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Pushouts, Pullbacks and Their Properties

Joonwon Choi

Abstract rewriting has numerous applications, such as software engineering and biology tech- niques. This technique is theoretically based on pushouts and pullbacks, which are involved with given categories. This paper deals with the definition of pushout and pullback, and their properties.

1 Introduction

Descriptions of biological systems are largely involved with graph rewriting techniques, and it has been used shortly after its invention. Formally, a graph rewriting system usually consists of a of graph rewriting rules of the form L R, with L being called pattern graph and R being called → replacement graph. A graph rewriting rule is applied to L, by searching for an occurrence of the pattern graph and by replacing the found occurrence by an instance of the replacement graph, and finally we get R. [2] There are several approaches to deal with graph rewritings, and the most well-known technique is the algebraic approach, which is based on theory. The algebraic approach is divided into some substructures - one is called the double-pushout approaches(DPO) and the other is called the single-pushout approach(SPO). This paper deals with the pushouts and pullbacks, the basic of above two approaches, and will not cover two approaches in detail. We first give some basic preliminaries in order to understand what pushouts and pullbacks are and definitions of them. Then we give some properties to understand the essense of two categorical structures. We note that contents of this paper is largely based on [1].

Overview In 2, we first give some preliminaries for understanding pushouts and pullbacks, including the basics of category theories. And we give formal definitions and universal properties of pushouts and pullbacks in section 3. We also give some properties of pushouts and pullbacks in section 4, for a better understanding of them.

2 Preliminaries

In this section, we give some preliminaries for understanding pushouts and pullbacks. We assume that readers have a general knowledge of following concepts:

1 R-modules, and • Kernels and of modules • Exact sequences and short exact sequences •

2.1 Categories

First of all, we define basic notions of category.

Definition 1 (category). A category consists of three ingredients: a obj( ) of ob- C C jects, a set of Hom(A, B) for every (A, B) of objects, and composition Hom(A, B) Hom(B,C) Hom(A, C), denoted by × → (f, g) gf, 7→ for every ordered triple A, B, C of objects. (We write f : A B instead of f Hom(A, B).) These → ∈ ingredients are subject to the following :

the Hom sets are pairwise disjoint; that is, each f Hom(A, B) has a unique domain A and • ∈ a unique target B;

for each object A, there is an 1A Hom(A, A) such that f1A = f and • ∈ 1Af = f for all f : A B; 7→ composition is associative: given morphisms • f : A B, g : B C, h : C D, → → → we get h(gf) = (hg)f.

We now introduce RMod, which is the category of all right R-modules. This category is the basis of pushouts and pullbacks, which will be defined later.

Definition 2 (left R-modules). The category RMod of all left R-modules (where R is a ) has as its objects all left R-modules, as its morphisms all R-homomorphisms, and as its composition the usual composition of functions. We denote the sets Hom(A, B) in RMod by

HomR(A, B).

We conclude this subsection by defining , in order to define isomorphisms, , and universal properties in later subsections.

Definition 3 (subcategory). A category is a subcategory of a category if they satisfy follow- S C ings:

obj( ) obj( ) • S ⊆ C

2 HomS (A, B) HomC(A, B) for all A, B . • ⊆ ∈ S

If HomS (A, B) = HomC(A, B), is called a full subcategory of a category . S C

2.2 Functors

In this section, we give the definition of functors of categories.

Definition 4 (). If and are categories, then a covariant functor F : consists C D C → D of the following functions:

F : obj( ) obj( ) C → D F : HomC(A, B) HomD(F (A),F (B)), (A, B ) → ∈ C satisfying

F (1A) = 1A for all A • ∈ C

F (gf) = F (g)F (f) for all f HomC(A, B), g HomC(B,C). • ∈ ∈

If F (gf) = F (f)F (g) for all f HomC(A, B), g HomC(B,C), then F is called a contravariant ∈ ∈ functor.

Definition 5 ( of functors). A morphism f : A B in a category is an isomor- → C phism if there exists a morphism g : B A in with gf = 1A and fg = 1B. → C The morphism g is called the inverse of f.

