ADHESIVE AND QUASIADHESIVE CATEGORIES

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1 Theoretical Inforatics and Applications Inforatique Théorique et Applications Will be set by the publisher ADHESIVE AND QUASIADHESIVE ATEGORIES STEPHEN LAK 1 AND PAWEŁ SOBOIŃSKI 2 Abstract. We introduce adhesive categories, which are categories with structure ensuring that pushouts along onoorphiss are well-behaved, as well as quasiadhesive categories which restrict attention to regular onoorphiss. Many exaples of graphical structures used in coputer science are shown to be exaples of adhesive and quasiadhesive categories. Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories Matheatics Subject lassification. 18D99, 18A30, 18A35, 68Q42, 68Q65. INTRODUTION Recently there has been renewed interest in reasoning using graphical ethods, particularly within the fields of obility and distributed coputing [15, 21] as well as applications of seantic techniques in olecular biology [4, 6]. Research has also progressed on specific graphical odels of coputation [20]. As the nuber of various odels grows, it is iportant to understand the basic underlying principles of coputation on graphical structures. Indeed, a solid understanding of the foundations of a general class of odels (provided by adhesive categories), together with a collection of general seantic techniques (for exaple [23]) will provide practitioners and theoreticians alike with a toolbox of standard techniques with which to construct the odels, define the seantics and derive proof-ethods for reasoning about these. ategory theory provides unifor proofs and constructions across a wide range of odels. The usual approach is to find a natural class of categories with the right structure to support the range of constructions particular to the application area. A well-known Keywords and phrases: adhesive categories, quasiadhesive categories, extensive categories, category theory, graph rewriting The first author acknowledges the support of the Australian Research ouncil. The second author acknowledges the support of BRIS, Basic Research in oputer Science ( funded by the Danish National Research Foundation. 1 School of Quantitative Methods and Matheatical Sciences, University of Western Sydney,Australia 2 BRIS, University of Aarhus, Denark c EDP Sciences 1999

2 2 TITLE WILL BE SET BY THE PUBLISHER exaple is the class of cartesian-closed categories, which provides odels for siply typed labda calculi [19]. In this paper we shall deonstrate that adhesive categories have structure which allows a developent of a rich general theory of double-pushout (DPO) rewriting [14]. DPO graph rewriting has been widely studied and the field can be considered relatively ature [8, 13, 22]. In DPO rewriting, a rewrite rule is given as a span L K R. Roughly, the intuition is that L fors the left-hand side of the rewrite rule, R fors the right-hand side and K, coon to both L and R, is the sub-structure to be unchanged as the rule is applied. To apply the rule to a structure, one first needs to find a atch L of L within. The rule is then applied by constructing the issing parts (E, D and arrows) of the following diagra L K E in a way which ensures that the two squares are pushout diagras. Once such a diagra is constructed we ay deduce that D, that is, rewrites to D. DPO rewriting is forulated in categorical ters and is therefore portable to structures other than directed graphs. There have been several attepts [9, 11] to isolate classes of categories in which one can perfor DPO rewriting and in which one can show that such rewriting graars satisfy useful properties. In particular, several axios were put forward in [11] in order to prove a local hurch-rosser theore for such general graars. Additional axios were needed to prove a general version of the so-called concurrency theore [16]. An iportant general construction which appears in uch of the literature on graphical structures in coputer science is the pushout construction. Soeties referred to as generalized union [9], it can often be thought of as the construction of a larger structure fro two saller structures by gluing the together along a shared substructure. One can think of adhesive categories as categories in which pushouts along onoorphiss are well-behaved, where the paradig for behaviour is given by the category of sets. An exaple of the good behaviour of these pushouts is that they are stable under pullback (the dual notion to pushout, which intuitively can often be thought of as a generalized intersection ). The idea is analogous to that of extensive categories [3], which have well-behaved coproducts in a siilar sense. Since coproducts can be obtained with pushouts and an initial object, and an initial object is well-behaved if it is strict, one ight expect that adhesive categories with a strict initial object would be extensive, and this indeed turns out to be the case. Various notions of graphical structures used in coputer science for adhesive categories. This includes ordinary directed graphs, typed graphs [1] and hypergraphs [11], aongst others. Indeed, it turns out that any eleentary topos is adhesive, although this shall be proved elsewhere [18]. The structure of adhesive category allows us to derive useful properties. For instance, the union of two subobjects is calculated as the pushout over their intersection, which corresponds well with the intuition of pushout as generalized union. R D

3 TITLE WILL BE SET BY THE PUBLISHER 3 The notion of adhesivity is too strong for several relevant exaples. These exaples otivate the study of quasiadhesive categories. As a topos is adhesive, so a quasitopos is quasiadhesive [18]; a fact which has guided our terinology. Roughly, instead of focusing on the behaviour of pushouts along arbitrary onoorphiss, quasiadhesive categories restrict attention to pushouts along regular onoorphiss. Quasiadhesive categories are shown to encopass several categories of interest to coputer scientists, including the category Spec of algebraic specifications and algebraic specification orphiss. Adhesive categories can be seen as a degenerate quasiadhesive category, naely one where all onoorphiss are regular. While the theory of quasiadhesive categories generalises the theory of adhesive categories, we believe it is nonetheless useful to study the stronger property of adhesivity because of its siplicity; helpful when dealing with exaples which do not require the extra generality. This paper is an expanded version of the extended abstract [17]. It extends this earlier version with full proofs, soe new results about adhesive categories and a ore coplete treatent of the relationship between adhesive categories and HLR-categories. It also introduces the class of quasiadhesive categories, foreshadowed in [17]. We shall consider adhesive graars and quasiadhesive graars, which are DPO graars on, respectively, adhesive and quasiadhesive categories. We prove the local hurch-rosser theore and the concurrency theore without the need for extra axios. We shall also exaine how adhesive and quasiadhesive categories fit within the previously conceived general fraeworks for rewriting [9, 11]. Many of the axios put forward in [11] follow elegantly as leas fro the axios of adhesive categories. Adhesive and quasiadhesive categories, therefore, provide an abstract setting in which it is possible to define a theory of DPO rewriting. They are atheatically elegant and arguably less ad-hoc than previous approaches. Structure of the paper. In 1 we recall the definition of extensive categories. The notion of van Kapen (VK) square is given in 2. VK squares are central in the definition of adhesive categories which are introduced in 3. In 4 we state and prove soe basic leas which hold in any adhesive category. In 5 we show that, in adhesive categories, the subobjects of any object for a distributive lattice, with the union of two subobjects constructed as the pushout over their intersection. Quasiadhesive categories are otivated and defined in 6. We develop double-pushout rewriting theory in adhesive categories in 7 and offer a coparison with High-Level Replaceent ategories in 8. We conclude in 9 with directions for future research. onventions. Throughout the paper we assue that the reader is failiar with basic concepts of category theory. We adopt the convention in diagras of usually not writing labels for identities, arrows to terinal objects, arrows fro initial objects and (binary) coproduct coprojections. When we do need to nae coprojections, we use i 1 and i 2 to refer to, respectively, the first and the second coprojection, letting the context deterine exactly which coproduct is discussed.

