Weak Inclusion Systems; part two 1

: New properties and implications of inclusion systems are investigated in the present paper. Many properties of lattices, factorization systems and special practical cases can be abstracted and adapted to our framework, making the various versions of inclusion systems useful tools for computer scientists and mathematicians.


Introduction
Computing Science concepts often take birth observing and studying practical situations and phenomena encountered in the process of development of software systems.For this reason, well established mathematical concepts and tools seem not to be perfectly suitable for some computing aspects.A concrete case is modularization GB92, DGS93, Ro s99], whose purpose is to formalize and give semantics to operations on software modules, such as importing, aggregation, hiding, parameterization, etc.Most of the operations on modules involve the notion of inclusion as a unique interpretation of a module into another; the \uniqueness up to an isomorphism" does not re ect the intuition behind these operations.Therefore, categorical notions like (mono) subobjects and factorization systems are not proper for some areas of computing.
At the authors' knowledge, the rst formal de nition of a factorization system of a category was given by Herrlich and Strecker4 HS73] in 1973, and a rst comprehensive study of factorization systems containing di erent equivalent de nitions was done by N emeti N em82] in 1982.However, the idea to form subobjects by factoring each morphism f as e; m, where e is an epimorphism and m is a monomorphism, seems to go back to Grothendieck Gro57] in 1957, and was intensively used by Isbell Isb64], Lambek Lam66], Mitchell Mit65], and many others.At our knowledge, Lambek was the rst to explicitly state and prove a diagonal-ll-in lemma for factorization systems Lam66] in 1966.
In general terms, this paper is a sequel to our paper Weak Inclusion Systems CR97].It develops the notion of inclusion in a categorical setting, emphasizing the idea of unique factorization.In CR97], we de ned the weak inclusion systems as a natural extension of inclusion systems and as an alternative to factorization systems.In this paper, we further explore weak inclusion systems' properties, our main goal being to present them as a real useful tool for computer scientists and, why not, for mathematicians in search of elegant, clear, and smooth proofs.
Section 2 just introduces some notations and basic categorical properties, and section 3 presents a bunch of easy but useful properties of weak inclusion systems.Section 4 introduces the notion of complete weak inclusion system as an alternative to inclusion systems.Section 5 explores relations between reachable and generated objects in a category.A criterion to say if a category has enough projectives is also given.Finally, section 6 is concerned with lifting (weak) inclusion systems to comma categories, functor categories, and categories of algebras and coalgebras of an endofunctor.
Acknowledgements We would like to thank to professor Sergiu Rudeanu for his remarks and suggestions on the application of inclusions systems in Birkholike axiomatizability results for generalizations of equational logics, and to professor Joseph Goguen for his comments on previous versions of this paper.

Preliminaries
The reader is supposed to be familiar with the basics of category theory (e.g., see Lan71, HS73]).In this section we present our formalism and remind the reader some notions used later in the paper.
Calligraphic letters denote categories and functors.If A is a category then jAj denotes its class of objects.The composition of morphisms is written in diagrammatic order, that is, if f : A ! B and g : B ! C are two morphisms, their composition is written f; g : A ! C. Sometimes, we use the word epic (monic, iso) instead of epimorphism (monomorphism, isomorphism).Some basic properties of epics and monics are supposed known, such as \f; g is an epic implies g is an epic", etc.
An important notion in category theory is that of subobject of an object.To be more precise and to avoid confusion with another kind of subobject introduced later in the paper, we call it mono subobject.A mono subobject of an object A in a category A is a coset (an equivalence class) of the equivalence relation de ned on monics of target A as follows: m m 0 if there exist two morphisms f and g such that f; m = m 0 and g; m 0 = m (actually, f and g are isomorphisms).
Inclusion systems are related to an old and useful concept in category theory, namely factorization systems (see HS73] and also N em82]).There are many equivalent de nitions of factorizations systems; we remind the reader the one we think is the closest to our approach: De nition 1.A factorization system of a category A is a pair hE; Mi, such that { E and M are subcategories of epics and monics, respectively, in A, { all isomorphisms in A are both in E and M, and { every morphism f in A can be factored as e; m with e 2 E and m 2 M \uniquely up to isomorphism", that is, if f = e 0 ; i 0 is another factorization of f then there is a unique isomorphism such that e; = e 0 and ; m 0 = m.There also are many equivalent de nitions of adjointness in the literature.Within this paper we adopt the following two: De nition 2. A functor F : X !A is a left adjoint of U : A ! X i for each pair of objects X 2 jXj, A 2 jAj there is a bijection X(X; U(A)) = A(F(X); A) which is natural in X and A.
De nition 3. A functor F : X !A is a left adjoint of U : A ! X i there exists a natural transformation : 1 X ) F; U having the universal property: for every X 2 jXj, A 2 jAj and every f : X !U(A) there is a unique f \ : F(X) !A such that X ; U(f \ ) = f. is called the unit of adjunction.Given an object A in A, let A denote the morphism 1 \ U(A) .

