A Taste of. Categorical Petri Nets

Transcription

1 A Taste of Categorical etri Nets Claudia Ermel, Alo Martini Bericht-Nr. 96-9

2 A Taste of Categorical etri Nets Claudia Ermel Alo Martini Abstract This report aims at providing introductory concepts for a categorical approach for the study of etri Nets. After motivating why a categorical approach for studying petri nets might be desirable, we show that \classical" placetransition Nets, usually seen as bipartite directed graphs, can be naturally given a monoid structure. Upon this idea we construct a category with placetransition Nets as objects and a suitable notion of placetransition Net morphisms as morphisms. We also verify the existence of ubiquitous categorical constructions in order to verify issues concerning cocompleteness. In the end, the usual notion of semantics of placetransition Nets by a marking graph construction is shown to be nothing else than an adjoint situation between two suitable functors. Further points of interest are pointed out in the bibliographic notes. Contents 1 Introduction 2 2 reliminaries 3 3 Categorical T-Nets Motivation etri nets are monoids The category T-Net Colimits in T-Net Free constructions and the semantics of categorical T-Nets arallel ring of transitions Reexive transitions and the marking graph Bibliographic notes 28 1

3 A Categorical Background 30 A.1 Categories and diagrams A.2 Special kinds of morphisms A.3 Universal constructions A.4 Functors and natural transformations A.5 General colimit constructions A.6 Free constructions and adjoints Introduction etri nets are used to describe processes as concurrent and interacting machines which engage in internal actions and communications with their environment or user. They are the rst model of concurrent systems which has been developed and, in their various evolutions, the most heavily used in many applications. etri nets are also a well-known specication technique for concurrent and distributed systems, the reasons being its simplicity, universality, graphical representation and support of computer tools. However, some main objections against etri nets both as a model for concurrency and as a specication technique from a software engineering point of view can be summarized by saying that etri nets lack abstraction, compositionality, data-type handling, renement and structuring. Data-type handling is usually analyzed in connection with High-Level Nets. The other ones can be properly formulated and studied by bringing the language of category theory into etri net theory. As a consequence, we are here concerned mainly with the link between category theory and etri Net theory, that is to say, we want to provide here a taste of a collection of denitions and results concerning categorical etri nets, and thus provide the reader with an adequate basis to follow the current literature dealing with questions such as structuring, abstraction, compositionality and renement. The slogan here is, after [8], "etri nets are monoids", i.e., a placetransition net is simply an ordinary, directed graph equipped with two algebraic operations. Especially, a net provides the generators of the algebraic structure. This observation allows one straightforwardly to bring categorical methods into etri net theory. In the rst section we review basic concepts of place transition nets needed for this presentation with the sole purpose of xing notation and terminology used throughout. In the second section we introduce categorical etri nets, the equivalence between the classical and the categorical formulation and a discussion about the natural representation of concurrent transition in the categorical approach. In the last one, the usual marking graph semantical construction for placetransition nets is shown to be nothing but a free construction within the categorical terminology. 2

4 2 reliminaries The only purpose of this section is to present basic concepts related to placetransition nets which we will need for further development of the material contained in this chapter. In case the reader is already familiar with them, a quick look might suce so as to x the notation used throughout. lacetransition nets are composed of two basic components: a set of places and a set of transitions. Besides, it is necessary to dene a relationship between places and transitions. Later on, we will need to considerer a distribution of tokens (a primitive concept) in the set of places, called marking, which has the purpose of controlling the execution of a placetransition net. These observations lead directly to the following Denition 2.1 A 4-tuple N = def where (; T; F; W ) is called a placetransition-net (T-Net) 1. and T are disjunct sets, whose elements are called places and transitions, respectively. 2. F ( T ) [ (T ) is a binary relation, the ow relation of N. 3. W : F! N n f0g is a mapping which associates to each edge of the net its weight. Remark 2.2 The usual denition of a T-Net requires imposing a (possibly) limited capacity to each place. We consider here that any place has an unlimited capacity, since every net with limited place capacity can uniquely be extended through the addition of the so-called complementary places to a net of an unlimited place capacity (see, e.g, [14]). The above denition of the ow relation may suggest that a T-Net may be viewed as a bipartite directed graph, where an edge is directed from a place p i to a transition t j if hp i ; t j i 2 F and conversely, there is an edge directed from t j to p i if ht j ; p i i 2 F. Before turning to the dynamic properties of a T-Net, we need the following auxiliary Denition 2.3 Let N be T-Net. For x 2 [ T, x = def fyjhy; xi 2 Fg is called the pre-condition of x, and x = def fyjhx; yi 2 Fg is called the post-condition of x. Remark 2.4 Note that a T-Net is a straightforward generalization of a nite state machine if we restrict the sets x and x for all x 2 T to be singletons and set to each edge of the net, the constant weight 1. Example 2.5 In the T-Net of Figure 1 (see 2.8) we have t 2 = fp 2 ; p 6 g; p 1 = ft 1 ; t 4 g, and so on. In addition to the static properties represented by a T-Net graph, a T-Net has dynamic properties that result from its execution, which is controlled by the position and movement of tokens. The use of tokens rather resembles a board game. The tokens are 3

5 moved by the ring of the transitions. A transition t must be enabled in order to re. It is enabled when each place p belonging to the set of its pre-condition has at least as many tokens as dened by the weight of the edge connecting p and t. The transition res by removing W (p; t) tokens from each place p of the pre-condition and producing new tokens by depositing in each place p belonging to the post-condition set of t the number of tokens determined by the edge connecting each p with t. More formally, we have the following Denition 2.6 (Marking and ring) Let N be a placetransition net. Then 1. A mapping M :! N is called a marking on N. 2. A transition t 2 T is M-enabled if and only if 8p 2 t : M(p) W (p; t). 3. A M-enabled transition t 2 T may yield a follower marking M 0 of M which is such that for each p 2 M 0 (p) = def 8 > <>: M(p)? W (p; t) M(p) + W (t; p) M(p)? W (p; t) + W (p; t) M(p) if p 2 t? t if p 2 t? t if p 2 t \ t otherwise We say t res from M to M 0, and we write M[tiM 0. Example 2.7 Considering M 0 :! N as in Figure 1. Then M 0 [t 1 im 1 is given by: M 1 (p 1 ) = M 0 (p 1 )? W (p 1 ; t 1 ) = 3? 1 = 2; M 1 (p 2 ) = M 0 (p 2 ) + W (t 1 ; p 2 ) = = 1; M 1 (p 3 ) = M 0 (p 3 ) = 0; M 1 (p 4 ) = M 0 (p 4 ) = 0; M 1 (p 5 ) = M 0 (p 5 ) = 0; M 1 (p 6 ) = M 0 (p 6 ) = 0: Example 2.8 (Readers-Writers roblem). Figure 1 shows a T-Net illustrating the readers-writers problem. There are several variants of the readers-writers problem, but the basic structure is the same. rocesses are of two types: readers processes and writer processes. All processes share a common le, variable, or data object. Reader processes never modify the object, while writer processes do modify it. Thus writer processes must mutually exclude all other reader and writer processes, but multiple reader processes can access the shared data simultaneously. The problem is to dene a control structure which does not deadlock or allow violations of the mutual exclusion criteria. Figure 1 depicts a solution where the number of processes is bounded by 3 (i.e., tokens are seen as processes), where in our graphical representation of T-Nets places are represented by circles and transitions by boxes. The edges f 2 F are labeled by W (f) if W (f) > 1, and a marking M is represented by drawing M(p) tokens on each place p 2. The formal description of this net, according to 2.1, can be given as follows: 4