4 4 TITLE WILL BE SET BY THE PUBLISHER 1. EXTENSIVE ATEGORIES We shall repeatedly use basic properties of pushouts and pullbacks, and in particular, the following well-known lea, soeties referred to as the pasting lea. Since pushouts and pullbacks are dual, there are two versions of the lea. Lea 1.1. Given a coutative diagra: A k B k E l u s D w F v Pullback version - If the right square is a pullback then the left square is a pullback if and only if the whole rectangle is a pullback. Pushout version - If the left square is a pushout then the right square is a pushout if and only if the whole rectangle is a pushout. In the reainder of this section we recall briefly the notion of extensive category [3]. Definition 1.2. A category is said to be extensive when (i) it has finite coproducts (ii) it has pullbacks along coproduct coprojections (iii) given a diagra where the botto row is a coproduct diagra X Z n Y r A A + B B the two squares are pullbacks if and only if the top row is a coproduct. The third axio states what we ean when we say that the coproduct A + B is wellbehaved : it includes the fact that coproducts are stable under pullback, and it iplies that coproducts are disjoint (the pullback of the coprojections is initial) and that initial objects are strict. 1 It also iplies a cancellativity property of coproducts: given an isoorphis A + B = A + copatible with the coprojections, one can construct an isoorphis B =. For an object Z of an extensive category, the lattice Sub(Z) of coproduct suands of Z is a Boolean algebra. This third axio is actually equivalent to requiring certain canonical functors to be equivalences of categories (Proposition 1.3, (i)). This is soeties taken to be the definition of extensive categories; apart fro elegance, it has the added advantage of not requiring pullbacks along coprojections. h s 1 An initial object is said to be strict precisely when any arrow with the initial object as its codoain is an isoorphis.

5 TITLE WILL BE SET BY THE PUBLISHER 5 Proposition 1.3. Given a category, the following conditions are equivalent for all objects A and B: (i) the functor /A /B /(A + B), which fors the coproducts of orphiss in, is an equivalence of categories; (ii) in a coutative diagra X Z n Y r A A + B B the top row is a coproduct diagra if and only if the squares are pullbacks; (iii) (when has pullbacks along coproduct coprojections) the functor /(A + B) /A /B h given by pullback along the coprojections A A + B and B A + B is an equivalence of categories. The following is a siple lea about pullbacks in extensive categories; it states that pullbacks coute with coproducts. Lea 1.4. Suppose that is an extensive category and that diagras (i) and (ii) below are pullbacks in. s A 1 f 1 B 1 A 2 f 2 B A 1 + A 2 f 1 + f 2 B 1 + B 2 g 1 u 1 1 v 1 D 1 (i) g 2 u 2 2 v 2 D 2 (ii) g 1 +g 2 u 1 +u v1 +v 2 D 1 + D 2 (iii) Then diagra (iii) is a pullback in. Proof. Diagras (i) and (ii) can also be seen as pullbacks in, respectively, /D 1 and /D 2. Diagra (iii) is then the iage of the two pullbacks under the equivalence /D 1 /D 2 /(D 1 + D 2 ) of Proposition 1.3, part (i). It is a pullback since equivalences preserve liits. 2. VAN KAMPEN SQUARES In this section we work in a category with pullbacks. The definition of adhesive category is stated in ters of soething called a van Kapen square, which can be thought of as a well-behaved pushout, in a siilar way to which coproducts can be thought of as well-behaved in an extensive category; essentially this eans that they satisfy soe of the properties of certain pushouts in the category of sets.

6 6 TITLE WILL BE SET BY THE PUBLISHER The nae van Kapen derives fro the relationship between these squares and the van Kapen theore in topology, in its coverings version, as presented for exaple in [2]. This relationship is described in detail in [18]. Definition 2.1 (van Kapen square). A van Kapen (VK) square is a pushout as in (VK1) A g f n D (V K1) B f c A a B g D n f A d B g n D (V K2) b which satisfies the following condition, for any coutative cube (VK2) of which (VK1) fors the botto face and the back faces are pullbacks: the front faces are pullbacks if and only if the top face is a pushout. (Another way of stating the only if condition is that such a pushout is required to be stable under pullback.) Another, equivalent, way of defining a VK square in a category with pullbacks is as follows. A VK square (VK1) is a pushout which satisfies the property that given a coutative diagra (i), the two squares are pullbacks if and only if there exists an object and orphiss A a g D d n B A g D B n b A a A c f B f B b f B n A g D (i) (ii) (iii) so that the squares in (ii) are pullbacks and (iii) is a pushout. By a pushout along a onoorphis we ean a pushout, as in diagra (VK1) above, in which is a onoorphis. A orphis is said to be regular onoorphis when it is an equalizer A B f g of two orphiss. (It follows easily fro the universal property of equalizers that any equalizer is a onoorphis.) Then if in diagra (VK1) is a regular onoorphis,