Basic De nitions and Properties
It is well-known that a small category can be associated to any partially ordered set: there exists exactly one object A for each element a in the set and there exists a morphism from A to B, written A , !B, if and only if a b.Furthermore, there is a bijection between partially ordered sets and small categories in which there is at most one morphism from A to B for every objects A and B (partiality), and if there is a morphism from A to B and a morphism from B to A then A = B (anti-symmetry).The correspondents of in mum and supremum are the product and the coproduct, respectively.Generalizing all these to categories which are not required to be small, we get: De nition 5.A category I is called a category of inclusions if and only if { I(A; B) has at most one element, and { I(A; B) 6 = ; and I(B; A) 6 = ; implies A = B. for every pair of objects A and B. If I(A; B) 6 = ; then let A , !B denote the unique morphism in I(A; B).It is called an inclusion and A is called a subobject of B. We say that I has ( nite) intersections i I has ( nite) products and we say that I has ( nite) unions i I has ( nite) coproducts.For every pair of objects A, B, let A \ B denote their product (also called their intersection) and let A B denote their coproduct (also called their union).
A small category of inclusions with nite intersections and nite unions corresponds to nothing else than a lattice Rud63, Bir67, Gr a71].Consequently, many properties of lattices hold in categories of inclusions.The following are only a few: Proposition 6.For any category of inclusions I and any objects A, B and C in jAj (assume that I has nite intersections and/or nite unions whenever \= appear), The union and intersection are commutative, associative and idempotent, ), De nition 7. A category of inclusions I which is a subcategory of A having the same objects as A is called a subcategory of inclusions of A (alternatively, we can say that A has inclusions I).I is a subcategory of strong inclusions of A (or A has strong inclusions I) i I is a subcategory of inclusions of A, I has nite intersections and unions, and for every pair of objects A, B, their union in I is a pushout in A of their intersection in I.
Example 1.We look at the following examples within the paper: Set the category of sets and functions, in which the inclusions are the ordinary inclusions of sets.It is easy to see that these inclusions are strong for Set.Top the category of topological spaces and continuous functions.The continuous inclusions form a subcategory of strong inclusions of Top: given A and B two topological spaces, their intersection is the set intersection of A and B together with the initial topology of its inclusions in A and B, and their union is the set union of A and B together with the nal topology of the inclusions of A and B in their union.
Sign the category of many sorted algebraic signatures and morphisms of signatures.The signature inclusions form a subcategory of strong inclusions of Sign.Alg the category of -algebras and morphisms of -algebras over a signature .The inclusions of -algebras form a subcategory of inclusions of Alg , but it is not strong.
Alg ;E the full subcategory of Alg containing all -algebras that satisfy the -equations E. Notice that depending on and E, Alg ;E can be any category of important structures in mathematics or computer science: monoids, modules, groups, abelian groups, rings, commutative rings, etc.The inclusions in Alg ;E are not strong either.
We will not insist on the notion of strong inclusions in the present paper.However, strong inclusions together with semiexactness DGS93] seem to play a major role in modularization.Abstract semantics is given for modularization in Ro s99], based on strong inclusions; no factorization (see De nition 8) is involved, which means that, perhaps, strong inclusions are good enough technical tools to handle complex modularization concepts.
De nition 8. hI; Ei is a weak inclusion system of A, or A has a weak inclusion system hI; Ei, i I is a subcategory of inclusions of A, E is a subcategory of A having the same objects as A, and every morphism f in A has a unique factorization f = e; i with e 2 E and i 2 I. hI; Ei is called an inclusion system if E contains only epics, and it is called a regular inclusion system if E contains only coequalizers.

Example 2. All structures in Example 1 have weak inclusion systems:
Set with I the set of inclusions and E the set of surjective functions.It is regular as each surjective function is a retract, so a coequalizer.