6 p1 t1 t4 p1: rocess with opportunity of acces p2 p4 p3: rocess is writing p4: rocess can read p2: rocess can write t2 t5 p5: rocess is reading 3 p6: Control process t1-t3: Actions of the writing process p3 p6 p5 t4-t6: Actions of the reading process 3 t3 t6 Figure 1: T-Net for a reader-writer system = fp 1 ; p 2 ; p 3 ; p 4 ; p 6 g; T = ft 1 ; t 2 ; t 3 ; t 4 ; t 5 ; t 6 ; g F = fhp 1 ; t 1 i; ht 1 ; p 2 i; hp 2 ; t 2 i; ht 2 ; p 3 i; hp 3 ; t 3 i; ht 3 ; p 1 i; hp 1 ; t 4 i; ht 4 ; p 4 i; hp 4 ; t 5 i; ht 5 ; p 5 i; hp 5 ; t 6 i; ht 6 ; p 1 i; hp 6 ; t 2 i; hp 6 ; t 5 i; ht 3 ; p 6 i; ht 6 ; p 6 ig; W : W (hp 1 ; t 1 i) = 1; W (ht 1 ; p 2 i) = 1; : : : ; W (hp 6 ; t 2 i) = 3; W (hp 6 ; t 5 i) = 1; W (ht 3 ; p 6 i) = 3; W (ht 6 ; p 6 i) = 1; M 0 : M 0 (p 1 ) = M 0 (p 6 ) = 3; M 0 (p 2 ) = M 0 (p 3 ) = M 0 (p 4 ) = M 0 (p 5 ) = 0 Two transitions are independently enabled, when the ring of one transition does not prevent the ring of the other, or in other words, each place belonging to the intersection of the pre-condition set of the two transitions contains "enough' tokens for the occurrence of both rings. Otherwise the enabled transitions are in conict. More precisely, we have: Denition 2.9 Let N be a placetransition net, t 1 ; t 2 2 T two transitions from N, and let M be a marking from N. Then, t 1 ; t 2 are said to be independently enabled when the following holds: 8p 2 t 1 \ t 2 : M(p) W (p; t 1 ) + W (p; t 2 ): Otherwise, t 1 and t 2 are said to be in conict. 5

7 Example 2.10 In the T-Net of Figure 1 the transitions t 1 and t 4 are independently enabled. By the parallel ring of the two transitions, we get a resulting marking where the transitions t 2 and t 5 are enabled. However, at this time, the two enabled transitions are in conict, since the ring of one necessarily prevents the ring of the other (as desired). In such independent situations, in which the \concrete" sequence of rings plays no role in the resulting system's state, it is reasonable to suggest that the corresponding transitions might execute in parallel. It is exactly in this sense that etri nets model parallel or concurrent systems. However, these systems may have properties which are not present in sequential ones. For instance, due to parallelism, the system can get into a \deadlock" situation, i.e., a situation in which no other transition can re because two or more processes simultaneously prevent the transition of one another. In etri nets, the concept of deadlock is normally studied in connection with the concept of liveness. Example 2.11 As an example of a deadlock situation, consider Figure 2 where two processes share two resources. In a rst step, process 1 need one resource (the token on place b1 ) and process 2 the other one (the token on place b2 ). In a second step, process 1 needs the resource that lies on b2 and deposits the rst resource again on b1. On the other hand, process 2 needs the resource from b1 and deposits the second on b2. The third step initializes the system to its initial state (initial marking). The system runs correctly whenever the two processes execute sequentially. A complication arises when the both independently enabled transitions T 1 and T 4 re in parallel or one after the other. In such a situation, no transition can re again and a deadlock arises. From a marking M, a set of transition rings is possible. The result of ring a transition in a marking M is a new marking M 0, where in this case, M 0 is a immediately reachable marking from M, resulting from ring a enabled transition in M. We can also say that a marking M 0 is reachable from M if it is immediately reachable from M or it is reachable from any marking that it is immediately reachable from M. If we take the reexive, transitive closure of this \immediate reachable" relation we get the reachability set of a marked T-Net. This is formally represented in the following Denition 2.12 Given a T-Net with a marking M, the reachable set [Mi, is the smallest set of markings such that M 2 [Mi and if M 1 2 [Mi and for some t 2 T M 1 [tim 2 then M 2 2 [Mi. A marking gives the actual system's state of a net. The marking graph of a T-Net represents all possible markings, i.e., all permissible system's states of the T-Net together with all possible ring transitions between these states, and hence can be seen as the semantics of a T-Net. Denition 2.13 The marking graph MG = def (MG V ; MG E ; s M G ; t M G ) of a T-Net is given by: 6

8 init rocess 1 rocess 2 T1 T4 1 b1 3 T2 T5 2 b2 4 T3 T6 Figure 2: T-Net for a deadlock situation MG V = def fmjm :! N is a marking for Ng; MG E = def f(m; t; M 0 )jt 2 T ^ M[tiM 0 ; M 2 MG V g; s M G (M; t; M 0 ) = def M; t M G (M; t; M 0 ) = def M 0 : Since the marking graph contains all markings and all possible transitions, we have also a number of markings and corresponding transitions which have no sense from the point of view of the modeled system. In the case of our reader-writer system, the only markings which have sense are those markings in which the sum of the tokens in the places p 1 ; p 2 ; p 3 ; p 4 and p 5 is exactly equal to three, since otherwise the principle of mutual exclusion would not work. This motivate us to present the following Denition 2.14 A marked T-Net N M0 = def (; T; F; W; M 0 ) is a T-Net N = def (; T; F; W ) together with a marking M 0 :! N, which will be called initial marking. Besides, when marked nets are treated, we are often not interested in the whole set of possible transitions, but only on those ones which are possible taking into account the initial marking. This information give us the reachability graph. The vertices of the reachability graph of a T-Net N are obtained from the reachable set of markings from the initial marking M 0 (see 2.12). The corresponding edges are given by the directed edges which have as sources (or targets) markings from [M 0 i. 7

