Adhesive Categories. Stephen Lack 1 and Paweł Sobociński 2. Australia 2 BRICS, University of Aarhus. Denmark

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1 Adheive ategorie Stephen Lack 1 and Paweł Sobocińki 2 1 School of Quantitative Method and Mathematical Science, Univerity of Wetern Sydney Autralia 2 BRIS, Univerity of Aarhu Denmark Abtract. We introduce adheive categorie, which are categorie with tructure enuring that puhout along monomorphim are well-behaved. Many type of graphical tructure ued in computer cience are hown to be example of adheive categorie. Double-puhout graph rewriting generalie well to rewriting on arbitrary adheive categorie. Introduction Recently there ha been renewed interet in reaoning uing graphical method, particularly within the field of mobility and ditributed computing [14, 20] a well a application of emantic technique in molecular biology [6, 4]. Reearch ha alo progreed on pecific graphical model of computation [19]. A the number of variou model grow, it i important to undertand the baic underlying principle of computation on graphical tructure. Indeed, a olid undertanding of the foundation of a general cla of model (provided by adheive categorie, introduced in thi paper), together with a collection of general emantic technique (for example [22]) will provide practitioner and theoretician alike with a toolbox of tandard technique with which to contruct the model, define the emantic and derive proof-method for reaoning about thee. ategory theory provide uniform proof and contruction acro a wide range of model. The uual approach i to find a natural cla of categorie with the right tructure to upport the range of contruction particular to the application area. A wellknown example i the cla of carteian-cloed categorie, which provide model for imply typed lambda calculi [18]. In thi paper we hall demontrate that adheive categorie have tructure which allow a development of a rich general theory of double-puhout (d-p) rewriting [13]. D-p graph rewriting ha been widely tudied and the field can be conidered relatively mature [21, 8, 12]. In D-p rewriting, a rewrite rule i given a a pan L K R. Roughly, the intuition i that L form the left-hand ide of the rewrite rule, R form the right-hand ide and K, common to both L and R, i the ub-tructure to be unchanged a the rule i applied. To apply the rule to a tructure, one firt need to find a match L of L within. The Baic Reearch in omputer Science ( funded by the Danih National Reearch Foundation.

2 rule i then applied by contructing the miing part (E, D and arrow) of the following diagram L K R E in a way which enure that the two quare are puhout diagram. Once uch a diagram i contructed we may deduce that D, that i, rewrite to D. D-p rewriting i formulated in categorical term and i therefore portable to tructure other than directed graph. There have been everal attempt [11, 9] to iolate clae of categorie in which one can perform d-p rewriting and in which one can develop the rewriting theory to a atifactory level. In particular, everal axiom were put forward in [11] in order to prove a local hurch-roer theorem for uch general rewrite ytem. Additional axiom were needed to prove a general verion of the ocalled concurrency theorem [15]. An important general contruction which appear in much of the literature on graphical tructure in computer cience i the puhout contruction. Sometime referred to a generalied union [9], it can often be thought of a the contruction of a larger tructure from two maller tructure by gluing them together along a hared ubtructure. One can think of adheive categorie a categorie in which puhout along monomorphim are well-behaved, where the paradigm for behaviour i given by the category of et. An example of the good behaviour of thee puhout i that they are table under pullback (the dual notion to puhout, which intuitively can often be thought of a a generalied interection ). The idea i analogou to that of extenive categorie [3], which have well-behaved coproduct in a imilar ene. Since coproduct can be obtained with puhout and an initial object, and an initial object i well-behaved if it i trict, one might expect that adheive categorie with a trict initial object would be extenive, and thi indeed turn out to be the cae. Variou notion of graphical tructure ued in computer cience form adheive categorie. Thi include ordinary directed graph, typed graph [1] and hypergraph [11], amongt other. The tructure of adheive category allow u to derive ueful propertie. For intance, the union of two ubobject i calculated a the puhout over their interection, which correpond well with the intuition of puhout a generalied union. We hall conider adheive grammar which are d-p rewrite ytem on adheive categorie. We how that the reulting rewriting theory i atifactory by proving the local hurch-roer theorem and the concurrency theorem without the need for extra axiom. We hall alo examine how adheive categorie fit within the previouly conceived general framework for rewriting [11, 9]. Many of the axiom put forward in [11] follow elegantly a lemma from the axiom of adheive categorie. Adheive categorie, therefore, provide a atifactory model in which to define a theory of rewriting on graph-like tructure. They are mathematically elegant and arguably le ad-hoc than previou approache. We firmly believe that they will prove ueful in the development of further theory in the area of emantic of graph-baed computation, and in particular, in the development of a contextual theory of graph rewriting. Structure of the paper. In 1 we recall the definition of extenive categorie. The notion of van Kampen (VK) quare i given in 2. VK quare are central in the definition of D