Top has two interesting weak inclusion systems (see CR97]
).One is hI 1 ; E 1 i, where I 1 is the set of continuous inclusions and E 1 is the set of nal continuous surjections, and the other one is hI 2 ; E 2 i, where I 2 is the set of initial continuous inclusions and E 2 is the set of continuous surjections.hI 1 ; E 1 i is not an inclusion system as there are continuous surjective functions that are not nal; hI 2 ; E 2 i is a regular inclusion system.Alg with inclusions of -subalgebras and surjective morphisms of -algebras is a regular inclusion system.
Alg ;E with inclusions of -subalgebras satisfying E and surjective morphisms of -algebras satisfying E is a regular inclusion system.If f : X !Y is a morphism in A, let e f ; i f denote its factorization and f(X) denote the factorization object of f, that is, the target object of e f .Moreover, we use the same notation, f(A), for the factorization object of the morphism A , !X; f, where A is a subobject of X.
Notice that every category A admits a trivial weak inclusion system in which I contains only identities and E = A. The following fact contains properties of weak inclusion systems proved in CR97]: Proposition 9.If hI; Ei is a weak inclusion system of A, e 2 E and i 2 I, then 1.I contains only monics.2. Each morphism in I \ E is an identity.3. right-cancellable: If f; i 2 I then f 2 I. 4. If f; i 2 E then i is an identity and f 2 E. 5.If f; g 2 E then g 2 E. 6.Any coequalizer is in E.
7. Any retract is in E. 8.All isomorphisms in A are in E. 9. diagonal-ll-in: If f; i = e; g then there is a unique morphism h 2 A such that e; h = f and h; i = g: The following proposition is also proved in CR97] and it says that inclusions are preserved under pullbacks: Proposition 10.If A has pullbacks and a weak inclusion system hI; Ei, and if i : B , !Y is an inclusion and f : X !Y is any morphism, then there is a unique pullback of the pair hi; fi such that the opposite arrow of i is an inclusion, too.
The pullback object given by the proposition above is written f ?1 (B).An immediate consequence is that f ?1 (B) is a subobject of X.
1. Let j be the inclusion A , !A 0 .Factor A , !X; f as e; i and A 0 , !X; f as e 0 ; i 0 , and let f(j) be the unique morphism given be the diagonal-ll-in lemma for the diagram (j; e 0 ); i 0 = e; i in the picture below: Since f(j); i 0 = i, by the right-cancellable property, f(j) is an inclusion, i.e., f(A) , ! f(A 0 ) , !Y .
2. Let j be the inclusion B , !B 0 , let hf ?1 (B) , !X; u f i denote the pullback of hB , !Y ; fi, and let hf ?1 (B 0 ) , !X; v f i denote the pullback of hB 0 , !Y ; fi.Then de ne f ?1 (j) : f ?1 (B) !f ?1 (B 0 ) as the unique morphism such that f ?1 (j); f ?1 (B 0 ) , !X = f ?1 (B) , !X and f ?1 (j); v f = u f ; j (this is because f ?1 (B 0 ) is a pullback object): By the right-cancellable property, f ?1 (j) is an inclusion, that is, f ?1 (B) , ! f ?1 (B 0 ) , !X. 3. It follows from the uniqueness of factorization for the morphism A , !X; f; g. 4. Let i Z be the inclusion C , !Z, and let hi Y ; u g i, hi X ; u f i and hi 0 X ; u f;g i denote the pullbacks of the pairs hi Z ; gi, hi Y ; fi and hi Z ; f; gi, respectively, as in the diagram below: Since hi X ; u f ; u g i is a cone of hi Z ; f; gi, there is a unique j such that j; u f;g = u f ; u g and j; i 0 X = i X .By the right-cancellable property, j is an inclusion.
On the other hand, since hi 0 X ; f; u f;g i is a cone of hi Z ; gi there is a unique v such that v; u g = u f;g and v; i Y = i 0 X ; f.Therefore, hi 0 X ; vi is a cone of hi Y ; fi, so there is a unique i such that i; u f = v and i; i X = i 0 X .By the right-cancellable property, i is an inclusion.Consequently, (f; g) ?1 (C) = f ?1 (g ?1 (C)).
5. Factor A , !X; f as e; i, and let hf ?1 (f(A)) , !X; u f i be the pullback of hi; fi as in the diagram below: By the pullback property, there is a unique morphism j : A ! f ?1 (f(A)) such that j; u f = e and j; f ?1 (f(A)) , !X = A , !X.By the rightcancellable property, j is an inclusion, and by 5. in Fact 9, u f 2 E. Therefore, A , ! f ?1 (f(A)) and E(f ?1 (f(A)); f(A)) 6 = ;.6.Let i be the inclusion B , !Y , let hf ?1 (B) , !X; u f i be the pullback of hi; fi, and let e 0 ; i 0 be the factorization of f ?1 (B) , !X; f i 0