9 T3 <1, 1, 0, 0, 0, 0, 1> T2 <1, 0, 1, 0, 0, 1, 0> M0 T1 T4 <2, 0, 0, 0, 0, 1, 1> <0, 1, 0, 1, 0, 0, 0> MD T4 T1 T6 <1, 0, 0, 0, 1, 0, 1, 0> T5 <1, 0, 0, 0, 1, 0, 0> Figure 3: Reachability graph for the T-Net of Figure 2 Denition 2.15 The sequential reachability graph EG = def a marked T-net N M0 is given by (EG V ; EG E ; s EG ; S EG ) of EG V = def [M 0 i; EG E = def f(m; t; M 0 jm[tim 0 with M; M 0 2 [M 0 i; t 2 Tg; s EG (M; t; M 0 ) = def M; t EG (M; t; M 0 ) = def M 0 : Remark 2.16 Note that the sequential reachability graph is in fact a subgraph of marking graph of Example 2.17 We present in Figure 3 the reachability graph from the T-Net of Figure 2, where the marking of the places is expressed in tuple form as hm( init); M( 1 ); M( 2 ); M( 3 ); M( 4 ); M( b1 ); M( b2 )i. It is easy to see that the marking M D represents a deadlock situation, since it is not the source of any edge. Besides, such a state is obtained after the ring of T 1 and T 4 (in any sequence). 3 Categorical T-Nets 3.1 Motivation etri nets are able to distinguish clearly the basic concepts in the behavior of processes. The graphical representation of nets visualizes these concepts and are also suitable for modeling of very detailed, machine-like view of processes. However, there are at least two serious problems with etri nets: Compositionality. Missing is a convenient way of composing or decomposing larger nets from or into smaller ones by a set of high level operators. Abstraction. The concept of etri nets is a low-level concept, comparable with machine level programming language. However, in specication of complex systems, there 8

10 is a need to abstract from internal transitions and focus on the communication behavior between processes, i.e., we may need sometimes to treat processes as \black-boxes" in the same way we treat modules in programming languages. On the other hand, the application of categorical methods in connection to T-Nets has the following advantages when comparing with the classical, more established approach: The ring of a unique transition as well as the concurrent ring of various transitions can be homogeneously treated and described. By bringing in T-Nets into the mathematical context of category theory, we have at our disposal the whole toolkit of categorical methods and concepts which can usually be immediately applied when approaching questions concerning structuring, compositionality and renement. The possibility of comparing and relating etri nets with other models for concurrent systems, like event-structures, transition systems, synchronization trees, etc., by giving to each one of them a categorical structure and using well-known categorical concepts like functors, natural transformations and adjoint situations in order to establish semantical relations amongst these models. In the following we introduce elementary concepts concerning monoids and show that the usual concept of etri nets can be given a \monoid structure". 3.2 etri nets are monoids Before we motivate a justication for the title of this section, we need the following auxiliary Denition Let be a set. Then = def ( ; ; ) is called the free monoid generated by, where is the set of all words over, such that for all u; v; w 2 the following equations hold: v = v =, u (v w) = (u v) w, that is to say, is a associative binary operation in with identity. Remark is also called the the Kleene closure over. 2. If is a free commutative monoid, i.e., if for all w; v 2 ; w v = v w, then any word of can be represented as a linear sum. For instance, if = def fa; b; c; dg, then a b c a c d c = a a b c c c d 2 can be represented as \2a b 3c d". In this case, we will stick to the notation in order to denote the free commutative monoid generated by. 9

11 T - Net T-Graph p1 p2 p3 p1 7p2 4p3 7 4 t t 2 3 p4 p5 p6 2p4 3p5 p6 Figure 4: Unmarked T-Net 3. In general, we have the following formal denition for any w 2 expressed in linear sum form: w = def nx i=1 k i a i ; k i 2 N; a i 2 : 4. Commutative monoids together with monoid homomorphisms dene a subcategory CMon of Mon, the latter one being the category of monoids and monoid homomorphisms. Consider now the \unmarked" T-Net on the left of Figure 4. It has a set of places = def fp 1 ; p 2 ; p 3 ; p 4 ; p 5 ; p 6 g. The main point is that this T-Net can be understood as a ordinary graph whose set of vertices are elements of the base set of the free commutative monoid over in the following way: The places p 1 ; p 2 ; p 3 belonging to the pre-condition of t together with their respective weights must described as an element of, i.e., p 1 7p 2 4p 3. This is the rst vertex of the graph. Likewise, the places belonging to the post-condition of t together with their respective weights must also be described as an element of, i.e., 2p 4 3p 5 p 6. This is the second vertex of the graph. The resulting graph is obtained by an edge labeled t whose source is the rst vertex and whose target is the second vertex, as described above. The resultant graph is depicted on the right of Figure 4. Such a graph will be called T-Graph from now on. Based on the above considerations we can now give a monoid structure to our classical T-Net, which is expressed in the following 10