3 adheive categorie which are introduced in 3. In 4 we tate and prove ome baic lemma which hold in any adheive category. We alo how that the ubobject of an object in an adheive category form a ditributive lattice, with the union of two ubobject contructed a the puhout over their interection. We develop double-puhout rewriting theory in adheive categorie in 5 and offer a comparion with High-Level Replacement ategorie in 6. We conclude in 7 with direction for future reearch. Many of the proof have been omitted. The intereted reader may wih to conult the full verion [16]. 1 Extenive categorie Throughout the paper we aume that the reader i familiar with baic concept of category theory. In thi ection we recall briefly the notion of extenive category [3]. Definition 1 A category i aid to be extenive when (i) it ha finite coproduct (ii) it ha pullback along coproduct injection (iii) given a diagram where the bottom row i a coproduct diagram X r m Z h n Y A i A+B B j the two quare are pullback if and only if the top row i a coproduct. The third axiom tate what we mean when we ay that the coproduct A+B i wellbehaved : it include the fact that coproduct are table under pullback, and it implie that coproduct are dijoint (the pullback of the coproduct injection i initial) and that initial object are trict (any arrow to an initial object mut be an iomorphim). It alo implie a cancellativity property of coproduct: given an iomorphim A+B = A+ compatible with the injection, one can contruct an iomorphim B =. For an object Z of an extenive category, the lattice Sub(Z) of coproduct ummand of Z i a Boolean algebra. 2 Van Kampen quare The definition of adheive category i tated in term of omething called a van Kampen quare, which can be thought of a a well-behaved puhout, in a imilar way to which coproduct can be thought of a well-behaved in an extenive category; eentially thi mean that they behave a they do in the category of et. The name van Kampen derive from the relationhip between thee quare and the van Kampen theorem in topology, in it covering verion, a preented for example in [2]. Thi relationhip i decribed in detail in [17].

4 Definition 2 (van Kampen quare) A van Kampen (VK) quare (i) i a puhout which atifie the following condition: given a commutative cube (ii) of which (i) form the bottom face and the back face are pullback, m f A B g n D (i) m f c A a g B D n m f A d B g n D (ii) b the front face are pullback if and only if the top face i a puhout. Another way of tating the only if condition i that uch a puhout i required to be table under pullback. Another, equivalent, way of defining a VK quare in a category with pullback i a follow. A VK quare (i) i a puhout which atifie the property that given a commutative diagram (iii), the two quare are pullback if and only if there exit an object and morphim A a g D d n B A g D B n b A a A m m c f B f B b m f B n A g D (iii) (iv) (v) o that the quare in (iv) are pullback and (v) i a puhout. By a puhout along a monomorphim we mean a puhout, a in Diagram (i) above, in which m i a monomorphim. Similarly, if m i a coproduct injection, we have a puhout along a coproduct injection. A crucial cla of example of VK quare i provided by: Theorem 3. In an extenive category, puhout along coproduct injection are VK quare. We have the following important propertie of VK quare: Lemma 4 In a VK quare a in (i), if m i a monomorphim then n i a monomorphim and the quare i alo a pullback.