2.3 Universal Objects

In this section, we give the definitions of initial and terminal objects. Existence of universal object is one of significant properties of a given category.

Definition 6 (universal object). U is called the initial object(universally repelling object) in ∈ C if HomC(U, A) = 1 for all A . U is called the terminal object (universally attracting C | | ∈ C ∈ C object) in if HomC(A, U) = 1 for all A . Both initial object and terminal object are called C | | ∈ C the universal object.

Following lemma explains the uniqueness of universal object in a given category. Note that uniqueness does not mean existence - universal object may not exist in some categories.

Lemma 7. If the initial object exists, then it is unique up to isomorphism. Similarly, if the termi- nal object exists, then it is unique up to isomorphism.

Proof. Let U, V be two initial object in . Then since HomC(U, V ) = 1 and HomC(V,U) = 1, C | | | | there exist morphisms f : U V and g : V U, and they are the unique morphisms between U → → and V . Then it indicates that fg = 1U and gf = 1V since HomC(U, U) = 1 and HomC(V,V ) = 1. | | | |

3 Now we can conclude that U and V are isomorphic in . Thus the initial object is unique up to C isomorphism. For a terminal object, we can prove it similarly.

3 Pushouts and Pullbacks 5.1 Categorical Constructions 223 In this section, we define pushouts and pullbacks.

DefinitionDefinition. 8 (pushout).GivenGiven two two morphisms morphismsf f: :AA BBandandgg: :AA CCinin a a category given category , a ,apushout (or fibered sum)isatriple(→D→, α, β) with β→g→ α f that is a C pushout(or fibered sum) is a triple (D, α, β) with βg = αf that satisfies the following universal solutionC to the universal mapping problem: for every triple (=Y, α , β ) with 0 0 0 0 " " property: forβ everyg α tripleg,thereexistsauniquemorphism(Y, α , β ) with β g = α f, thereθ : D existsY amaking unique the morphism diagram θ : D Y " = " → → making the diagramcommute. commute. The pushout The is pushout often denoted is often by denotedB A C by. B A C. ∪ ∪ g g A / C A / C 11 f f β 1  1  α  11 B B / D 1 β" QQQ Cθ 11 QQQ C 1 QQ C 11 α QQQ 1 " QQC!  Q( Y

Pushouts are unique to isomorphismFigure 1: Pushout when they exist, for they are initial objects in a suitable category.

Figure 1 represents the diagram of pushout and its . For pullbacks, we just Proposition 5.13. The pushout of two maps f : A Bandg: A Cin need to know following definition. → → RMod exists. DefinitionProof. 9 (pullback)It is easy. pullback to see thatis the notion of a pushout.

We close this section afterS provingf ( thata), theg(a pushout) B existsC : a in RAMod. = − ∈ ⊕ ∈ Theorem 10. The pushout of two!" maps f : A # B and g : A C$in RMod exists. is a submodule of B C.DefineD→ (B C)/S→,defineα : B D by ⊕ = ⊕ → Proof. Let Sb = (fb,(a0)), gS(,definea)) Bβ : C : a DAby,c then( it0, isc) easyS;itiseasytoseethat to see that S is a submodule of βg'→{ α f , for+− if a ∈ A,then⊕ →α fa∈ }βga'→ ( fa,+ ga) S S.Given B C. Now let D== (B C)/S, α∈ : B D by b − (b, 0)+=S, β : −C D+by c= (0, c)+S. Then with ⊕ another triple⊕ (X, α", β") with→ β"g 7→α" f ,define → 7→ simple calculation, we see that βg = αf. Thus= for all a A, we get αfa βga = (fa, ga)+S = S. 0 0 0 0 ∈ − − 0 0 Now given another tripleθ (:X,D α , βX) withby βθg: (=b,αc)f, letS θ :αD"(b) Xβby"(c). (b, c) + S α (b) + β (c). → + '→ →+ 7→ Then it is easy to see that θ is unique. We let the reader prove commutativity of the diagram and uniqueness of θ. •

