⇔ m n m ∼ n ⇔ m = n. (c) the continuity property: for every morphism f : Since M is an arbitrary class of morphisms and X has quasi right Mfactorizations, Definition 3.1 is a generalization of the closure operator that is defined in [10].
Remark 3.2. Suppose that M has X -pullbacks (that is, m ∈ M and f ∈ X with the same codomains, implies the pullback, f −1 (m), of m along f is in M) and X has quasi right M-factorizations. For every object X in X consider the preordered class M/X. We have the adjunction, see [22]. In the presence of (b), by adjunction (3.1), the continuity condition can equivalently be expressed as: (c ) for every morphisms f : (1) Let X be a pointed category with finite products, and K be a non-empty class of objects of X such that for any pair of isomorphic objects either both are in K or both are not; and let M be the class of all spilt epis with kernel in K. Then M is a quasi right factorization structure in X which is closed under composition with isomorphisms on both sides. A morphism f : It is easy to see that π 2 is a spilt epi with kernel in K. Since M is a collection of split epis, any family C = (c X ) X∈X where c X is a map from M/X to itself, forms a general closure operator on X with respect to M.
(2) Consider the category R M S of (R, S)-bimodules, where R and S are commutative rings and suppose that there exists a ring homomorphism σ : R G G S such that σ(1 R ) = 1 S . Thus S is an R-module by r ·s = σ(r)s and hence S ∈ R M S . Suppose that C is a full subcategory of R M S whose objects are (R, S)-bimodules M such that for each r ∈ R, s ∈ S and m ∈ M we have s(rm) = (s · r)m. Let M be the class of all split epis in C. One can easily verify that M is a quasi right factorization structure in C. A morphism f : X G G Y in C can be factored as For each morphism ϕ : M G G G G X in M, define its closureφ to be the unique map making the diagram Let C be the category of torsion free modules, [11], and M be the class of all split epis. Then M is a quasi right factorization structure in C, [22]. Suppose that m : There is a torsion free precover ϕ : T G G X. Since M is torsion free, there is a map ψ : M G G T such that ϕψ = m. Now define the closure of m to be the map ϕ.
(4) Let C be an abelian category with enough injectives, [13]. The collection M of all epis whose kernels are injective is a quasi right factorization structure. A morphism f : X G G Y can be factored as π 2 i, f , where i : X G G E is the mono from X to an injective object E and π 2 : E × Y G G Y is the projection to the second factor. Now for each morphism m : M G G G G X define its closure to be the map mπ 2 : K ⊕ M X, where K = Ker(m).
(5) Let C be a closed model category, [24]. The collection M of fibrations in C is a quasi right factorization structure. For each object X ∈ C we have a trivial fibration p X : Q(X) G G X with Q(X) cofibrant. Now for each morphism m : M G G X in M define its closure to be the map mp m : Q(M ) G G X. (6) As a special case of (5), in the category Top, of topological spaces and continuous maps, the collection M of Serre fibrations is a quasi right factorization structure. Now the closure of a morphism m : M G G X in M is as in (5).
(7) Let C be a model category. For the category of fibrant objects, C f , the collection M of fibrations is a quasi right factorization structure. Define the closure of m : M G G X in M to be the projection to the first factor, π 1 : X × M G G X.
(8) As a special case of (7), in the category Top, in which all the objects are fibrant, the collection M of Serre fibrations is a quasi right factorization structure. Define the closure of m : M G G X in M to be the projection to the first factor, π 1 : X × M G G X. (9) In the cofibrant category (Top, cofibrations, homotopy equivalences), the collection M of homotopy equivalences is a quasi right factorization structure. A morphism f can be factored as where r f is a homotopy equivalence, i f is a cofibration and Z f is the mapping cylinder of f , [20]. For each morphism M m G G X in M define its closure to be the map, Z m rm G G X.
is a homotopy equivalence, k f is a fibration and P f is the mapping path space of f , [20]. For each morphism m : M G G X in M define its closure to be the map k m : P m G G X. (11) In the Kleisli category Set P , where P is the power set monad P = (P, η, µ), for each morphismf : X G G Y in Set P , let f : X G G P (Y ) be its associated morphism in Set and be the (Epi, M ono) factorization of f . The class M = { m f :f ∈ Set P } is a quasi right factorization structure, see [22].
For each morphism . LetÃ be the full subcategory of A consisting of those objects A such that A : F G(A) G G A the component of the counit (of the above adjunction) at A, is a split epi. The class M consisting of those split epis in A whose domain is F (X) for some object X in X is a quasi right factorization structure onÃ. Each morphism f inÃ factorizes as where s is any splitting of B . If in addition X has binary coproducts, for each morphism m : F (X) G G A in M define its closure to be the unique morphism c X (m) : F (X + X) G G A, corresponding by adjunction Since θ X,A (m) =m and θ X,A is one-to-one, m = A F (m). Suppose that ι 1 , ι 2 : X G G X + X are the canonical injections of the coproduct X + X. (14) As a special case of (13), consider Proj(R-Mod) as the full subcategory of the category R-Mod, consisting of all projective R-modules.