12 Denition Given a (classical) T-Net N = def (; T; F; W ), a categorical T- Net N = def (; ; T; pre; post : T! ) is a graph with a set of vertices, a set of edges T, as well the pre- and post-condition mappings (source and target mappings, respectively) pre and post, respectively. Remarks We will use interchangeably the terms \categorical T-Net" and \T-Graph". 2. A T-Graph usually contain countably innite isolated vertices (all the elements of which don't correspond to any pre- or post-condition of transitions in the T-Net). 3. For each transition t in the T-Net, there is a corresponding edge in the T-Graph with label t and where the mappings pre(t); post(t) are dened. 4. For every transition without pre-condition, rep. without post-condition, the corresponding edge of the T-Graph has source, resp. target, the -vertex. Example As a more substantial example, let us now consider our readerswriters system from Figure 1 without the initial marking. The corresponding categorical T-Net N is dened by the mappings pre; post in the following way: pre(t 1 ) = p 1 post(t 1 ) = p 2 pre(t 4 ) = p 1 post(t 1 ) = p 4 pre(t 2 ) = p 2 3p 6 post(t 2 ) = p 3 pre(t 3 ) = p 3 post(t 3 ) = 3p 6 p 1 pre(t 5 ) = p 4 p 6 post(t 5 ) = p 5 pre(t 6 ) = p 5 post(t 6 ) = p 6 p 1 The corresponding T-Graph is depicted in Figure 5 (note that isolated vertices suggest the \innity" of the set of vertices in ). The above remarks and example demand the following Theorem Classical T-Nets and categorical etri nets are in one-to-one correspondence to each other. roof: First we show that for each classical T-Net N = def (; T; F; W ), there is a categorical T-Net N = def (; ; T; pre; post) with pre; post : T!. Now, for all t 2 T; pre; post are dened as: (hp;ti2f pre(t) = def W (p; t) p if 9p 2 : hp; ti 2 F otherwise (ht;pi2f W (t; p) p if 9p 2 : ht; pi 2 F post(t) = def otherwise 11

13 ... t1 p1 t4 p2 p4 p2 3p6 p4 p6 t2... t5 s p5 t t6 3p6 p1 p6 p1 Figure 5: T-Graph for the readerswriters T-Net Now, to each categorical T-Net N = def (; ; T; pre; post) with pre; post : T!, (; T; F; W ), where the ow relation F there is a corresponding classical T-Net N = def and the weight W are dened as follows: F = def fhp; tijt 2 T ^ pre(t) = [fht; pijt 2 T ^ post(t) = nx i=1 W (hp; ti) = def k j ; if pre(t) = W (ht; pi) = def k j ; if pre(t) = nx i=1 k i p i ^ 9j 2 f1; : : : ; ng : p = p j ^ k j 6= 0g k i p i ^ 9j 2 f1; : : : ; ng : p = p j ^ k j 6= 0g nx i=1 nx i=1 k i p i ^ 9j 2 f1; : : : ; ng : p = p j ^ k j 6= 0g k i p i ^ 9j 2 f1; : : : ; ng : p = p j ^ k j 6= 0g 2 Remark Due to the above equivalence, it is not necessary to dierentiate the qualifying terms \classical" and \categorical". Now consider the T-Net in Figure 6. If we conceive the marking of a T-Net also as a linear sum from, then we have M = 3pz 2pl, and we can describe the ring of a etri net also in an algebraic way. This is reected in the following Denition A transition t 2 T is enabled under a marking M, denoted M[ti, when pre(t) M (where is dened componentwise on the summands of the linear sum). 12

14 pz tl ts M = 3pz 2pl pl ps Figure 6: A marked T-Net 3.3 The category T-Net Denition Let N i = def ( i ; T i; pre i ; post i : T i! i ); i = 1; 2 be T-Nets. Then a T-Net morphism f : N 1! N 2 = def hf ; f T i is a pair of mappings f : 1! 2 ; f T : T 1! T 2, such that the following diagram commutes componentwise: pre 1 T 1 pre 2 1 ft f post 1 T 2 post 2 2 where f is the free extension of f, i.e., for every w = p 1 : : : p n 2 1 ; n 0, f (w) = f (p 1 ) : : :f (p n ) Remark The free extension above must be a monoid homomorphism, i.e., the following equations should necessarily hold for all w 1 ; w : f () = f (w 1 w 2 ) = f (w 1 ) f (w 2 ) Example Consider the two T-Nets in Figure 7 together with their associated T- Graphs. Then f = def fp 1 7! p 5 ; p 2 7! p 5 ; p 3 7! p 6 ; p 4 7! p 6 g and f T = def ft 1 7! t 2 g is a T-Net morphism according to With the concept of T-morphism available, we can now introduce the desired roposition The class of all T-Nets N = def (; ; T; pre; post : T! ) together with the set of all T-Net morphisms dene a category T-Net. roof: First, given T-Nets N i ; i = 1; : : : ; 3 according to and corresponding =T -net morphisms f : N 1! N 2 ; g : N 2! N 3 according to 3.3.1, we have to show that gf : N 1! N 3 is also a T-Net morphism. We put h(g f) ; (g f) T i = def hg f ; g T f T i. This means 13

15 p1 p2 p5 p1 2p2 3p5 2 3 t1 t1 t2 t p3 p4 3p3 2p4 p6 5p6 Figure 7: An example of a T-Net morphism that we have to show that the following diagram commutes (note that here we are already using the fact of the existence of a free functor (?) : Set! CMon of A.6.6, such that (g f ) = g f ): ft gt pre 1 T 1 post 1 pre 2 T 2 post 2 pre 3 T post 3 3 But by assumption, the two squares commute, and hence the whole diagram commutes. Now, given any T-Net N 1 = def ( 1 ; ; T 1 1; pre 1 ; post 1 : T 1! 1 ), we dene the identity on N 1 as id N1 = def hid : 1! 1 ; id T : T 1! T 1 i such that this pair becomes a T-Net morphism. Now, for any f : N 1! N 2, we have that f id N1 = id N1 and id N2 f = f, since these equalities hold componentwise for the base mappings, which are just morphisms in Set. It remains to show associativity of composition, i.e., given morphisms f : N 1! N 2 ; g : N 2! N 3 ; h : N 3! N 4, we have to show that h (g f) = (h g) f. But this equality again holds trivially, since componentwise composition is associative because the base mappings are morphisms in Set. 2 f g 3.4 Colimits in T-Net We now turn our attention to the internal structure of the category T-Net, but with special attention on how to appropriately compose T-Nets. Since structuring concepts in specication theory (e.g., [4],[3]) can be suitable modeled by colimit constructions, we ask the natural question whether T-Net is colimit complete. According to A.5.9, we have to show that T-Net has products and coequalizers. This is nothing but the task of our next two propositions. 14