T T T T T T T T T T T T T T T T
By the pullback property, there is a unique j : f(f ?1 (B)) !B such that e 0 ; j = u f and j; i = i 0 .By the right-cancellable property, j is an inclusion, that is, f(f ?1 (B)) , !B. 7. If A , ! f ?1 (B) then by 1., f(A) , ! f(f ?1 (B)), and by 6., one gets f(A) , !B. On the other side, if f(A) , !B then by 2., f ?1 (f(A)) , ! f ?1 (B), and by 5., A , ! f ?1 (B).De nition 12.If hI; Ei is a weak inclusion system of A and D is a subcategory of A then let I D and E D denote the restrictions of I and E, respectively, to D. Sub(A) is the full subcategory of A generated by all subobjects of an object A; it is called the subobject category of A and we write I A and E A instead of I Sub(A) and E Sub(A) .As usual, a subcategory D of A is closed under subobjects i A 2 jDj whenever A , !B and B 2 jDj.Proposition 13.Let hI; Ei be a weak inclusion system of A. Then 1.If D is a full subcategory of A closed under subobjects then hI S ; E S i is a weak inclusion system of S. 2. If A is an object in A then hI A ; E A i is a weak inclusion system of Sub(A).
Proof.It is straightforward that I D is a subcategory of inclusions of D. If f is a morphism in D then e f and i f belong to D because D is full and closed under subobjects, so f admits a factorization in hI D ; E D i. Furthermore, the factorization is unique as it is unique in A. 2 follows from 1 observing that Sub(A) is a full subcategory of A closed under subobjects.Theorem 14.If hI; Ei is a weak inclusion system for A then every morphism f : X !Y yields a pair of adjoint functors, f : I X !I Y and f ?1 : I Y !I X , where f : I X !I Y is a left adjoint of f ?1 : I Y !I X .Proof.1., 2., 3. and 4. in Lemma 11 say nothing else than f : I X !I Y and f ?1 : I Y !I X are functors.An easy way to show that f is a left adjoint to f ?1 is to use De nition 2: 7. in Lemma 11 says that for every A 2 I X and B 2 I Y there is a bijection I X (A; f ?1 (B)) = I Y (f(A); B); this bijection is natural in A and B because there is at most one inclusion between any two objects both in I X and I Y .Corollary 15.Any category A admitting a weak inclusion system is an existential Lawvere doctrine.
Proposition 16.In the context of Proposition 14, 1. f; f ?1 ; f = f in Cat, 2. f ?1 ; f; f ?1 = f ?1 in Cat, and 3.If f is an isomorphism and f : Y !X is its inverse, then f ?1 = f as functors I Y !I X .Proof. 1.It su ces to show that f(f ?1 (f(A))) = f(A) for every A , !X.
Composing with f on the right we get (f ?1 ; f); f = f , that is, f ?1 = f .