5 Proof. Suppoe that the bottom face of the cube m f B f B m f A n B g n D i VK. Then the top and bottom quare are puhout, while the back quare are pullback if m i a monomorphim. Thu the front face will be pullback: the front right face being a pullback mean that n i a monomorphim, and the front left face being a pullback mean that the original quare i a pullback. 3 Adheive categorie We hall now proceed to define the notion of adheive category, and provide variou example and counterexample. Definition 5 (Adheive category) A category i aid to be adheive if (i) ha puhout along monomorphim; (ii) ha pullback; (iii) puhout along monomorphim are VK-quare. Jut a the third axiom of extenive categorie (Definition 1) enure that coproduct are well-behaved, it i the third axiom of adheive categorie which enure that puhout along monomorphim are well-behaved. Thi include the fact that uch puhout are table under pullback. Since every monomorphim in Set i a coproduct injection, and Set i extenive, we immediately have: Example 6 Set i adheive. Oberve that the retriction to puhout along monomorphim i neceary: there are puhout in Set which are not VK quare. onider the 2 element abelian group Z 2 (the following argument work for any non-trivial group). In the diagram Z 2 π1 Z 2 Z 2 π Z 2 Z

6 both the bottom and the top face are eaily verified to be puhout and the rear face are both pullback. However, the front two face are not pullback. Even with the retriction to puhout along a monomorphim, many well-known categorie fail to be adheive. ounterexample 7 The categorie Po, Top, Gpd and at are not adheive. Since the definition of adheive category only ue pullback, puhout, and relationhip between thee, we have the following contruction involving adheive categorie: Propoition 8 (i) If and D are adheive categorie then o i D; (ii) If i adheive then o are / and / for any object of ; (iii) If i adheive then o i any functor category [X,]. Since Set i adheive, part (iii) of the propoition implie that any preheaf topo [X, Set] i adheive. In particular, the category Graph of directed graph i adheive. Indeed, if i adheive, then o i the category Graph()=[, ] of internal graph in. Part (ii) implie that categorie of typed graph [1], coloured (or labelled) graph [5], ranked graph [14] and hypergraph [11], conidered in the literature on graph grammar, are adheive. A a conequence, all proof technique and contruction in adheive categorie can be readily applied to any of the aforementioned categorie of graph. In fact, more generally, we have: Propoition 9 Any elementary topo i adheive. Thi i omewhat harder to prove than the reult for preheaf topoe; the proof can be found in [17]. Part (ii) of Propoition 8 alo allow u to contruct example of adheive categorie which are not topoe. Example 10 The category Set = 1/Set of pointed et (or equivalently, et and partial function) i adheive, but i not extenive, and therefore, i not a topo. 4 Baic propertie of adheive categorie Here we provide everal imple lemma which hold in any adheive category. Lemma 11 demontrate why adheive categorie can be conidered a a generaliation of extenive categorie. Lemma 12, 13, 15 and 16 hed ome light on puhout in adheive categorie. Lemma 11 An adheive category i extenive if and only if it ha a trict initial object.

7 Proof. In an extenive category the initial object i trict [3, Propoition 2.8]. On the other hand, in an adheive category with trict initial object, any arrow with domain 0 i mono. onider the cube 0 X Y m Z n r 0 A t B i j A+B in which the bottom quare i a puhout along a monomorphim, while the back quare are pullback ince the initial object i trict. By adheivene, front quare are pullback if and only if the top quare i a puhout; but thi ay that the front quare are pullback if and only if the top row of thee quare i a coproduct (Z=X+Y). The concluion of the following two lemma are ued extenively in literature on algebraic graph rewriting. Indeed, they are uually aumed a axiom (ee [9] and 6 below) in attempt at generaliing graph rewriting. They hold in any adheive category by Lemma 4: Lemma 12 Monomorphim are table under puhout. Lemma 13 Puhout along monomorphim are alo pullback. The notion of puhout complement [13] i vital in algebraic approache to graph rewriting. Definition 14 Let m : A and g : A B be arrow in an arbitrary category (m i not aumed to be mono). A puhout complement of the pair (m, g) conit of arrow f : B and n : B D for which the reulting quare commute and i a puhout. We hall ometime refer to puhout complement of mono, thi refer to puhout complement of pair (m,g) where m i mono. The concluion of the following lemma i a crucial ingredient in many application of graph rewriting. It ha alo been aumed a an axiom [11] in order to prove the concurrency theorem (cf. Theorem 27). It i important mainly becaue it aure that once an occurrence of a left hand ide of a rewrite rule i found within a tructure, then the application of the rewrite rule reult in a tructure which i unique up to iomorphim (cf. 5). In other word, rewrite rule application i functional up to iomorphim. Lemma 15 Puhout complement of mono (if they exit) are unique up to iomorphim. Proof. Suppoe that the following diagram m f A g B n D m f A g B D n