16 ; O c o : : O O d d o o ; O c o roposition T-Net has coproducts. roof: Let N i = def ( i ; i ; T i; pre i ; post i : T i! i ); i = 1; 2 be T-Nets. We claim that N 1 + N 2 = def ( 1 ] 2 ; ( 1 ] 2 ) ; [pre 1 ; pre 2 ]; [post 1 ; post 2 ] : T 1 ] T 2! ( 1 ] 2 ) ) is a coproduct in T-Net (where ] is the disjoint union of sets). First note that there exists a free functor (?) : Set! CMon (see A.6.6) such that we put (?) ( 1 ] 2 ) = ( 1 ] 2 ), where ( 1 ] 2 ) = +, since 1 2 (?), by A.6.5, preserves colimits and hence coproducts. Moreover, we dene [pre 1 ; pre 2 ] and [post 1 ; post 2 ] in the following way [pre 1 ; pre 2 ](t) = def ( pre1 (t) if t 2 T 1 pre 2 (t) if t 2 T 2 [post 1 ; post 2 ](t) = def ( post1 (t) if t 2 T 1 post 2 (t) if t 2 T 2 This shows that N 1 + N 2 is a well dened T-Graph. Now, in order to show that N 1 + N 2 is a coproduct in CMon, we have to show that for given T-Graph morphisms f : N 1! M; g : N 2! M, where M = def ( M ; ; T M M; pre M ; post M : T! M ) there is a unique h : N 1 + N 2! M such that the following diagram commutes: N 1 ttttttttt f t in N 1 M N 1 + N 2 (1) J g h JJJJJJJJJ N 2 inn 2 Now consider the following coproduct diagrams in Set. M vvvvvvvv H f g [f ;g ] v HHHHHHHH 1 1 ] 2 2 in 1 in 2 (2) T M wwwwwwww G ft gt [ft ;gt ] w GGGGGGGG T 1 T 1 ] T 2 T 2 int 1 int 2 (3) Since the free functor (?) : Set! CMon preserves colimits we have the following commutative diagram in CMon (the translation of (2) along (?) ). 1 tttttttt M f t J g 1 [f ;g 2 1 ] 2 t JJJJJJJJJ ( 1 ] 2 ) in 2 in 1 2 (3) 15

17 where in 1 = def (?) (in 1 ); in 2 = def (?) (in 2 ); f = def (?) (f ); g = def (?) (g ). We dene h = hh ; h T ] : N 1 + N 2! N were h : 1 ] 2! M ; h T : T 1 ] T 2! T M by h = def h[f ; g ]; [f T ; g T ]i. To see that h is indeed a morphism in T-Net, we have to show that the following diagram commutes: We distinguish two cases: Case t 2 T 1 [pre 1 ;pre 2 ] T 1 ] T 2 [post 1 ;post 2 ] [ft ;gt ] ( 1 ] 2 ) prem T M M postm [f ;g ] [f ; g ] [pre 1; pre 2 ](t) = [f ; g ](pre 1(t)) (by denition of [pre 1 ; pre 2 ]) = f (pre 1(t)) (by denition of [f ; g ]) = pre M (f T (t)) (f is a T-Graph morphism) = pre M [f T ; g T ](t) (by denition of [f T ; g T ]) Case t 2 T 2 : [f ; g ] [pre 1; pre 2 ](t) = [f ; g ](pre e(t)) (by denition of [pre 1 ; pre 2 ]) = g (pre 2(t)) (by denition of [f ; g ]) = pre M (g T (t)) (g is a T-Graph morphism) = pre M [f T ; g T ](t) (by denition of [f T ; g T ]) Applying a similar argument we get [f ; g ][post 1; post 2 ] = post M [f T ; g T ]. This shows that the above diagram diagram commutes and hence that h is a T-Graph morphism. Now, in order to see that (1) commutes observe that as well as h in N1 = h[f ; g ]; [f T ; g T ]i hin 1 ; in T1 i (by denition) = h[f ; g ] in 1 ; [f T ; g T ] in T1 i (by composition) = hf ; f T i (by (2) and (3)) = f (by denition) h in N2 = h[f ; g ]; [f T ; g T ]i hin 2 ; in T2 i (by denition) = h[f ; g ] in 2 ; [f T ; g T ] in T2 i (by composition) = hg ; g T i (by (2) and (3)) = g (by denition) It remains to show uniqueness of h = def h[f ; g ]; [f T ; g T ]i. Therefore, suppose there is another h 0 = def h[f 0 ; g0 ]; [f 0 T ; g0 ]i : N T 1 + N 2! M such that (4) commutes. But this implies, by the above calculations, that [f 0 ; g0 ] and [f 0 T ; g0 T ] would also make, respectively, (1) and 16

18 ! p1 p2 p1 2p2 2 N1 3 t1 t1 p3 3p3 p4 p4 N2 t2 t2 2 p5 2p5 p1 p2 p4 p1 2p2 p4 2 N1+N2 t1 t2 t1 t2 3 2 p3 p5 3p3 2p5 Figure 8: A coproduct of T-Nets (2) commute. Now [f ; g ] and [f T ; g T ] are unique in (2) and (3) by the couniversal property of coproducts and therefore, [f ; g ] = [f 0 ; g0 ] as well as [f T ; g T ] = [f 0 T ; g0 T ], i.e., h = h0, as required. 2 Example Figure 8 shows an example of a coproduct from two T-Nets N 1 and N 2. Note that the coproduct N 1 + N 2 of two T-Nets is the result of a composition operation without synchronization, i.e., both nets are put together without any kind of cooperation. roposition The category T-Net has coequalizers. roof: Given two T-Net morphisms f; g : N 1! N 2 we must dene a T-Net N C and a morphism c : N 2! N C with c f = c g such that for any other N D and d : N 2! N D with d f = d g, there is a unique h : N C! N D such that h c = d according to the following diagram: f N 1 g c N 2 CC CCCC C d (1) N C h C N D 17

19 ! Now consider the following diagram, where the coequalizer c T is dened by the following diagram in Set (knowing that Set is cocomplete): ft T 1 gt ct T 2 B BBBBBB T C dt (2) ht B T D where T C = def T 2 =, with the equivalence relation induced by fhf T (t); g T (t)ijt 2 T 1 g, that is to say, c T : T 2! T C is the natural mapping t 7! [t]. Now, consider the equalizer c : 2! C in the category Set according to the following diagram: f 1 g c 2 B BBBBBB C d (3) h B D where again C = def 2 =, with the equivalence relation induced by fhf (p); g (p)ijp 2 1 g, that is to say, c : 2! C is the natural mapping p 7! [p]. Since there is a free functor (?) : Set! CMon, we dene = def (?) ( C ); c = def (?) (c ) and obtain c as a coequalizer in CMon according to the following diagram (since (?) preserves colimits and hence coequalizers): 1 f g 2 c A C AAAAAA d (4) A D h Now consider the following diagram: 18