Inclusion Systems vs. Complete Weak Inclusion Systems
The notion of weak inclusion system is too general because it catches very uninteresting cases (for example, the case where I contains the identities and E contains all the morphisms).For this reason, stronger results usually require a stronger version of weak inclusion systems, such as inclusion systems.This section presents an alternative of the inclusion system, called complete weak inclusion system, which does not require the morphisms in E be epimorphisms, still having much of the power of inclusion systems.For example, Ro s96] presents Birkho -like axiomatizability results for a categorical generalization of equational logic strongly based on inclusion systems; all the results in that paper could be very well done in a framework based on complete weak inclusion systems instead of inclusion systems.
Proposition 17.The following assertions are equivalent in any category A admitting a weak inclusion system hI; Ei: 1.Each monomorphism in E is an isomorphism, 2. Each monomorphism is factored as an isomorphism and an inclusion, 3.Each mono subobject contains exactly one inclusion.
Proof.1: ) 2: Let m be a monic and let e m ; i m be its factorization.Then e m is a monic, too, and by hypothesis it is an isomorphism.
2: ) 3: Each mono subobject contains at most one inclusion because if i i 0 then there exist f and g such that f; i = i 0 and g; i 0 = i, so by the right cancellable property f and g are inclusions, that is, i = i 0 .To show that each mono subobject contains at least one inclusion, let m be a monic and consider e m ; i m its factorization; then e m is an isomorphism.Thus m i m .
3: ) 1: Let m be a monic in E and let i be the unique inclusion such that m i.Then there exist f and g such that f; m = i and g; i = m.Since m is in E, i is an identity (see 4. in Fact 9).Therefore f; m = 1; moreover, (m; f); m = m implies m; f = 1 because m is a monic.Consequently, m is an isomorphism.
The fact above presents conditions under which the inclusions given by a weak inclusion system give a complete and independent system of representatives of the mono subobjects.The following de nition introduces formally the notion of complete weak inclusion system: De nition 18.A (weak) inclusion system hI; Ei verifying the equivalences in the proposition above is called a complete (weak) inclusion system.Example 3.Among the structures in Example 2, only Top with hI 2 ; E 2 i is not complete because there are monics which are continuous surjections but which are not isomorphisms.
The following propositions show relations between inclusion systems and complete weak inclusion systems: Proposition 19.Any regular inclusion system is both an inclusion system and a complete weak inclusion system (i.e., it is a complete inclusion system).
Proof.It follows immediately from the fact that any monomorphism which is a coequalizer, actually is an isomorphism.
De nition 20.We let Mono and Epi denote the subcategories of A containing all monics and epics of A, respectively.A category in which any morphism that is both an epic and a monic is an isomorphism is called a balanced category.Proposition 21.Let hI; Ei be a weak inclusion system of A. Then 1.If A is balanced then Epi E, 2. If A is balanced and E = Epi then hI; Ei is complete, 3.If A is balanced and hI; Ei is an inclusion system then hI; Ei is complete, 4. If hI; Ei is a complete weak inclusion system and hE; Mi is a factorization system of A then E E, 5.If hI; Ei is a complete weak inclusion system and A has a factorization system then hI; Ei is an inclusion system, 6.If A is balanced and has a factorization system then hI; Ei is an inclusion system i hI; Ei is a complete weak inclusion system, 7. If hI; Ei is a complete inclusion system then hE; Monoi is a factorization system of A.
Proof. 1.Let g be any epic and let e g ; i g be its factorization.Then i g is an epic, too.Since i g is a monic (1. in Fact 9) and A is balanced, i g is an iso.Hence i g is an identity (see 8. and 2. in Fact 9), so g = e g 2 E. 2. Let m be a monic in E = Epi.Then m is an iso as A is balanced.
3. It is an immediate consequence of 1. and 2.. 4. Let e 2 E and let e 0 ; m 0 be a factorization of e in hE; Mi.By 5. in Fact 9 m 0 is in E, and since hI; Ei is complete, m 0 is an isomorphism.So e = e 0 ; m 0 is in E.

7.
Every morphism f admits a factorization in hE; Monoi, namely e f ; i f , because i f is a mono (1. in Fact 9).Now, let us consider two factorizations e; m = e 0 ; m 0 and let m = e m ; i m and m 0 = e m 0; i m 0 be factorizations in hI; Ei; then (e; e m ); i m = (e 0 ; e m 0); i m 0, so i m = i m 0 (the factorization is unique in hI; Ei).Since hI; Ei is complete, e m and e m 0 are isomorphisms.
Consequently, there exists an isomorphism = e m ; e ?1 m 0 such that e; = e 0 and ; m 0 = m.Since E Epi, we conclude that hE; Monoi is a factorization system.
Example 4. Looking to the categories in Example 2, it can be easily seen that Set and Alg are balanced.Top is not balanced.Alg ;E can be either balanced or not.For example, groups form a balanced category but rings do not: the inclusion of integers in rationals is both a monic and an epic without being an isomorphism.