4 PropertiesExample 5.14.

In this section,(i) we If giveB and someC are properties submodules of pushouts(pullbacks). of a left R-module U First,thereareinclusions example is that we can get the pushout from thef : B inclusionC B maps.and g : B C C.Thereaderwillenjoyprovingthat ∩ → ∩ → the pushout D exists in RMod and that D is B C. Proposition 11. If B and C are submodules of a left R-module+U, there are inclusions f : B C ∩ → B and g : B (ii)C If CB.and In thisC are case, subsets theof pushout a set UD,thereareinclusionsexists in RMod andf :DB= BC+ C.B ∩ → and g : B C C.ThepushoutinSets is the union B C.∩ → ∩ → ∪

(iii) If f : A B is a in RMod,thencokerf is the pushout → 4 Proof. By Theorem 10, we just need to show that (B C)/S is isomorphic to B + C, where ⊕ S = (a, a): a B C . Let φ :(B C)/S B+C by (b, c)+S b+c. If (b, c)+S = (b0, c0)+S, { − ∈ ∩ } ⊕ → 7→ then there exists a0 B C such that (b, c) = (b0 + a, c0 a). Then we get b + c = b0 + a + c0 a = ∈ ∩ − − b0 + c0, which means that φ is well-defined. For injectivity, if b + c = b0 + c0 in B + C, we define a0 = b0 b. Then it is easy to prove that (b, c) + S = (b0, c0) + S. For surjectivity, it is obvious since − φ((b, c) + S) = b + c. to prove the existence of the pushout of any such diagram. Second one states that how cokernels are involved with pushouts. Since the proof of the following Proposition 1. Let f : X Y and g : X Z be R-module homomorphisms. If we proposition is almost similar to! the previous proposition,! we omit the proof of it. set T = (f224(x), gSetting(x)) Y the StageZ : x X Ch.,5 then (Y Z) /T , together with the maps ã(y)= { Ä 2 à 2 } à Proposition 12. If f : A B is a homomorphism in RMod, then of f, coker f is the (y, 0) + T and å(z)=(0,z)+→ T , is a pushout of f and g. pushout of the followingof the first diagram. diagram below. Proof. With the definition of ã and å, we have ã f = å g since (f(x), 0) (0,g(x)) mod T . f f é é ë A / B A / B We then have to verify the mapping property. Set F7 =(Y Z) /T , and suppose there is 0 77 à a module G and maps ã0 : Y G and0 å0 : Z 7G7 with ã0 f = å g. To define 0 !  ! 77 é é ' : F G, we have the canonical ã0 0 TåT0 : Y/ cokerZ f 7G7 , given by (y, z) ã0(y)+å0(z), ! à TTTT à !I 7 7! that arises from the mapping property of a .TTTT Iθ This77 map sends (f(x), g(x)) to TTTT I 7 TTTTI7 Ä ã0(f(x)) å0(g(x)) = 0 since ã0 f = å0 g. Therefore, ã0 T*$ å0 factors through T to give a Ä é é à X map f : F G,The defined verification by f (( thaty, zcoker)+Tf )=is aã pushout0(y)+ iså0 similar(z). It to is that easy in Exam-to see that ã0 = f ã ! Figure 2: Pushouts and cokernels é and å0 = f å.ple Moreover, 5.12(iii). the definition of f is forced upon us by the requirement that é ã0 = f ã and å0 = f å. Thus, F , together with ã and å, is a pushout of f,g. é (iv) Pushoutsé exist in Groups,andtheyarequiteinteresting;forexample, The last property in this paper is about the relationship between pushouts and short exact We now provethe a technicalpushout of resulttwo injective about homomorphisms pullbacks that is we called will a usefree in Section 3.4 of Weibel sequences. with amalgamation. If K1 and K2 are subcomplexes of a connected [1]. simplicial complex K with K1 K2 K and K1 K2 connected, then Proposition 13. Suppose we have following∪ two short= exact sequences∩ in RMod with an assump- van Kampen’s Theorem says that π1(K ) is the of π1(K1) Propositiontion that all 2. diagramsandSupposeπ1( areK2) we commutative:with haveπ1( aK1 commutativeK2) amalgamated diagram (see Spanier, Algebraic ∩ ,p.151). ! i õ Here is another dual0 pair ofM useful constructions.P A 0 å ã id Definition. Given two morphisms f, g : B C,thentheircoequalizer is → an ordered pair (Z, e)0with efB egj thatX is universalú A with0. this property: if p : C X satisfies pf pg,thenthereexistsaunique= p : Z X with → = $ → Then X is thep$e pushoutp. of B andFigureP with 3: Pushouts respect andto i sequencesand å. = f e B / C / Z Proof.ThenWeX firstis the note pushout that of BX and= ãP(Pwith)+g respectj(B).AA to Toi and prove β. this, let x X. Then ú(x)=õ(p) A AA p$ 2 for some p P . Therefore, x ã(p) ker(úp) =AA im( j), so x ã(p)=j(b) for some b B. Proof. First,2 from the definitionÄ of short2 ,  it is easyÄ to see that X = α(P ) + j(B).2 Thus, x = ã(p)+j(b), as desired. We have j X.å = ã i by the assumption that the And since all diagrams are commutative, we get jβ =éαi. é diagram is commutative.More generally, if To( fi verify: B theC)i mappingI is a family property, of morphisms, suppose thenG theis co- an R-module with Now we prove the universal property.→ Suppose∈ G is an R-module with homomorphisms α0 : P G equalizer is an ordered pair (Z, e) with efi efj for all i, j I that is homomorphisms0 ã : P G and0 j 0 : B G such that ã i = j å. We0 define0 ' : →X G and j : BuniversalG 0such with thatthis property.α i = j0 β. Let φ : X =G by α0 (p) + j(b0 )∈ α (p) + j (b). To see → ! ! → é é7→ ! by '(ã(p)+j(b)) = ã0(p)+j0(b). If we show that ' is well defined, then0 we0 get ' ã = ã0 that φ is well-defined,We can prove we just the existenceneed to show of that if α(p) in +Setsj(b)using = 0 then the notionα (p) +ofj (b) = 0.é Let and ' j = j0 by alternatively setting b = 0 and p = 0 in the definition of '. Furthermore, é orbit . if û is another map from X to G with û ã = ã0 and û å = å0, then é é Definition. Let be an equivalence on a set X.Theorbit space ∼ X/ is theû set(ã( ofp)+ all equivalencej(b)) = '( classes:ã(p))5 + '(j(b)) = ã0(p)+j0(b) ∼ X/= '(ã([xp]:)+x j(bX)),, ∼ ={ ∈ } where [x]istheequivalenceclasscontainingx.Thefunctionν : X X/ , showing thatdefinedû = ' by.ν Also,(x) it[x is], is clear called that the natural' will bemap a. homomorphism→ once∼ we know that it is well defined. To see that =' is well defined, it is enough to show that if ã(p)+j(b) = 0, then