20 T pre 1 post 1 ft gt g f pre 2 T 2 post 2 ct T C C c ht h pred T D postd D where, by assumption, we have that c is a coequalizer of g ; f and c T a coequalizer of f T ; g T. But now note that c pre 2 f T = c f pre (f is a T-morphism) = c g pre (c is a coequalizer) = c pre 2 g T (g is a T-morphism) This shows that (c pre 2)f T = (c pre 2)g T. But, by assumption, c T is the coequalizer of f T ; g T, and hence, there must be a unique pre C : T C! C such that pre C c T = c pre 2. By a similar argument, there is a unique post C : T C! C such that post C c T = c post 2. This implies the commutativity of the middle square in the following diagram. ft ct ht pre 1 T 1 post 1 gt g pre 2 T 2 post f c prec T C postc C h pred T D postd D Now we put N C = def ( C ; T C; pre C ; post C : T C! C ), which is by the above arguments a well-dened T-Net. Moreover, the commutativity of the middle square above asserts that hc ; c T i : N 2! N C becomes a T-Net morphism. We must now show that the bottom square also commutes so that also h becomes a T-Net morphism. To see this, note that pre D h T c T = d T pre D (by (2)) = d pre 2 (d is a T-Net morphism) = h c pre 2 (by (4)) = h pre C c T (c is a T-Net morphism) 19

21 p1 p2 p3 p1 3p2 3p3 2 3 t1 t2 t1 t p4 3p5 4p6 p4 p5 p6 Figure 9: arallel ring in a T-Graph Now, since c T is a coequalizer, it is a epimorphism (see A.3.7), and hence (pre D h T )c T = (h pre C) c T ) pre D h T = h pre C which is the desired equality to show that h = def hh ; h T i is a T-Net morphism. It still remains to show equality of (1) and uniqueness of h. To see that (1) commutes note that h c = hh ; h T i hc ; c T i (by denition) = hh c ; h T c T i (composition) = hd ; d T i (c and c T are coequalizers) = d (by denition) Finally, to see uniqueness of h, suppose that there is another i = def hi ; i T i such that ic = d. But this means that hi ; i T ihc ; c T i = hi c ; i T c T i = hd ; d T i, i.e., i c = d and i T c T = d T. But h T and h are unique in (2) and (3) such that we must have, by uniqueness, i = h and i T = h T, that is to say, i = h. 2 4 Free constructions and the semantics of categorical T- Nets 4.1 arallel ring of transitions The parallel ring of several transitions in T-Nets can be naturally expressed in T- Graphs. Both the pre-condition as well as the post-condition of the ring transitions can be expressed as a common pre- or post-condition as elements of. At the same time, the simultaneous transition of more than one transition can also be expressed as a linear sum. For instance, in Figure 9, the possible parallel ring of t 1 and t 2 can be expressed by the corresponding T-Graph on the right. Therefore, a monoid structure is dened both for the vertices as well as for the edges of a T-Graph. Obviously, the parallel ring of transitions can only be realized when they are enabled by the current marking. This leads to the next Denition Given a T-Net N = def (; ; T; pre; post : T! ), we say that two 20

22 p2 4p6 p4 2p1 3p1... t2 t5 t1 t4... p3 p5... p2 p4 4p1... Figure 10: Subgraph of a parallel T-Graph for the reader-writer system transitions t 1 ; t 2 2 T are independently enabled when pre(t 1 )pre(t 2 ) M. The parallel ring of the transitions t 1 and t 2 build a parallel ring step t 1 t 2. Example As an example, consider the readers-writers system of Figure 1. Under M 0 = 3p 1 3p 6, the transitions t 4 and t 1 are independently enabled, since we have pre(t 1 ) pre(t 4 ) = 2p 1 M 0. On the other hand, the transition t 5 and t 2 are not independently enabled for any marking in the reachable set [M 0 i, since pre(t 5 ) pre(t 2 ) = p 2 4p 6 p 4 is bigger that every possible marking in [M 0 i (since the number of tokens in p6 is never bigger than 3). The above example motivates us to present the following Denition A free T-Monoid N = def (; ; T ; pre ; post : T! ) consists of a T-Net, such that the T is the free commutative monoid generated by T, and post ; post : T! are the unique extensions of pre; post : T!, given for all t 1 : : : t n 2 T by: pre (t 1 : : : t n ) = pre(t 1 ) : : : pre(t n ); that is to say, pre and post are free monoid homomorphisms. Remark We speak simply about a (non-free) commutative monoid T when the monoid of transitions is not necessarily free. Thus, the pre- and post-conditions functions pre; post : T! are simple monoid homomorphisms. Example As an example of a subgraph of a parallel graph for our reader-writer system consider Figure 10. Note that additional subgraphs and innitely many isolated vertices are only suggested. Moreover, a T-Net in monoid representation with parallel ring of transitions (called parallel T-Graph) contain all possible combination of transitions, since the current marking plays no role for the generation of the such a parallel graph. 21

23 Denition Let N i = def ( i ; i ; T i ; pre i ; post i : T i! i ); i = 1; 2 be free T- Monoids. A free T-Monoid morphism is a T-Net morphism f = def hf ; f T i where f : 1! 2 ; f T : T 1! T 2, and f : 1! ; f : T 2 T 1! T 2 are free monoid homomorphisms, such that the following diagram commutes componentwise: T 1 pre 1 post 1 1 f T f T 2 pre 2 post 2 2 With the available concepts of free T-Monoids and free T-Monoid morphisms we can establish our next roposition The class of all free T-Monoids together with set of all free T- Monoid morphisms constitute a category, called FM-Graph. roof: Let N ; N ; N be free T-Monoids according to and f = def hf ; f T i : N 1! N ; g = 2 def hg ; g T i : N 2! N 3 be free T-Monoid morphisms according to To say that g f : N 1! N 3 is again a free T-Monoid morphism amounts to say that the following diagram is componentwise commutative, T 1 pre 1 post 1 1 f T f T 2 pre 2 post 2 2 g T g T 3 pre 3 post 3 3 which follows directly from the commutativity of the two squares by assumption. Now for any free T-Monoid N we have an identity T-Monoid morphism id 1 N 1 = def hid ; id T i : N 1! N, such that for any T-Monoid morphism f : N 1 1! N 2 the equations id N2 f = f and f id N1 hold because they hold componentwise in Set. The same argument applies for the associativity axiom Reexive transitions and the marking graph We now turn our attention to construct a marking graph in the sense of 2.13 from our parallel T-Graph. If we examine more closely our parallel T-Graph, we will see that there are transitions which exist in a marking graph but are still missing in our parallel T-Graph. 22