Reachability and Projectivity
De nition 22.Let I be a subcategory of inclusions of A. An object A is Ireachable i it has no proper subobjects, i.e., B , !A implies B = A. If A admits an initial object I and E is a class of morphisms in A, then A is Egenerated i the unique morphism A : I !A is in E. Proposition 23.If A admits an initial object and hI; Ei is a weak inclusion system of A then an object is I-reachable if and only if it is E-generated.
Proof.Let A be a an I-reachable object and let e; i be the factorization of the unique morphism A : I ! A. Since A is I-reachable, i is an identity.Hence A is equal to e 2 E. Conversely, if A is E-generated and B is a subobject of A (let i denote the inclusion B , !A) then B ; i = A 2 E, so i is an identity (see 4. in Fact 9).
It is well-known that a category A admits an initial object i U : A ! f g has a left adjoint, where f g is the category having one object and one morphism and U takes every object/morphism to the object/morphism of f g.Generalizing that, from now on in this section, let U : A ! X be a functor having a left adjoint F : X ! A. De nition 24.Let I and E be classes of morphisms in A. An object A in A is (U; I)-reachable i there exists no proper subobject B of A such that U(B , !A) is an isomorphism.A is (U; E)-generated i there exist some free objects F and some morphisms e : F !A in E. Theorem 25.Let hI; Ei be a weak inclusion system of A. Then 1.A is (U; E)-generated i A 2 E, 2. A is (U; I)-reachable i A is (U; E)-generated.Proof. 1.If A : F(U(A)) !A is in E then obviously A is (U; E)-generated as F(U(A)) is a free object.Conversely, if A is (U; E)-generated then there exist an X in jXj and a morphism e : F(X) !A in E. Since e = F(e ); A , by 5. in Fact 9, A is in E.
2. Firstly, assume that A is (U; E)-generated and let B , !A such that U(B , !A) is an isomorphism.Since : U; F ) 1 A is a natural transformation, B ; (B , !A) = F(U(B , !A)); A , and since F(U(B , !A)) is an isomorphism and E contains all isomorphisms, B ; (B , !A) is in E. By 5. in Fact 9, B , !A is in E, so by 2. in Fact 9, B = A. Therefore A is (U; I)-reachable.
Conversely, let A be (U; I)-reachable and factor A as e; i.Then by Fact 4, 1 U(A) = A = e ; U(i).Let B be the factorization object of A , i.e., the target of e.

U(A)
e F(U(A)) Since i is a monic, B = F(U(i); e ); B , so U(i); e = B = 1 U(B) .Therefore e ; U(i) = 1 U(A) and U(i); e = 1 U(B) , that is, U(i) is an isomorphism.Since A is (U; I)-reachable, i is an identity, so A = e 2 E, which means that A is (U; E)-generated.
2. follows immediately from 1.. Theorem 28.If hI; Ei is a weak inclusion system of A such that every object in X is U(E)-projective, then A has enough projectives whenever every object in A is (U; I)-reachable.
Notice that X is Set in most practical situations, and that testing if an object is (U; I)-reachable is easy.Therefore, Theorem 28 can be viewed as an easy criterion to say if a category has enough projectives.Example 5.All categories in Example 2 have enough projectives.

Weak Inclusion Systems for Complex Categories
In this section, (weak) inclusion systems are built for comma categories, functor categories and categories of algebras and coalgebras, from (weak) inclusion systems for the categories involved.

Comma Categories
Given F : C !A and G : D !A, the comma category (F=G) has triples hC; f : F(C) !G(D); Di as objects, where C and D are objects in C and D, respectively, and f is a morphism in A. A morphism from hC; f : F(C) !G(D); Di to hC 0 ; f 0 : F(C 0 ) ! G(D 0 ); D 0 i in (F=G) is a pair (c : C ! C 0 ; d : D !D 0 ) of morphisms in C and D, respectively, such that f; G(d) = F(c); f 0 .Proposition 29.Let hI C ; E C i, hI D ; E D i and hI A ; E A i be weak inclusion system for C, D and A, respectively, and let F : C !A and G : D !A be functors such that F preserves the E C -morphisms and G preserves the inclusions.Then 1. hI; Ei is a weak inclusion system for the comma category (F=G), where I = f(i C ; i D ) 2 (F=G) j i C 2 I C ; i D 2 I D g E = f(e C ; e D ) 2 (F=G) j e C 2 E C ; e D 2 E D g 2. hI; Ei is an inclusion system if hI C ; E C i and hI D ; E D i are inclusion systems.Proof. 1.It is straightforward that I and E can be organized as subcategories of (F=G) having the same objects as (F=G); also, it can be easily seen that I is a partial order.It remains to show that every morphism (c; d) can be factored uniquely as a morphism in E composed with a morphism in I.