7 TITLE WILL BE SET BY THE PUBLISHER 7 we have a pushout along a regular onoorphis, while if it is a coprojection then we have a pushout along a coprojection. A crucial class of exaples of VK squares is provided by: Theore 2.2. In an extensive category, pushouts along coproduct coprojections are VK squares. Proof. If : A is a coprojection, say +E, then the diagras (VK1) and (VK2) have the for + E f +E B + E (i) f B f c + E B u Z v c+e f + E h B f +E B + E (ii) b where the unlabelled arrows are coprojections. If the top face is a pushout then we ay take Z = B +E, it then follows that h = b+e. The front right face of the cube is then a pullback, using extensivity. The front left face is a pullback using the conclusion of Lea 1.4; it is constructed by adding together two pullbacks, naely the back right face of the cube and the pullback of the identity on E and e : E E. onversely, suppose that the front faces are pullbacks. Then, as the botto row of the following diagra E e + E c+e u Z h v B b E + E f +E B + E B is a coproduct diagra and all the squares are pullbacks, we ay deduce that the top row is a coproduct diagra, that is Z = B +E. Thus the top face of the cube is a pushout. We have the following iportant properties of VK squares: Lea 2.3. In a VK square as in (VK1), if is a onoorphis then n is a onoorphis and the square is also a pullback.

8 8 TITLE WILL BE SET BY THE PUBLISHER Proof. Suppose that the botto face of the cube f B f B f A n B g n D is VK. Then the top and botto squares are pushouts, while the back squares are pullbacks if is a onoorphis. Thus the front faces will be pullbacks: the front right face being a pullback eans that n is a onoorphis, and the front left face being a pullback eans that the original square is a pullback. There is an alternative way of defining a VK square which involves requiring a certain functor to be an equivalence of categories. This is akin to the alternative way of presenting the ain axio of extensive categories (see Proposition 1.3). f Definition 2.4. Given a span A B, let /A / /B denote the category with: objects: coutative diagras, as illustrated below, where both squares are pullbacks; A f B a A c f B b arrows: given two such diagras, as illustrated below, an arrow is a triple p : A A, q : and r : B B so that a p = a, c q = c and b r = b, while p = q and f q = r f. f B r f B a c b A B A p q a c A f b oposition and identities are obvious.

9 TITLE WILL BE SET BY THE PUBLISHER 9 For a orphis u : U V we shall write u : /V /U for the functor given by pulling back along u. Now, suppose that we have a pushout diagra as below. A Then the functors n and g induce a functor g f n D B Pb : /D /A / /B which takes an arrow d : D D and gives the object of /A / /B obtained by taking the rear faces of the cube (VK2). One constructs the cube by taking pullbacks, first in order to construct the front faces and then the back faces. It is easy to verify that this definition defines the functor On the other hand, if has pushouts (or pushouts along onoorphiss, if we assue to be a onoorphis) we can define a functor Po : /A / /B /D as follows: starting with the back faces of diagra (VK2) we construct the pushout of and f and obtain a unique arrow d : D D given by the universal property of pushouts. Proposition 2.5. Pb is right adjoint to Po. The coposite PoPb is given by pulling back d : D D and then foring a pushout; thus the counit of the adjunction is invertible if and only if, in the cube, if the vertical faces are pullbacks then the top face is a pushout; in other words, if the pushout (VK1) is stable under pullback. On the other hand, the unit of the adjunction is invertible if and only if, whenever the back faces are pullbacks, and the top (and botto) faces are pushouts, then the front faces are also pullbacks. We ay suarize all this as follows: Proposition 2.6. For the pushout diagra (VK1), the following conditions are equivalent: (i) Pb is an equivalence; (ii) Po is an equivalence; (iii) diagra (i) is a VK-square; (iv) the pushout is stable under pullback, and the functor Pb is essentially surjective on objects. 2 Reark 2.7. It is iportant to consider what underlying conditions are required of a category for Pb and Po to exist. 2 A functor F : D is said to be essentially surjective on objects when, for every object D D, there exists an object such that F = D.

10 10 TITLE WILL BE SET BY THE PUBLISHER (i) ondition (i) of the proposition akes sense without assuing all pushouts, and could be taken as the definition of van Kapen square when not all pushouts are assued to exist. Furtherore, if in diagra (VK1) is a onoorphis, then Po will exist provided that pushouts along onoorphiss do so, and the proposition will hold in that generality; (ii) ondition (ii) akes sense without assuing all pullbacks, however, arbitrary pullbacks along and g in the square (VK1) ust exist for Pb to exist; recall that we ake no assuptions about g. We can put a preorder on onoorphiss into an object Z of an arbitrary category by defining a onoorphis a : A Z to be less than or equal to a onoorphis b : B Z precisely when there exists an arrow c : A B such that bc = a. We shall denote such a preorder by Sub(Z). A subobject (of Z) is an equivalence class with respect to the equivalence generated by this preorder. For exaple, subobjects in Set are subsets while subobjects in Graph are subgraphs. Let g : A D be a orphis along which pullbacks of onoorphiss exist, and let g : Sub(D) Sub(A) be the resulting functor. A right adjoint to g, if it exists, is called g. Even if g does not exist, it ay exist partially: given a subobject of A, we write g () for a subobject of D for which D g () if and only if g (D ). Lea 2.8. In a VK square (VK1) with a onoorphis, n = g (). Proof. Let d : D D be an arbitrary onoorphis, and for the cube f c A g n D a B f A d B g n D in which the top and botto squares are pushouts and the reaining faces are pullbacks. Observe that g (d : D D) is a : A A. Now D B if and only if n is invertible, while g (D ) if and only if is invertible. But the top face is both a pushout and a pullback, so is invertible if and only if n is invertible. Thus n : B D has the universal property of g ( : A). b 3. ADHESIVE ATEGORIES We shall now proceed to define the notion of adhesive category, and provide various exaples and counterexaples. Definition 3.1 (Adhesive category). A category is said to be adhesive if (i) has pushouts along onoorphiss; (ii) has pullbacks;