24 p1 t1 t4 p2 p4 Figure 11: Subnet of the reader-writer system p1 2p1 3p1 4p1 p1 2p1 3p1 4p1 t1 t4 t1 t4 t1 t4 p1 t1 t4 2p1 p2 p4 p2 p4 p2 p4 p1 p2 p4 2p1 p2 p4 p2 p4 p2 p4 p1 p2 p4 2p1 Figure 12: Reexive parallel T-Graph of the reader-writer system For instance, in the parallel T-Graph corresponding to the Figure 10, the ring step t 4 t 1 can re from 2p 1 to p 2 p 4 but not from the marking 3p 1, since 3p 1 is an isolated vertex in the parallel T-Graph. Therefore, in order to derive the marking graph from the T-Graph, it is rst necessary to extend our notion of T-Monoid in the following way. For each place x 2 we introduce a reexive transition with the same name, which has x itself as both pre- and post-condition (essentially an identity loop). As an example, consider a subnet of our reader-writer system in Figure 11. A subnet of the reexive monoid with the reexive transitions can be seen in Figure 12. Observe that we now have the \missed" transition from 3p 1, namely 3p 1 [t 1 t 4 p 1 ip 2 p 4 p 1. Note that the ring of a transition under a specic marking which is bigger than the pre-condition of the respective transition (containing therefore the so-called \idle" tokens) corresponds, in the reexive T-Monoid, to the parallel composition of the ring transition with so many idle tokens as the ones which are not \consumed\ by the transition. As an example, consider the above mentioned transition t 1 t 4. Its pre-condition is pre (t 1 t 4 ) = pre(t 1 ) pre(t 4 ) = 2p 1. Now taking into account the vertex (marking) 4p 1 we have that 4p 1? 2p 1 = 2p 1, i.e., we have two idle tokens p 1. This means that the actual parallel transition from the pre-condition 4p 1 should be t 1 t 4 2p 1. Its respective post-condition is post (t 1 t 4 2p 1 ) = post(t 1 ) post(t 4 ) post(2p 1 ) = p 2 p 4 2p 1, where post(2p 1 ) = 2p 1 holds because every vertex in the reexive T-Monoid should also be equipped with an identity loop as discussed previously. This discussion motivates us to introduce the following 23

25 O 2init b1 b2 T6 init b2 T1 init b2 T4 init b2 init 1 b2 init 3 b1 T4 1 T1 3 T2 init 1 3 T5 init init b1 2 init 4 b2 Figure 13: Reexive T-Monoid corresponding to the sequential reachability graph of the deadlock system of Figure 3 Denition A reexive T-Monoid R = def (; ; T = def (U; ; +); pre; post : (U; ; +)! ) is a T-Monoid together with a monoid homomorphism idle :! (U; ; +), which associates an identity transition (edge) to each vertex from such that pre idle = post idle = id. Example A reexive T-Monoid corresponds in some sense to the marking graph of denition 2.13, with the crucial dierence that in the reexive T-Monoid all possible parallel rings are also present. On the other hand, since marking graphs are essentially innite, we take, as an example, the nite, sequential reachability graph of the deadlock system of Figure 2. The corresponding reexive T-Monoid is depicted in Figure 13. This graph is, up to \loops", isomorphic to the sequential reachability graph. Denition Let R i = def ( i ; i ; (U i; ; +); pre i ; post i : (U i ; ; +)! i ; idle i : i ), i = 1; 2 be reexive T-Monoids. A reexive =T?Monoid morphism is a pair of functions : 1! 2 and f T : (U 1 ; ; +)! (U 2 ; ; +) is a monoid homomor- f = def hf ; f T i, where f phism such the following diagram should commute componentwise, i.e., f pre = 1 pre 2 f T (the same for post ) and idle 2 f = f T idle 1 : idle 1 (U 1 ; ; +) f T GF (U 2 ; ; pre 1 post 1 pre 2 post 2 ED 1 2 BC f idle 2 Remark In general, reexive T-Monoids are nothing else than reexive graphs. A reexive graph (G; id G : V! E) where G = def (V; E; s; t : E! V ) is a graph such that for 24

26 O O each v 2 V there exists a corresponding identity edge (loop) id G (v) such that s(id G (v)) = t(id G (v)) = v. Now, reexive graph morphisms are graph morphisms f = def hf V ; f E i : G 1! G 2 which should additionally preserve identity loops, i.e., such that the equation f E id G1 (v) = id G2 f V (v) should hold for all v 2 V 1. roposition The class of all reexive T-Monoids together with the set of all reexive T-Monoid morphisms constitute a category, called RM-Graph roof: Given reexive T-Monoid morphisms f : N 1! N ; g : N 2 2! N 3, we have to show that the following diagram commutes as well as the preservation of reexive transitions. idle 1 (U 1 ; ; +) f T GF (U 2 ; ; +) g (T 3 ; ; pre 1 post 1 pre 2 post 2 idle 2 pre 3 post 3 ED 1 f 2 BC 1 BC g idle 3 Now observe that the bigger square commute (componentwise) by assumption, since f and g are reexive T-Monoid morphisms. reservation of identity transitions hold by the following trivial calculation: idle 3 g f = f T idle 2 f (g is a ref. T-Monoid morph.) = g T f T idle 1 (f is a ref. T-Monoid morph.) This shows that g f = def hg f ; g T f T i is a reexive T-Monoid morphism. The associativity law holds because for all f : N 1! N ; g : N 2 2! N ; h : N 3 3! N 4 we have that h (g f) = (h g) f since the components of reexive T-Monoid morphisms are functions in Set, respectively, monoid homomorphisms in CMon. The identity law holds also by a similar argument. 2 Denition Marking Graph Given a T-Net N = def (; ; T; pre; post : T! ), the corresponding marking graph M[N] is given by M[N] = def (; ; T + ; [pre ; in ]; [post ; in ] : T +! ; idle :! T + ), where in is the inclusion of into : T +. Example Figure 14 shows a subset of the marking graph corresponding to the example of the deadlock system. This graph is also a subgraph of the marking graph in the sense of The complete (innite) marking graph contains additionally many similar \parallel reachability graphs", corresponding to others initial markings. Especially, this graph corresponds to the \initial" marking M = def 3 init 2 b1. 25