ã0(p)+j0(b) = 0. So, suppose that ã(p)+j(b) = 0. Applying ú gives 0 = ú(ã(p)) = õ(p).

2 α(p) + j(b) = 0, then 0 = τ(α(p)) = σ(p). Thus p = i(m) for some m M. Thus we get ∈ 0 = α(i(m)) + j(b) = j(β(m)) + j(b). Since j is injective, β(m) + b = 0. Thus we get

α0(p) + j0(b) = α0(i(m)) + j0( β(m)) = (α0i j0β)(m) = 0. − − Therefore, φ is well-defined. Now we get φα = α0 and φj = j0 by setting b = 0 and p = 0 respectively. So the only thing that we need to show is its uniqueness. if ψ : X G satisfies the → universal property, we get ψα = α0 and ψβ = β0, then

ψ(α(p) + j(b)) = ψ(α(p)) + ψ(j(b)) = α0(p) + j0(b) = φ(α(p) + j(b)).

This shows the uniqueness of φ. Note that obviously φ is a homomorphism due to its well- definedness.

References

[1] J.J. Rotman. An Introduction to . Universitext - Springer-Verlag. Springer Science+Business Media, LLC, 2009.

[2] Grzegorz Rozenberg, editor. Handbook of graph grammars and computing by graph transfor- mation: volume I. foundations. World Scientific Publishing Co., Inc., River Edge, NJ, USA, 1997.

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