11 TITLE WILL BE SET BY THE PUBLISHER 11 (iii) pushouts along onoorphiss are VK-squares. Reark 3.2. (i) By defining VK-squares as in Proposition 2.6 (i) we could leave out the assuption of pushouts along onoorphiss, but the existence of such pushouts would then be a consequence (see Reark 2.7 (i)); (ii) Siilarly, by defining VK-squares as in Proposition 2.6 (ii) we could drop the assuption of arbitrary pullbacks, but arbitrary pullbacks would then be a consequence (see Reark 2.7 (ii)); Just as the third axio of extensive categories (Definition 1.2) ensures that coproducts are well-behaved, it is the third axio of adhesive categories which ensures that pushouts along onoorphiss are well-behaved. This includes the fact that such pushouts are stable under pullback. Since every onoorphis in Set is a coproduct coprojection, and Set is extensive, we iediately have: Exaple 3.3. Set is adhesive. Observe that the restriction to pushouts along onoorphiss is necessary: there are pushouts in Set which are not VK squares. onsider the underlying set of the 2-eleent abelian group Z 2 (the arguent works for any non-trivial group). In the diagra Z 2 π 1 Z 2 Z 2 π 2 + Z 2 1 Z both the botto and the top faces are easily verified to be pushouts and the rear faces are both pullbacks. However, the front two faces are not pullbacks. Even with the restriction to pushouts along a onoorphis, any well-known extensive categories fail to be adhesive. Exaple 3.4. The categories Pos, Top, Gpd and at are not adhesive. Proof. Write [n] for the ordered set { n 1}. The pushout square [1] 1 0 [2] [2] [3] in Pos is not van Kapen, since it is not stable under pullback along the ap [2] [3] sending 0 to 0 and sending 1 to 2. Thus Pos is not adhesive. The sae pushout square,

12 12 TITLE WILL BE SET BY THE PUBLISHER regarded as a pushout of categories, shows that at is not adhesive. For the case of Gpd, one siply replaces the poset [n] by the groupoid with n objects and a unique isoorphis between each pair of objects. Finally consider the category Top of topological spaces. A finite poset induces a finite topological space on the sae underlying set: the topology is deterined by specifying that y is in the closure of x if and only if x y. Applying this process to the previous exaple yields an exaple showing that Top is not adhesive. Since the definition of adhesive category only uses pullbacks, pushouts, and relationships between these, we have the following constructions involving adhesive categories: Proposition 3.5. (i) If and D are adhesive categories then so is D; (ii) If is adhesive then so are / and / for any object of ; (iii) If is adhesive then so is any functor category [X,]. Because Set is adhesive, part (iii) of the proposition iplies the following. orollary 3.6. Any presheaf topos [X,Set] is adhesive. In particular, the category Graph of directed graphs is adhesive. Indeed, if is adhesive, then so is the category Graph() = [, ] of internal graphs in. Part (ii) iplies that categories of typed graphs [1], coloured (or labelled) graphs [5], ranked graphs [15] and hypergraphs [11], considered in the literature on graph graars, are adhesive. As a consequence, all proof techniques and constructions in adhesive categories can be readily applied to any of the aforeentioned categories of graphs. In fact, ore generally, we have: Proposition 3.7. Any eleentary topos is adhesive. This is soewhat harder to prove than the result for presheaf toposes; the proof can be found in [18]. Part (ii) of Proposition 3.5 also allows us to construct exaples of adhesive categories which are not toposes. Exaple 3.8. The category Set = 1/Set of pointed sets (or equivalently, sets and partial functions) is adhesive, but is not extensive, and therefore, is not a topos. Proof. In the category of pointed sets, the initial object is the one-point set 1. Since every non-initial object has a ap into 1, the initial object is not strict, and so the category is not extensive [3, Proposition 2.8]. 4. BASI PROPERTIES OF ADHESIVE ATEGORIES In this section we prove several siple leas which hold in any adhesive category. Lea 4.1 describes the relationship between adhesive categories and extensive categories. Leas 4.2, 4.3, 4.5, 4.6 and 4.7 shed soe light on pushouts in adhesive categories.

13 TITLE WILL BE SET BY THE PUBLISHER 13 Lea 4.1. An adhesive category is extensive if and only if it has a strict initial object. Proof. In an extensive category the initial object is strict [3, Proposition 2.8]. On the other hand, in an adhesive category with strict initial object, any arrow with doain 0 is a onoorphis. onsider the cube 0 X Y Z n r 0 A t i j A + B in which the botto square is a pushout along a onoorphis, while the back squares are pullbacks since the initial object is strict. By adhesiveness, front squares are pullbacks if and only if the top squares is a pushout; but this says that the front squares are pullbacks if and only if the top row of these squares is a coproduct (Z=X+Y). As explained previously, there are adhesive categories which are not extensive for instance Set (Exaple 3.8) which doesn t have a strict initial object. onversely, there are any extensive categories which are not adhesive; for instance Pos, Top and at (Exaple 3.4). The following two leas are used extensively in the literature on algebraic graph rewriting. Indeed, they are usually assued as axios (see [9] and 8 below) in attepts at generalizing graph rewriting. They hold in any adhesive category by Lea 2.3. Lea 4.2. Monoorphiss are stable under pushout in any adhesive category. Lea 4.3. In any adhesive category, pushouts along onoorphiss are also pullbacks. The following definition introduces the notion of pushout copleent [14], which is vital in algebraic approaches to graph rewriting. Definition 4.4. Let : A and g : A B be arrows in an arbitrary category ( is not assued to be a onoorphis). B s A g D f n B A pushout copleent of the pair (,g) consists of arrows f : B and n : B D for which the resulting square coutes and is a pushout, as illustrated in the diagra