27 9 3init 2b1 2T1 init init 21 T1 2init b1 T1 init 1 2init 1 b1 Figure 14: Subgraph of the marking graph of the deadlock T-net as a reexive T-Monoid The construction of the marking graph can be described functorially, more specically by a free functor, which for each T-Graph associates its corresponding marking graph. The precise mathematical formulation of this construction is nothing but the task of our next Theorem There is a free functor F :T-Net!RM-Graph, which delivers a marking graph M[N] in RM-Graph for each etri Net N in T-Net. Moreover, this free functor is left adjoint with respect to the forgetful functor V :RM-Graph! T-Net, which forgets reexive transitions and parallel composition of transitions. roof: We must show that for each T-Net N = def (; ; T; pre; post : T! ), for each reexive T-Monoid M = def (R; R ; (U; ; +); pre M ; post M : (U; ; +)! R ; idle M : R! (U; ; +)) and for each T-Net morphism f = def hf ; f T i : N! V (M), where V (M) = def (R; R ; U; pre M ; post M : U! R ), f :! R; f T : T! U, there is a unique f ] : F (N)! M such that the following diagram commutes: N f N V (F (N)) rrrrrrrrr r V (f ] ) V (M) We dene F (N), according to 4.2.6, as F (N) = def (; ; T + ; [pre ; in ]; [post ; in ] : T +! ; idle :! T + ) and V (F (N)) = def (; ; T + ; [pre ; in ]; [post ; in ] : V (T + )! ). We rst dene the universal morphism N as N = def hid ; in T i such that the following diagram trivially commutes, i.e., such that N becomes a T-Net morphism. T pre post int V (T + ) [pre ;in ] [post ;in ] id 26

28 O : O d o o To existence of f ], note that, by assumption, we have a T-Net morphism f = def hf ; f T i such that the following diagram commutes (where U = V ((U; ; +))). ft T U pre post f prem postm R Now consider the following (coproduct) diagram in Set: in ] T o int T Since F preserve colimits (according to A.6.5), we also obtain the following coproduct diagram in CMon: in + T in T T This means that, given CMon morphisms idle M f :! R! (U; ; +); f T : T! (U; ; +), there is a unique h = def [idle M f ; f T ], by the couniversal property of + T, such that the following diagram commutes: idlem uuuuuuuu f u in (U; ; +) h + T I f T IIIIIIII T in T Now, we dene f ] = def hf ; [idle M f ; f T ]i, and to verify that this is well-dened, the following diagram has to be shown commutative, which we will leave as an exercise for the reader: GF idle [pre ;id ] ED [post ;id ] T + [idlem f ;f T ] f prem (U; ; +) postm BC idlem To see that V (f ] ) N = f, note rst that V (f ] ) = f ], since the base sets of F (N) are kept after the application of the forgetful functor V. Moreover, we have that 27

29 V (f ] ) N = f ] N (by the above remark) = hf ; [idle M f ; f ]i T hid ; in T i (by denition of f ] and N ) = hf ; f T i (identity and the denition of the induced coproduct morphism) = f (by denition) lease, note that the second equality above holds, for all t 2 T, by the following trivial calculation: [idle M f ; f T ] in T (t) = [idle M f ; f T ](t) (by denition of inclusion) = f T (t) (by denition and remark below) = f T (t) (by denition of a free extension) To see that f ] is the unique morphism that satises such commutativity, consider another g = def hg ; g T i; g :! R; g T : + T! (U; ; +) which fullls the same requirement, i.e., V (g) N = f. Since we have V (g) = g we have that that hg ; g T i hid ; in T i = hf ; [idle M f ; f T ]i and then, by composition and the identity law, g = f ; g T = g T in T = [idle M f ; f T ]. This, after all, means that g = f ], as required. 2 5 Bibliographic notes The starting point for etri net theory dates from Dr. etri's h.d. dissertation [13]. Good introductions on etri nets in general can be found in [11, 12, 14], where the rst is a tutorial and the other two are textbooks. Especial thanks for Claudia Ermel's highly readable Studienarbeit [5], from which I have \borrowed" the examples and the basis for this presentation. Other topics closely related to the discussion in this text are the notions of Algebraic High- Level Nets (see [10, 16, 15]) where tokens are \structured entities" (instead of black dots), more precisely, terms from an algebraic specication. The milestone paper which set the basis for an algebraic approach to the study of etri Nets is [8]. A very recent work connecting category theory and semantic foundations for a unifying view of several classes of etri Nets is Julia adberg's hd thesis [9]. Acknowledgments: This work was developed within a German-Brazilian cooperation on Information Technology, in the context of GRAHIT project, and partially supported by a CNq-grant for Alo Martini. This is part of a work which was initiated by rof. Dr. Harmut Ehrig's Kloosterman Lecture \On the role of Category Theory in 28

30 Computer Science" in Leiden, Holland, September, 1993, as well as the encouragement of rof. Dr. Daltro Nunes concerning categorical methods in computer science. A very special \thank you" goes to Uwe Wolter, who carefully read this report, found numerous signicant errors, and made very useful suggestions on how to correct them. 29

31 A Categorical Background In this appendix we summarize some basic notions from category theory which are used in this report as well as a set of elementary results (propositions and theorems) concerning categorical constructions that are referenced in the proofs. The complete proofs of the following elementary results, may be found, for instance, in [7]. Detailed introductions to this subject may be found in [1], [2], and [6]. A.1 Categories and diagrams Denition A.1.1 A category C comprises 1. a collection Ob(C) of objects; 2. a collection M or(c) of morphisms; 3. two operations dom; cod : M or(c)! Ob(C) assigning to each morphism f two objects, called respectively domain and codomain of f; 4. a composition operator : Mor(C) Mor(C)! Mor(C) assigning to each pair of morphisms hf; gi with dom(g) = cod(f) a composite morphism g f : dom(f)! cod(g), such that the following associative law holds: For any morphisms f; g; h in Mor(C) such that cod(f) = dom(g) ^ cod(g) = dom(h): h (f g) = (h g) h 5. for each object A, an identity morphism id A : A! A, such that the following identity law holds: For any morphism f such that dom(f) = A ^ cod(f) = B: Remarks A.1.2 id B f = f ^ f id A = f 1. Category theory is based on composition as a fundamental operation, in much the same way that classical set theory is based on the \element of" or membership relation. 2. Categories will be denoted by uppercase boldface letters A; B; C; : : : from the beginning of the alphabet. 3. We use letters A; B; : : : ; Y; Z from the alphabet (with subscripts when appropriate) to denote objects and lowercase letters a; b; c; : : : ; f; g; h; : : : ; y; z (occasionally with subscripts) to denote morphisms in any category. 4. If dom(f) = A ^ cod(f) = B we write f : A! B to denote that f is a morphism from A to B. 30