8 are puhout and that m i mono. onider the cube m h U k f B m f A n g B n D in which the front right face i a pullback, h : U i the map induced by f and f, and the unnamed arrow are identitie. Then the front face and the back left face are pullback, hence the back right face i alo a pullback; and the bottom face i a puhout, hence the top face i a puhout. But thi implie that k i invertible, ince it i the puhout of 1. By ymmetry, o too i l. The induced iomorphim j = kl 1 : B B atifie n j = n and j f = f. The final lemma of thi ection will be ued in Section 6 to how that adheive categorie are high-level replacement categorie: Lemma 16 onider a diagram A l k B r E u D w F in which the marked morphim are mono, the exterior i a puhout and the right quare i a pullback. Then the left quare i a puhout, and o all quare are both pullback and puhout. Proof. Thi amount to tability of the exterior puhout under pullback along w : D F. v l 4.1 Algebra of ubobject We can put a preorder on monomorphim into an object Z of an arbitrary category by defining a monomorphim a : A Z to be le than or equal to a monomorphim b : B Z preciely when there exit an arrow c : A B uch that bc = a. A ubobject (of Z) i an equivalence cla with repect to the equivalence generated by thi preorder. For example, ubobject in Set are ubet while ubobject in Graph are ubgraph. Here we hall demontrate that, in adheive categorie, union of two ubobject can be contructed by puhout over their interection. Thi provide further evidence of how puhout behave in adheive categorie a well a making more precie the intuition that the puhout operation glue together two tructure along a common ubtructure. A a corollary, it follow that in an adheive category the lattice of ubobject are ditributive.

9 Let be an adheive category, and Z a fixed object of. We write Sub(Z) for the category of ubobject of Z in ; it ha product (=interection), given by pullback in. It ha a top object, given by Z itelf. If ha a trict initial object 0, then the unique map 0 Z i a monomorphim, and i the bottom object of Sub(Z). Theorem 17. For an object Z of an adheive category, the category Sub(Z) of ubobject of Z ha binary coproduct: the coproduct of two ubobject i the puhout in of their interection. Since puhout are table it follow that interection ditribute over union: orollary 18 The lattice Sub(Z) i ditributive. 5 Double-Puhout Rewriting Here we hall recall the baic notion of double-puhout rewriting [13, 21] and how that it can be defined within an arbitrary adheive category. Henceforth we hall aume that i an adheive category. Definition 19 (Production) A production p i a pan L l K r R (1) in. We hall ay that p i left-linear when l i mono, and linear when both l and r are mono. We hall let P denote an arbitrary et of production and let p range over P. In order to develop an intuition of why a production i defined a a pan, we hall retrict our attention to linear production rule. One may then conider K a a ubtructure of both L and R. We think of L and R a repectively the left-hand ide and the right-hand ide of the rewrite rule p. In order to perform the rewrite, we need to match L a a ubtructure of a redex. The tructure K, thought of a a ubtructure of L, i exactly the part of L which i to remain invariant a we apply the rule to. Thu, an application of a rewrite rule conit of three tep. Firt we mut match L a a ubtructure of the redex ; econdly, we delete all of part of the redex matched by L which are not included in K. Thirdly, we add all of R which i not contained in K, thereby producing a new tructure D. The deletion and addition of tructure i handled, repectively, by finding a puhout complement and contructing a puhout. Definition 20 (Gluing ondition) Given a production p a in (1), a match in i a morphim f : L. A match f atifie the gluing condition with repect to p preciely when there exit an object E and morphim g : K E and v : E uch that L f l v i a puhout diagram. (In other word, there exit a puhout complement of (l, f) in the ene of Definition 14.) K E g