14 14 TITLE WILL BE SET BY THE PUBLISHER above. We shall soeties refer to pushout copleents of onoorphiss: this refers to pushout copleents of pairs (,g) where is a onoorphis. The following lea is a crucial ingredient in any applications of graph rewriting. It has also been assued as an axio [11] in order to prove the concurrency theore (cf. Theore 7.11). It is iportant ainly because it assures that once an occurrence of a left hand side of a rewrite rule is found within a structure, then the application of the rewrite rule results in a structure which is unique up to isoorphis (cf. 7). In other words, rewrite rule application is functional up to isoorphis. Lea 4.5. Pushout copleents of onoorphiss (if they exist) are unique up to isoorphis. In other words, given two pushouts along a onoorphis as illustrated below, there exists an isoorphis ϕ : B B such that n ϕ = n and ϕ f = f. f B f B A g n D A g D n Proof. Using Lea 2.8, both n and n have the universal property of g (), and so we have n = n in Sub(D), eaning that there is an isoorphis ϕ : B B such that n ϕ = n. The fact that ϕ f = f follows because n is a onoorphis. The following lea will be used in Section 8 to show that adhesive categories are high-level replaceent categories: Lea 4.6. onsider a diagra A l k B r u s D w E F v in which the arked orphiss are onoorphiss, the exterior is a pushout and the right square is a pullback. Then the left square is a pushout, and so all squares are both pullbacks and pushouts.

15 TITLE WILL BE SET BY THE PUBLISHER 15 Proof. This aounts to stability of the exterior pushout under pullback along w : D F. We illustrate this in the diagra below, where we leave the identity orphiss unlabelled. A l k u A B k c D B r B u s r D D E w w v F The following lea is of a siilar nature (the difference between the diagras of Lea 4.6 and Lea 4.7 is that different orphiss are assued to be ono) and was used in [24]; the proof is straightforward and siilar to the proof of Lea 4.6. Lea 4.7. onsider a diagra A k B r E l u s D w F v in which the arked orphiss are onoorphiss, the exterior is a pushout, the right square is a pullback, and orphiss k,r,u and w are onoorphiss. Then the left square is a pushout. The following lea shows that all onoorphiss in adhesive categories are regular. Lea 4.8. Monoorphiss are regular. Proof. Given : A B, construct the pushout of with itself: A B n 1 B n2. Since is a onoorphis, the diagra above is also a pullback (Lea 4.3). It is now easy to show that is the equalizer of n 1 and n 2. Indeed, suppose we re given a orphis p : X B such that n 1 p = n 2 p. Using the pullback property, there exists a unique orphis h : X A such that h = p.

16 16 TITLE WILL BE SET BY THE PUBLISHER The following lea is an easy corollary of the forer, but we include a siple direct proof. Lea 4.9. Adhesive categories are balanced. That is, a orphis which is both a onoorphis and an epiorphis is an isoorphis. Proof. Suppose that f : A B is both a onoorphis and an epiorphis. Because it is an epiorphis, the diagra f A B f B 1 B is a pushout. Since f is a onoorphis, using Lea 4.3 we can conclude that it is also a pullback, but this iplies that f is invertible ALGEBRA OF SUBOBJETS Recall that the onoorphiss into an object Z for a preorder Sub(Z), and that a subobject refers to an equivalence class with respect to the resulting equivalence relation. Thus the subobjects of Z organise theselves in a poset. Subobject intersection (resp. union) refers to the eet (resp. join) in this poset. If a poset has both (binary) eets and joins we say that it is a lattice, a lattice is distributive when the eets distribute over the joins. Here we shall deonstrate that, in adhesive categories, the union of two subobjects can be constructed as the pushout over their intersection. This provides further evidence of how pushouts behave in adhesive categories, as well as aking ore precise the intuition that the pushout operation glues together two structures along a coon substructure. As a corollary, it follows that in an adhesive category the lattices of subobjects are distributive. Let be an adhesive category, and Z a fixed object of. Then Sub(Z) has products (=intersections), given by pullback in. It has a top object, given by Z itself. If has a strict initial object 0, then the unique ap 0 Z is a onoorphis, and is the initial object of Sub(Z). Theore 5.1. For an object Z of an adhesive category, the category Sub(Z) of subobjects of Z has binary coproducts: the coproduct of two subobjects is their pushout in over their intersection. Proof. We shall show how to for binary coproducts (=unions) in Sub(Z). Let a : A Z and b : B Z be subobjects of Z, and for the intersection A B Z, with projections p : A B A and q : A B B; and now the pushout A B q B p A u v

17 TITLE WILL BE SET BY THE PUBLISHER 17 in. Let c : Z be the unique ap satisfying cu = a and cv = b. We shall show that c is a onoorphis, and so that is the coproduct A B in Sub(Z) of A and B. Suppose then that f,g : K satisfy c f = cg. For the following pullbacks L 1 f 1 l 1 K f l 2 L 2 A u B v f 2 M 1 g 1 1 K g 2 M 2 A u B v g 2 N l 11 l 21 M 1 12 N 12 1 l12 l 2 L 1 l 1 K L 2 N M N 22 l 22 and note that each of the following pairs are the coprojections of a pushout, hence each pair is jointly epiorphic: (l 1,l 2 ), ( 1, 2 ), ( 11, 12 ), and ( 21, 22 ). We are to show that f = g; to do this, it will suffice to show that f 1 = g 1 and f 2 = g 2 ; we shall prove only the forer, leaving the latter to the reader. To show that f 1 = g 1 it will in turn suffice to show that f 1 11 = g 1 11 and f 1 12 = g First note that a f 1 l 11 = cu f 1 l 11 = c f l 1 l 11 = cgl 1 l 11 = cg 1 11 = cug 1 11 = ag 1 11, so that f 1 l 11 = g 1 11 since a is onic; thus f 1 11 = f l 1 l 11 = u f 1 l 11 = ug 1 11 = g 1 11 as required. On the other hand, b f 2 l 12 = cv f 2 l 12 = c f l 2 l 12 = cgl 2 l 12 = cg 1 12 = cug 1 12 = ag 1 12, so by the universal property of the pullback A B, there is a unique ap h : N 12 A B satisfying ph = g 1 12 and qh = f 2 l 12. Now f 1 12 = f l 2 l 12 = v f 2 l 12 = vqh = uph = ug 1 12 = g 1 12, and so f 1 = g 1 as claied. As proised, we leave the proof that f 2 = g 2 to the reader, and deduce that f = g, so that c is onic. Since pushouts are stable it follows that intersections distribute over unions: orollary 5.2. The lattice Sub(Z) is distributive. Proof. It is easy to verify that the front and back faces of the cube below are pullbacks. Because the botto face is a pushout, we use adhesivity in order to conclude that the top face is a pushout, which in turn iplies that A (B ) = (A B) (A ). A B A B A A (B ) B B B