Functor Categories
The category C D has functors D !C as objects and natural transformations between them as morphisms.
Proposition 30.Let hI C ; E C i be a weak inclusion system for C and let D be any category.Then 1. hI; Ei is a weak inclusion system for the functor category C D , where I = fi : F ) G j (8F; G 2 jC D j)(8D 2 jDj) i D 2 Ig E = fe : F ) G j (8F; G 2 jC D j)(8D 2 jDj) e D 2 Eg 2. hI; Ei is an inclusion system whenever hI C ; E C i is an inclusion system.Proof. 1.It is straightforward that I and E are subcategories of C D with the same objects as C D , and that I is a partial order.It remains to show that every natural transformation can be uniquely factored as e; i with e in E and i in I. Let : F ) G be a natural transformation and : D !D 0 be a morphism in D.
Denote by H D and H D 0 the objects D (F (D)) and D 0 (F (D 0 )), respectively, and by H the unique morphism given by the diagonal-ll-in lemma, such that e ; H = F( ); e D 0 and H ; i D 0 = i D ; G( ).The reader may check that H : D !C de ned by H(D) = H D and H( ) = H is a functor.Thus, we got the factorization = e ; i , where e : F ) H and i : H ) G are the natural transformations e = fe D j D 2 jDjg and i = fi D j D 2 jDjg.The uniqueness of this factorization comes from the uniqueness of factorizations of each D with D 2 jDj. 2. It follows from the fact that e : F ) G is an epic in C D whenever each e D : F(D) !G(D) is an epic in C, for each D in D.

Algebras and Coalgebras
Given a functor F : A ! A, a pair ( A ; A) is an F-algebra i A : F(A) !A is a morphism in A. Giving two F-algebras ( A ; A) and ( B ; B), f : A ! B is a morphism of F-algebras i F(f); B = A ; f.F-algebras together with morphisms of F-algebras give a category, Alg(F).Dually, (A; A ) is an Fcoalgebra i A : A ! F(A), and f : A ! B is a morphism of F-coalgebras i f; B = A ; F(f).The category of F-coalgebras is written CoAlg(F).Proposition 31.Let hI; Ei be a weak inclusion system of A. Then hI; Ei is a weak inclusion system of 1. CoAlg(F) if F preserves inclusions, 2. Alg(F) if F preserves E-morphisms.Proof.We prove only 1., 2. being dual.Let f : ( A ; A) ! ( B ; B) be a morphism in CoAlg(F) and let e f ; i f be its factorization in A, with e f : A ! f(A) and i f : f(A) , !B. Since F preserves inclusions we get that F(i f ) is in I, so by the diagonal-ll lemma there is a unique morphism, let us denote it f(A) , from f(A) to F(f(A)) such that e f ; f(A) = A ; F(e f ) and i f ; B = f(A) ; F(i f ).Therefore e f : (A; A ) ! (f(A); f(A) ) and i f : (f(A); f(A) ) ! (B; B ) are morphisms of F-algebras and they give a factorization of f in CoAlg(F).Obviously, this factorization is unique.
that (c; d) can be uniquely factored as (e c ; e d ); (i c ; i d ).Let us show that (e c ; e d ); (i c ; i d ) is a correct factorization of (c; d), i.e., (e c ; e d ) and (i c ; i d ) are morphisms in (F=G) and (c; d) = (e c ; e d ); (i c ; i d ).Since F and G are morphisms of weak inclusive categories, by diagonal-ll-in lemma there exists a unique h : F(c(C)) !G(d(D)) such that f; G(e d )) = F(e c ); h and h; G(i d ) = F(i c ); f 0 .This certi es that (e c ; e d ) and (i c ; i d ) are morphisms in (F=G), from hC; f; Di to hc(C); h; d(D)i and from hc(C); h; d(D)i to hC 0 ; f 0 ; D 0 i, respectively.Since (e c ; e d ); (i c ; i d ) = (e c ; i c ; e d ; i d ) = (c; d), we deduce that it is a factorization of (c; d).Now, suppose that (e C ; e D ); (i C ; i D ) is another factorization of (c; d).Then e C ; i C = c and e D ; i D = d, and because of the uniqueness of factorizations in C and D, we deduce that (e C ; e D ) = (e c ; e d ) and (i C ; i D ) = i c ; i d ).Therefore, hI; Ei is a weak inclusion system for (F=G).2. It follows from the fact that (e C ; e D ) is an epic whenever e C and e D are epics.