The collection M of all epis with free domains is a quasi right factorization structure, see [22]. For each morphism m : F G G P in M define its closure to be the map [m, m] : F ⊕ F G G P . Now on instead of saying C is a general closure operator on the category X with respect to M we will say C is a closure operator.
Definition 3.5. Suppose that M is a class of morphisms in X and X has quasi right M-factorizations. Also suppose that C is a closure operator and m ∈ M/X, where X is an object in X . We say m is (a) quasi C-closed in X, if c X (m) ∼ m (see [22]); A morphism f in X is called quasi C-dense, whenever m f is quasi Cdense in X . We denote by E QC , the class of all quasi C-dense morphisms in X . Let M QC be the class of quasi C-closed members of M.
Remark 3.6. Let f and g be two morphisms in the category X .
(a) f ∼ 1 X is equivalent to 1 X f which is equivalent to f being a split epi.
(b) If f g and f is a split epi, then g is a split epi. For each morphism Proof. (a) By Propositions 1.3 and 1.6 and the continuity property of C,  Suppose that X has quasi right M-factorizations and C is a closure operator. C is called (a) quasi idempotent, if for each X ∈ X and m ∈ M/X, c X (c X (m)) ∼ c X (m) (see [22]).  With (E, M)-factorization structure as defined in [2], we have Theorem 4.3. [22] Suppose that X has (E, M)-factorization structure and C is a quasi idempotent closure operator. Then M QC is a quasi right factorization structure for X .   Also suppose X has quasi right M-factorizations and C is a closure operator. If

Quasi factorization structures
In this section the notations H and H are introduced and after studying some of their properties, the notion of quasi factorization structure in a category X is given. We will see that weak factorization structures as defined in [1] are quasi factorization structures, but the converse is not true as we will show by some examples. Finally we state the relation between a quasi factorization structure and a quasi idempotent and quasi weakly hereditary closure operator. Saying H has X -pushouts if the pushout of each morphism in H exists and is in H, we have: Proof. The proofs of (i) and (ii) follow directly from the definition.
(iii) Suppose that the commutative diagram such that s is a section and h ∈ H is given. Thus, h is a section and since h is a quasi epi, by 2.4(b), h ∈ Iso(X ). Put w = h −1 and so sw = v.
(iv) Suppose that E e 2 G G E 1 e 1 G G X are composable morphisms in H and the following commutative triangle is given Since H has X -pullbacks, we have Therefore we have the commutative triangle Proof. Suppose that mu = e, where e ∈ E QC and m ∈ M QC . Consider the quasi right M-factorization of e as Since e ∈ E QC , by Proposition 4.7, we have c X (m e ) is an isomorphism and so c X (m e ) ∈ M QC . We have m e = c X (m e )j and so e = c X (m e )(je 1 ). If e n with n quasi closed, then m e n. Therefore c X (m e ) c X (n) ∼ n.
Thus the factorization e = c X (m e )(je 1 ) is a quasi right M QC -factorization of e. So we have the commutative diagram. Proof. Suppose that m = ve, where e ∈ E QC and m ∈ M QC . First we G G X is a quasi right Mfactorization of m. For this reason suppose that the unbroken commutative diagram M with n ∈ M, is given. By Theorem 4.11, the quasi right M-factorization of u is a quasi left E QC -factorization of u and so e 1 ∈ E QC . Now consider the unbroken commutative diagram Let E and M be classes of morphisms in X . We say that E and M are closed under composition with isomorphisms (i) if α ∈ Iso(X ) and e ∈ E, and αe exists, then αe ∈ E; (ii) if α ∈ Iso(X ) and m ∈ M, and mα exists, then mα ∈ M.

Proof. Consider the factorization
Thus, m is a split epi. Since m is a quasi mono, by 2.4(a), m ∈ Iso(X ) and hence f ∈ E.
(ii) If f ∈ E , then by 2.2(b), e is a quasi mono and there exists w : M G G X such that f w ∼ m. Thus, mew ∼ m. Since m is a quasi mono, by 2.2(a), ew ∼ 1 M and hence e is a split epi. Therefore, by 2.4(a), we have e ∈ Iso(X ) and hence f ∈ M.
In the following definition X need not have quasi right M-factorizations.
Definition 5.7. A quasi factorization structure in a category X is a pair (E, M) of classes of morphisms such that; (a) every morphism f has a factorization as where e ∈ E and m ∈ M. Since f ∈ E ⊆ M, there exists Y w G G M such that mw ∼ 1 Y and hence m ∈ Iso(X ). On the other hand, f ∈ M ⊆ E and so there exists w : M G G X such that f w ∼ m ∼ 1 M . Therefore f is a split epi and hence f is an isomorphism.
(iii) If E is closed under composition with isomorphisms, then E = M and so, by 5.2(ii), Ret(X ) ⊆ E. Moreover, if M has X -pullbacks, then by 5.2(iv) E is closed under composition.