10 Definition 21 (Derivation) Given an object and a et of production P, we write p, f D for a production p P and a morphim f : L if (a) f atifie the gluing condition with repect to l, and (b) there i a diagram f L l K g E v r R h w D in which both quare are puhout. The object E in the above diagram can be thought of a a temporary tate in the middle of the rewrite proce. Returning briefly to our informal decription, it i the tructure obtained from by deleting all the part of L not contained in K. Recall from Lemma 15 that if l i mono (that i, if p i left-linear) then E i unique up to iomorphim. Indeed, if p i a left-linear production, p, f D and p, f D then we mut have D = D. Thi i a conequence of Lemma 15 and the fact that puhout are unique up to iomorphim. Definition 22 (Adheive Grammar) An adheive grammar G i a pair, P where i an adheive category and P i a et of linear production. Auming that all the production are linear allow u to derive a rich rewriting theory on adheive categorie. Henceforward we aume that we are working over an adheive grammar G. 5.1 Local hurch-roer theorem A hall be explained in ection 6, adheive categorie with coproduct are high-level replacement categorie. In particular, we get the local hurch-roer theorem [15, 9]. Before preenting thi theorem we need to recall briefly the notion of parallelindependent derivation and equential-independent derivation. The reader may wih to conult [5] for a more complete preentation. A parallel-independent derivation i a pair of derivation p1, f 1 D 1 and p2, f 2 D 2 a illutrated in diagram (2) which atify an additional requirement, namely the exitence of morphim r : L 1 E 2 and : L 2 E 1 which render the diagram commutative, in the ene that v 2 r = f 1 and v 1 = 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 L 2 f 2 v 2 l 2 K 2 v 1 E 2 g 2 r 2 R 2 h 2 (2) w 2 D 2 Similarly, a equential-independent derivation, illutrated in diagram (3), i a derivation p1, f 1 D 1 p2, f 2 D

11 where there additionally exit arrow r : R 1 E 3 and : L 2 E 1 uch that w 1 = 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 L 2 l 2 K 2 w 1 D 1 E 3 v 3 g 2 r 2 R 2 The tatement of the theorem below differ from thoe previouly publihed in the literature in that we do not need coproduct to etablih the equivalence of the firt 3 item. Theorem 23 (Local hurch-roer). The following are equivalent 1. p1, f 1 D 1 and p2, f 2 D 2 are parallel-independent derivation p1, f 1 D 1 and D 1 p2, f 2 D 2 and D 2 p2, f 2 D are equential-independent derivation p1, f 1 D are equential-independent derivation. If moreover i extenive then we may add the o-called parallelim theorem 4. p1 +p 2,[ f 1, f 2 ] D i a derivation. In fact, the proof that (1) (2) remain valid more generally in the context of leftlinear production, but the proof of the convere require linearity. w 3 D h 2 (3) 5.2 oncurrency Theorem The original concurrency theorem were proved for graph grammar [7] and later generalied to high-level replacement categorie (cf. 6) in [11] which atify additional axiom et, there called HLR2 and HLR2*. Roughly, the concurrency theorem tate that given two derivation in a equence, together with information about how they are related, one may contruct a ingle derivation which internalie the two original derivation and perform them concurrently. Moreover, one may revere thi proce and decontruct a concurrent derivation into two related equential derivation. Here we tate and prove the concurrency theorem for adheive grammar without the need for extra axiom. We hall firt need to recall the notion of dependency relation, dependent derivation and concurrent production. Definition 24 (Dependency Relation) Suppoe that p 1 and p 2 are linear production. A dependency relation for p 1, p 2 i an object X together with arrow : X R 1 and t : X L 2 for which r 1,, t, and l 2 can be incorporated into a diagram K 1 g 1 E 1 X r 1 R 1 h 1 in which all three region are puhout. w 1 t L 2 l 2 f 2 K 2 D E 2 v 2 g 2 (4)