18 18 TITLE WILL BE SET BY THE PUBLISHER 6. QUASIADHESIVE ATEGORIES The notion of adhesivity is too strong for soe relevant exaples, and therefore, it is useful to study weaker notions. In this section, we introduce the class of quasiadhesive categories. Quasiadhesive categories have well-behaved pushouts along regular 3 onoorphiss, that is, only such pushouts are required to be VK-squares. Definition 6.1. A category is said to be quasiadhesive if (i) has pushouts along regular onoorphiss; (ii) has pullbacks; (iii) pushouts along regular onoorphiss are VK-squares. Since regular onoorphiss are onoorphiss, the class of quasiadhesive categories includes the class of adhesive categories. In fact, we have that Proposition 6.2. Adhesive categories are precisely the quasiadhesive categories in which every onoorphis is regular. The conclusion of the Proposition 6.2 raises the question of whether it is useful to study the stronger property of adhesivity when quasiadhesive categories are strictly ore general. We believe that the answer is positive due to the added siplicity of not having to consider regularity when it is not required. Although quasiadhesive categories are strictly ore general than adhesive categories, the categories entioned in Exaple 3.4 still fail to be quasiadhesive: in each case the pushout in question is actually the pushout of two regular onoorphiss. Just as an adhesive category is extensive iff it has a strict initial object (Lea 4.1), we have: Lea 6.3. A quasiadhesive category is extensive if and only if it has a strict initial object PROPERTIES OF QUASIADHESIVE ATEGORIES It will be useful to exaine the properties of regular onoorphiss in quasiadhesive categories. First, recall soe basic facts about regular onoorphiss which hold in any category. Proposition 6.4. The following hold in any category : (i) if n is a regular onoorphis and is arbitrary then n is a regular onoorphis; (ii) if has pullbacks then the class of regular onoorphiss is stable under pullback. An iportant property of regular onoorphiss in quasiadhesive categories is that the coposite of regular onoorphiss is a regular onoorphis; this is not the case in general (non-quasiadhesive) categories for exaple it is not true in at. Another crucial property which holds in quasiadhesive categories is that regular onoorphiss are stable under pushouts. 3 Recall that a regular onoorphis is one that is an equalizer of two orphiss.

19 TITLE WILL BE SET BY THE PUBLISHER 19 Lea 6.5. The following hold in any quasiadhesive category : (i) regular onoorphiss are stable under pushout; (ii) regular onoorphiss are closed under coposition. Proof. (i). Suppose that diagra (a) below is a pushout and that is a regular onoorphis. f A f B g n D p n A B g q A g n D r D (a) A h (b) Then is the equalizer of p and q, obtained by constructing the pushout illustrated as the back left face of diagra (b). Now there exists a (unique) orphis t : A A such that t p = id A and tq = id A. 4 It follows that p and q are regular (for instance, p is easily checked to be the equaliser of id A and pt). Hence we are able to coplete the cube in (b) so that all faces are pushouts. Indeed all except possibly the front right face are pushouts along regular onoorphiss, hence VK squares, hence pullbacks. Since the botto square is VK, it follows that the front right face is also a pullback, which iplies that n is the equalizer of r and s. (ii). Recall fro the proof of Lea 4.8 that if a pushout of a orphis with itself is also a pullback then the orphis is a regular onoorphis. Now, if and n are regular onoorphiss, for the pushouts in n A B p n D B q t D E u v s F w G and use the fact (i) that regular onoorphiss are stable under pushout to deduce that all four squares are VK, hence pullbacks, and so that the coposite square is a pullback, and so finally that n is the equalizer of vr and ws. One also has the closure properties shown for adhesive categories. In particular: Proposition 6.6. Let and D be quasiadhesive categories, X an arbitrary category, and an object of. Then: (i) D is quasiadhesive; (ii) / is quasiadhesive; 4 Thus p and q are said are split onoorphiss; a orphis is said to be a split onoorphis precisely when it has a left inverse. It follows easily that such a orphis is ono and, oreover, regular. r s

20 20 TITLE WILL BE SET BY THE PUBLISHER If oreover has equalizers then also (iii) / is quasiadhesive; (iv) the functor category [X,] is quasiadhesive. Proof. We rely on the facts that pushouts and pullbacks are constructed pointwise in all of the above cases, the class of regular onoorphiss in D is the product of the regular onoorphiss of and D, the regular onoorphiss of / coincide with those of. We use the hypothesis that has equalizers to ensure that the regular onoorphiss of / and [X,] are fored as in. We can also say soething about the subobjects in quasiadhesive categories. Proposition 6.7. Given any two regular subobjects in the category of subobjects Sub(Z), their coproduct is calculated as the pushout over their intersection. In general, however, there sees to be no reason why the resulting subobject should be regular EXAMPLES Binary relations. A siple, yet iportant, exaple of a quasiadhesive category is the category BRel of binary relations. The objects of this category are injective functions : E V V, where E,V are arbitrary sets and V V denotes the cartesian product of V with itself. Morphiss are coutative squares E f E E V V fv f V V V. oputer scientists ay be failiar with this category in another guise, that is, the category of graphs without edge identities; or in other words, graphs with at ost one edge fro one vertex to another. More forally, the category BRel is easily seen to be equivalent to the category Gr where the objects are the class { V,E E V V } and the orphiss are ordinary graph hooorphiss. It is routine to verify that BRel is coplete and cocoplete, indeed, BRel is a full reflective subcategory of Graph, the category of ordinary graphs and graph orphiss. Liits are calculated pointwise, and a consequence, a ono in BRel is a orphis where both f E and f V are injective. Regular onoorphiss in BRel are easily characterised as precisely the graph onoorphiss which reflect edges, or ore abstractly: Proposition 6.8. A onoorphis is regular in BRel iff the corresponding square of onos is a pullback diagra. Notice that a siple corollary of Proposition 6.8 is that BRel is not a topos, since in toposes every onoorphis is regular. For the sae reasons, it is not adhesive (Lea 4.8). Indeed, we shall deonstrate a siple counterexaple of a pushout along a ono which fails to be VK, let 1 denote the object (the discrete graph on one