(iv) If E ⊆ QM (X ) and M is closed under composition with isomorphisms, then M = E . Also if E ⊆ QE(X ), then by 5.2(iii) Sec(X ) ⊆ M. Moreover, if E has X -pushouts, then 5.2(iv) implies that M is closed under composition.
In the following example (E, M) is a quasi factorization structure which is not a weak factorization structure.
where i is a trivial cofibration and p is a trivial fibration. Thus, [17, Put w = ds, so mw = 1 X . Therefore E ⊆ M. Similarly, we can show that M ⊆ E . Since E ∩ M IsoX , the system is not weak.
(2) As a special case of (1), in the category Top, in which all the objects are cofibrant, the collections E of homotopy equivalences and M of Serre fibrations form a quasi factorization structure. factorization off . As proved in [22] we can see the following commutative diagram has no diagonal: (5) Let X be a category with coproducts, is the coproduct inclusion to the first factor} and M be any collection of split epis. Then (E, M) is a quasi factorization structure. Since Iso(X ) E, the system is not weak. (6) Let X be an abelian category. Define is the second projection}. Then (E, M) is a quasi factorization structure. Since Iso(X ) M, the system is not weak.
Theorem 5.10. Suppose that X has a quasi right M-factorizations such that M is closed under composition, M ⊆ QM (X ) and for each f ∈ QM (X ), f is an isomorphism whenever m f is an isomorphism. Then there is a class E such that (E, M) is a quasi factorization structure in X .
Proof. Let E be the class M and such that e ∈ E be given. Thus, by 2.2(b) we have e ∈ QM (X ). Let Therefore there exists a morphism E w G G M 1 such that m e w ∼ 1 E and hence m e is a split epi. Since m e ∈ QM (X ), m e is an isomorphism and so by hypothesis we have e is an isomorphism. Let e −1 be the inverse of e, so me −1 = v. Thus, m ∈ E .
Theorem 5.11. Suppose that (E, M) is a quasi factorization structure in X .
(a) If M has X -pullbacks, then X has a quasi right M-factorization structure.
(b) If M ⊆ M on(X ) and E has X -pushouts, then X has a quasi left E-factorization structure.
Proof. (a) Let f : X G G Y be a morphism in X and consider the quasi factorization of f as where e f ∈ E and m f ∈ M. Suppose the unbroken square, So there is a morphism t such that the triangles in the following diagram commute: where e f ∈ E and m f ∈ M. Suppose the unbroken square So there exists a morphism t such that the triangles in the following diagram commute: Y Since e ∈ E and M ⊆ E , there exists w : In [10], it is proved that if the category X has (E, M)-factorization structures and C is a closure operator on X , then C is idempotent and weakly hereditary if and only if X has (E C , M C )-factorizations. In the following we prove a similar result under weaker conditions. Proof. (i) Let f = me be a quasi factorization of f , where e ∈ E and m ∈ M. By Propositions 5.11 and 5.5, e ∈ E QC . Since C is quasi weakly hereditary, there exists a quasi C-dense morphism j m : where c Y (m) ∈ M QC . Put d : def = j m e. Since E QC is closed under composition, d ∈ E QC . Thus every morphism f has a factorization such that its left part is in E QC and its right part is in M QC . Now we show that E QC = (M QC ). By 5.3, E QC ⊆ (M QC ). Let h ∈ (M QC ). Thus there exist morphisms e 1 ∈ E QC and n 1 ∈ M QC such that Thus there is a morphism w 1 as in the following diagram G G X such that m 1 w 1 ∼ n * (m * 2 ) −1 and hence there exists l : N G G N such that m 1 w 1 l = n * (m * 2 ) −1 . Thus m 2 m 1 (w 1 l) = m 2 n * (m * 2 ) −1 = n and hence n ≤ m 2 m 1 . Since m 2 m 1 = ne, m 2 m 1 ≤ n and so m 2 m 1 ∼ n. Therefore m 2 m 1 ∈ M QC . Now we prove that C satisfies (QCD). Let X e 1 G G Y e 2 G G Z be morphisms such that e 1 , e 2 ∈ E QC . We show that e 2 e 1 ∈ (M QC ). Let X e 1 G G Y e 2 G G Z = X u G G M m G G Z such that m ∈ M QC . Consider the pullback diagram Thus m * ∈ M QC and since E QC ⊆ (M QC ), there exists t 1 : Y G G K such that m * t 1 ∼ 1 Y . Therefore m * is a split epi and since m * ∈ M, m * is an isomorphism. Thus we have the diagram v v such that mt 2 ∼ 1 Z . Thus e 2 e 1 ∈ (M QC ). It is easy to see that M QC is closed under composition with isomorphisms and since M QC has X -pullbacks, by 5.11(a), X has quasi right M QC -factorization. So, by 3.9(a), E QC is closed under composition with isomorphisms. Thus, by 5.6(i), (M QC ) ⊆ E QC and hence e 2 e 1 ∈ E QC . Therefore C satisfies the property (QCD).