12 Definition 25 (Dependent Derivation) onider a derivation p1, f 1 D 1 p2, f 2 D a illutrated in (i) below L 1 f 1 l 1 K 1 g 1 E 1 v 1 r 1 R 1 h 1 L 2 f 2 l 2 K 2 g 2 w 1 D 1 E 3 v 3 (i) r 2 R 2 w 3 D h 2 K 1 g 1 E 1 e 1 X t r 1 l 2 R 1 h L 2 1 f 2 K 2 w 1 D d v 2 g 2 E 2 e 2 E 1 w 1 D 1 E 2 v 2 and a dependency relation X for p 1, p 2. The derivation i aid to be X-dependent if h 1 = f 2 t and there exit morphim e 1 : E 1 E 1 and e 2 : E 2 E 2 atifying e 1 g 1 = g 1 and e 2 g 2 = g 2, and if moreover the unique map d : D D 1 atifying dh 1 = h 1 and d f 2 = f 2 alo atifie dw 1 = w 1e 1 and dv 2 = v 2e 2 (ee (ii)). Definition 26 (oncurrent Production) Given a dependency relation X for p 1, p 2, the X-concurrent production p 1 ; X p 2 i the pan v 1 u P w 2 v D obtained by taking the bottom row of the following extenion of Diagram (4) l L 1 1 K 1 f 1 g 1 E 1 v 1 X t r 1 l 2 r 2 R 1 h L 2 1 f 2 K 2 R 2 g 2 D E 2 D in which and are puhout and i a pullback. Theorem 27 (oncurrency Theorem). u w 1 P 1. Given an X-dependent derivation p1, f 1 D 1 p2, f 2 D there exit an X- concurrent derivation p1 ; X p 2 D 2. Given an X-concurrent derivation p1 ; X p 2 D, there exit an X-dependent derivation p1, f 1 D 1 p2, f 2 D. 6 Relation with High-Level Replacement ategorie High-level replacement categorie [9 11] or HLR-categorie encompa everal attempt to iolate general categorical axiom which lead to categorie in which one can define double-puhout graph rewriting and prove ueful theorem uch a the local hurch-roer theorem and the concurrency theorem. HLR-categorie uually have axiom which are parametried over an arbitrary cla of morphim M. Here we give a implified verion of the definition which appear in [9]. The implification i that we take M to be the cla of monomorphim: we jutify thi by noting that thi i the cae in the majority of example. v 2 v w 2 h 2 (ii)

13 Definition 28 (HLR-categorie) A category S i an HLR-category if it atifie the following axiom: 1. pair A B with at leat one of the arrow mono have a puhout; 2. pair B D with both morphim mono have pullback; 3. mono are preerved by puhout; 4. finite coproduct exit; 5. puhout of mono are pullback; 6. puhout-pullback decompoition hold: that i, given a diagram A l k B r E u D w F if the marked morphim are mono, the whole rectangle i a puhout and the right quare i a pullback, then the left quare i a puhout. Lemma 29 Any adheive category with an initial object i an HLR-category. Proof. Thi follow immediately from Lemma 12, 13, and 16. The axiom lited above are enough to prove the local hurch-roer theorem (cf. Theorem 23), but not the concurrency theorem (cf. Theorem 27). To prove the latter, extra axiom had to be introduced in [11], uch a the concluion of the following lemma. Interetingly, it i almot the dual of the main axiom of adheive categorie. Lemma 30 (ube-puhout-pullback-lemma [11]) Given a cube in which all arrow in the top and bottom face are mono, if the top face i a pullback and the front face are puhout, then the bottom face i a pullback if and only if the back face are puhout. Proof. Since the front face are puhout along monomorphim, they are alo pullback. If the bottom face i a pullback, then the back face are puhout by tability of the puhout on the front face. Suppoe converely that the back face are puhout; ince they are puhout along monomorphim, they are alo pullback. One now imply rotate the cube : ince the front right and back left face are puhout, and the top and back right face are pullback, it follow by adheivene that the bottom quare i a pullback. An HLR-category which ha the concluion of Lemma 30 a an additional axiom i ometime referred to a an HLR2-category [11]. It i immediate, therefore, that any adheive category with an initial object i an HLR2-category. The tronget axiom ytem for general rewriting i enjoyed by the o-called HLR2*- categorie [11]. Thee are HLR2-categorie which, additionally, have the concluion of Lemma 15 a an axiom, that i, puhout complement of mono are, if they exit, unique up to iomorphim. Finally, they atify an axiom known a the twited-triple-puhoutcondition. We believe that thi axiom doe not hold in an arbitrary adheive category, although it doe hold, for intance, in any topo. Indeed, it i poible to extend the definition of adheive categorie in a natural way o that the twited-triple-puhoutcondition hold [17]. v