21 TITLE WILL BE SET BY THE PUBLISHER 21 vertex) and let 1 denote the object (the coplete graph on one vertex). Then clearly, the diagra below is a pushout along a ono in BRel, but it is not a pullback. Lea 6.9. BRel is quasiadhesive. Proof. See Reark Algebraic specifications. Here we shall consider the category Spec [9] of algebraic specifications and algebraic specification orphiss. We begin by recalling the definition of Spec as given in [9]. First we recall a siple notion of ulti-sorted signature for an algebraic specification. Definition A signature Σ = S,P,do,cod where S is a set of sorts, P is a set of operator sybols, do : P S is a ap giving each operator sybol a possibly epty string over S and cod : P S gives each operator a sort. An operator σ P with do(σ) = s 1...s n and cod(σ) = s shall be written σ : s 1...s n s. A signature orphis f : Σ 0 Σ 1 consists of a aps f S : S 0 S 1 and f P : P 0 P 1 so that given σ : s 1...s n s we have f P (σ) : f S (s 1 )... f S (s n ) f S (s). Let Sig denote the category of signatures and signature orphiss. Fixing a particular signature Σ and a set X = S s S X s of variables for each sort s of Σ, one constructs the set of all ters of each sort s with variables fro X, denoted T Σ (X) s, in a standard way. Definition An equation is a pair of ters t 1,t 2 T Σ (X) s, denoted t 1 = t 2. A positive conditional equation is an expression of the for e 1 e n e where e 1,...,e n,e are equations. A signature orphis f : Σ 0 Σ 1 induces a ap T Σ0 (X) s T Σ1 (X) fs (s), also called f, in the obvious way. This ap clearly extends to equations, so that given positive conditional equation e of sort s we have a positive conditional equation f (e) of sort f S (s). Definition An algebraic specification S = Σ,E consists of a signature Σ and a set E of positive conditional equations over Σ. An algebraic specification orphis f : S 0 S 1 consists of a signature orphis f Σ : Σ 0 Σ 1 so that for every e E 0 we have f (e) E 1. Algebraic specifications and their orphiss for the category Spec. An algebraic specification orphis is injective if and only if the underlying signature orphis is injective which is the sae as saying the underlying functions on the sets of sorts and operators are injective. An algebraic specification orphis f : S 0 S 1 is strict if, given an arbitrary positive conditional equation e, we have f (e) E 1 then e E 0. Proposition The classes of regular onoorphiss and strict injective orphiss in Spec coincide.

22 22 TITLE WILL BE SET BY THE PUBLISHER The reader is directed to [9, Fact 6.3.6] for details of how pushouts are constructed in Spec. Roughly, one constructs a pushout pointwise on the sorts and operators of the signature, as well as the equations. Pullbacks of arbitrary orphiss are siilarly constructed pointwise. Lea Spec is quasiadhesive. Proof. See Reark Reark We have oitted the proofs of Leas 6.9 and While it is possible to prove the VK property directly for pushouts along regular onos using only the descriptions of pushouts, pullbacks, and regular onos in BRel and Spec, the resulting proofs are tedious and unenlightening. Moreover, it turns out that both the exaples fit into a general fraework of quasiadhesive categories [18], aking such proofs unnecessary. 7. DOUBLE-PUSHOUT REWRITING Here we shall recall the basic notions of double-pushout rewriting [14, 22] and show that it can be carried out within an arbitrary quasiadhesive category. In the quasiadhesive context, we work with regular onoorphiss; recall that if we happen to be in an adhesive category, every onoorphis is regular. Definition 7.1 (Production). A production p is a span L l K r R (1) in. We shall say that p is left-linear when l is a regular onoorphis, and linear when both l and r are regular onoorphiss. We shall let P denote an arbitrary set of productions and let p range over P. In order to develop an intuition of why a production is defined as a span, we shall restrict our attention to linear production rules. One ay then consider K as a substructure of both L and R. We think of L and R as respectively the left-hand side and the right-hand side of the rewrite rule p. In order to perfor the rewrite, we need to atch L as a substructure of a redex. The structure K, thought of as a substructure of L, is exactly the part of L which is to reain invariant as we apply the rule to. Thus, an application of a rewrite rule consists of three steps. First we ust atch L as a substructure of the redex ; secondly, we delete all parts of the redex atched by L which are not included in K. Thirdly, we add all of R which is not contained in K, thereby producing a new structure D. The deletion and addition of structure is handled, respectively, by finding a pushout copleent and constructing a pushout. Definition 7.2 (Gluing onditions). Given a production p as in (1), a atch in is a orphis f : L. A atch f satisfies the gluing conditions with respect to p precisely

23 TITLE WILL BE SET BY THE PUBLISHER 23 when there exists an object E and orphiss g : K E and v : E such that L f l v K E g is a pushout diagra. (In other words, there exists a pushout copleent of (l, f ) in the sense of Definition 4.4.) Definition 7.3 (Derivation). Given an object and a set of productions P, we write p, f D for a production p P and a orphis f : L if (a) f satisfies the gluing conditions with respect to l, and (b) there is a diagra in which both squares are pushouts. L l K r R f v g E w D The object E in the above diagra can be thought of as a teporary state in the iddle of the rewrite process. Returning briefly to our inforal description, it is the structure obtained fro by deleting all the parts of L not contained in K. Recall fro Lea 4.5 that if l is a regular onoorphis (that is, if p is left-linear) then E is unique up to isoorphis. Indeed, if p is a left-linear production, p, f D and p, f D then we ust have D = D. This is a consequence of Lea 4.5 and the fact that pushouts are unique up to isoorphis. Definition 7.4 (Adhesive and quasiadhesive graars). A quasiadhesive graar G is a pair,p where is a quasiadhesive category and P is a set of linear productions. If is in fact adhesive, we call G an adhesive graar. Assuing that all the productions are linear allows us to derive a rich rewriting theory on quasiadhesive categories. Henceforth we assue that we are working over an arbitrary quasiadhesive graar G. h 7.1. LOAL HURH-ROSSER THEOREM As we shall explain in section 8, adhesive and quasiadhesive categories naturally fall into the fraework of high-level replaceent categories. In particular, we get the local hurch-rosser theore [9, 16] in the setting of quasiadhesive or adhesive graars. Before presenting this theore we recall briefly the notions of parallel-independent derivation and sequential-independent derivation. The reader ay wish to consult [5] for a ore coplete presentation. Definition 7.5. A parallel-independent derivation is a pair of derivations p1, f 1 D 1 and p2, f 2 D 2

24 24 TITLE WILL BE SET BY THE PUBLISHER as illustrated in diagra (2) which satisfy an additional requireent, naely the existence of orphiss r : L 1 E 2 and s : L 2 E 1 which render the diagra coutative, in the sense that v 2 r = f 1 and v 1 s = f 2. R 1 h 1 r 1 K 1 g 1 D 1 E w 1 1 l 1 L r 1 f 1 s L 2 f 2 l 2 K 2 v 1 E 2 v 2 g 2 r 2 R 2 h 2 (2) w 2 D 2 Definition 7.6. Siilarly, a sequential-independent derivation, illustrated in diagra (3), is a derivation p1, f 1 D 1 p2, f 2 D where there additionally exist arrows r : R 1 E 3 and s : L 2 E 1 such that w 1 s = f 2 and v 3 r = h 1. L 1 f 1 l 1 K 1 g 1 E v 1 1 r 1 R r 1 h 1 f 2 s L 2 l 2 K 2 w 1 D 1 E 3 v 3 g 2 r 2 R 2 w 3 D h 2 (3) If the underlying category has coproducts and given a pair of productions l p 1 = L 1 1 K 1 we ay construct a production r 1 l R 1 and p 2 = L 2 2 K 2 r 2 R 2, l 1 +l p 1 + p 2 = L 1 + L 2 2 K 1 + K 2 r 1 +r 2 R 1 + R 2. Note that if l 1 and l 2 are regular onoorphiss, l 1 + l 2 need not even be a onoorphis, and siilarly for r 1 and r 2. If this is required one could further suppose to be extensive, in which case the coproduct of two regular onoorphiss would indeed be a regular onoorphis. The stateent of the theore below differs fro those previously published in the literature in that we do not need coproducts to establish the equivalence of the first 3 ites. Theore 7.7 (Local hurch-rosser). The following are equivalent: (1) p1, f 1 D 1 and p2, f 2 D 2 are parallel-independent derivations (2) (3) p1, f 1 D 1 and D 1 p2, f 2 D 2 and D 2 p2, f 2 D are sequential-independent derivations p1, f 1 D are sequential-independent derivations. If oreover has coproducts then we ay add the so-called parallelis theore, which states that ite (4), below, is equivalent to the above three ites. (4) p1 +p 2,[ f 1, f 2 ] D is a derivation.

25 TITLE WILL BE SET BY THE PUBLISHER 25 Proof. (1) (2): Let p1, f 1 D 1 and p2, f 2 D 2 be parallel-independent (see Definition 7.5 and associated diagra) derivations. For the pullback (i) below. E 1 e 1 G L 1 l 1 K 1 L 1 l 1 K 1 K 1 r 1 R 1 v 1 E 2 v 2 e 2 r E 2 v 2 g 1 g 2 K 2 E v 1 1 L s 2 l 2 r ( ) k 1 E 2 e 2 v 2 ( ) G e 1 k 2 ( ) K 2 E v 1 1 L s 2 l 2 r 2 k 1 K 2 k 2 G e 3 E 3 e 4 w 3 R 2 E u 4 w4 D t (i) (ii) (iii) (iv) The two regions in (ii) are pushouts [cf. diagra (3)]. obining the two diagras gives (iii), with k 1 and k 2 obtained by the universal property of (i) and satisfying e 2 k 1 = g 1 and e 1 k 2 = g 2. Regions ( ), ( ), and ( ) are pushouts by Lea 4.6, and one now goes on to construct (iv) by taking successive pushouts. The sequential-independent (see Definition 7.6 and associated diagra) derivation p1, f 1 D 1 p2, f 2 D ay now be constructed with the pushout squares below. L 1 r E 2 v 2 l 1 e 2 K 1 k 1 G e 1 E v 1 1 r 1 R 1 e 3 t E 3 v3 w 1 D 1 L 2 s E 1 w 1 l 2 e 1 K 2 k 2 G e 3 r 2 R 2 e 4 D 1 E 3 v 1 w3 u E 4 w4 D (2) (1): Suppose that p1, f 1 D 1 and D 1 p2, f 2 D are sequential-independent derivations. For the pullback (v) below. F e 3 E 3 K 1 r 1 R 1 K 1 ( ) r 1 R 1 K 1 l 1 L 1 e 1 v 3 E 1 w 1 D 1 g 1 K 2 g 2 l 2 E 3 L 2 s E 1 w1 D 1 r v 3 K 2 k 2 ( ) l 2 k 1 F e 1 e 3 ( ) r E 3 v 3 L 2 s E 1 w1 D 1 k 1 K 2 k 2 F e 2 E 2 e 4 w 3 R 2 E u 4 w4 D 2 r 2 t (v) (vi) (vii) (viii) The two regions in (vi) are pushouts [cf. diagra (3)]. obining the two diagras gives (vii), with k 1 and k 2 obtained by the universal property of (i) and satisfying e 3 k 2 = g 2 and e 1 k 1 = g 1. Regions ( ), ( ), and ( ) are pushouts by Lea 4.6, and one now goes on to construct (viii) by taking successive pushouts.