14 7 oncluion and future work We introduced the notion of van Kampen (VK) quare and adheive category. VK quare are well-behaved puhout, and a category i adheive when puhout along mono are VK. Adheive categorie are cloely related to extenive categorie. Double-puhout (d-p) rewriting can be defined in an arbitrary adheive category. We introduced adheive grammar, which are adheive categorie with a et of linear production. Adheive grammar have ufficient tructure for the development of a rich rewriting theory. In particular, we proved the local hurch-roer and the o-called concurrency theorem within the etting of adheive grammar. We have alo hown that adheive categorie atify many of the axiom [9, 11] which were propoed in order to prove thee theorem. Thu, we have arrived at a cla of categorie which upport uch a theory of d-p rewriting, however, we believe that adheive categorie are mathematically elegant and le ad-hoc than previou propoal. In order to back thi claim and to further develop the theory of adheive categorie, we have demontrated a number of ueful propertie. For intance, ubobject union i formed a a puhout over the interection, and ubobject interection ditribute over ubobject union. We have provided ome cloure propertie which allow the contruction of new adheive categorie from old. Any elementary topo i adheive, but there are example of adheive categorie which are not topoe. Adheive categorie include many well-known notion of graph tructure ued in computer cience and are intance of HLR2-categorie [11]. We believe that adheive categorie will be ueful in the development of pecific graphical model of computation and the development of emantic technique for reaoning about uch model. The rewriting theory need to be developed further, with, for example, the contruction of canonical dependency relation from derivation [11]. A related tak i to clarify the relationhip of adheive categorie and the HLR2*- categorie [11]. Another poible direction for future work i to examine whether adheive categorie have enough tructure o that groupoidal relative puhout [22] can be contructed in copan bicategorie over adheive categorie. Such copan bicategorie provide a way of undertanding graph in a modular fahion and will provide a general cla of model which hould include bigraph [19] a example. A further quetion to be reolved i whether demanding the good behaviour of puhout only along ome cla of monomorphim will reult in further intereting categorie. Acknowledgement. The econd author would like to thank Vladimiro Saone for many dicuion in the early tage of thi project. Thank alo go to Marco arbone, Mogen Nielen, Paolo Oliva, Mikkel Nygaard Ravn and the anonymou referee for reading early draft and providing many valuable comment and uggetion. Reference 1. P. Baldan, A. orradini, H. Ehrig, M. Löwe, U. Montanari, and F. Roi. oncurrent emantic of algebraic graph tranformation. In H. Ehrig, H.-J. Kreowki, U. Montanari, and G. Rozenberg, editor, Handbook of Graph Grammar and omputing by Graph Tranformation, volume 3, chapter 3, page World Scientific, R. Brown and G. Janelidze. Van Kampen theorem for categorie of covering morphim in lextenive categorie. J. Pure Appl. Algebra, 119: , 1997.

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