P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, Math. Struct. in Comp. Science 10 (2000), 481-524.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Algebraic Topology, including directed paths and homotopies, have recently appeared within the analysis of concurrent processes; such notions have been developed for classical combinatorial structures, like 4 simplicial and cubical sets, for topological spaces with local orders and for Chuspaces [6, 7, 8, 9, 13, 22]. Higher fundamental n categories P n (X) have been developed by the author for simplicial sets [12] On a more formal level, it can be noted that a setting based on the (co)cylinder functor can be effectively adapted to a situation where reversion is missing, as already showed in [10] ....

P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, Math. Struct. in Comp. Science 10 (2000), 481524.

....1 (X) corresponds to the existence of morphisms in C[ between elements of these classes. The component category contains more information. It allows to compare factorisations of two given morphisms and to discuss in which parts of the po space they agree and in which they di er. P. Gaucher [19] has a quite di erent categorical approach to branching and merging, not only for dipaths, but also for their higher dimensional analoga. 5.3 Components of the state space Next, we shift attention to the entire state space of a concurrent program, modelled by an lpo space X with a minimal ....

P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, Math. Structures Comput. Sci. 10 (2000), no. 4, 481-524.

....Work supported by MIUR Research Projects. ordinary homotopies and fundamental n groupoids are replaced with directed homotopies and fundamental n categories. Its applications deal with domains where privileged directions appear, like concurrent processes, traffic networks, space time models, etc. [FGR, Ga, GG, Go, G3]. Following our setting, in [G6] a directed topological space, or d space, is a topological space X equipped with a set dX of directed paths [0, 1] X, containing all constant paths and closed under increasing reparametrisation and concatenation. Such objects, called directed spaces or ....

P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, in: Geometry and concurrency, Math. Structures Comput. Sci. 10 (2000), no. 4, pp. 481-524.

....becomes much more intricate in dimensions 2 than in the case of string rewriting, so that even asking the correct questions seems far from obvious. Finally, very similar motivations and techniques appear in recent works on concurrency, and we expect fruitful interactions in that direction (see [9, 10]) Acknowledgment I wish to thank Albert Burroni for his invaluable help. A. A category of paths Given an X, we define an # graph HX with identities and compositions and show how HX becomes itself an # category. We define sets (HX) n by induction, together with maps: HX) n 1 (HX) ....

P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Mathematical Structures in Computer Science, 10(4):481--524, 2000.

.... tetrahedra; this advantage appears clearly when studying singular homology based on cubical chains, cf. Massey [28] Various works have proved the importance of adding, to the ordinary structure provided by faces and degeneracies, the connections (introduced in Brown Higgins [4, 5, 6] see also [33, 1, 12] and their references) Finally, the interest of adding interchanges and reversions can be seen in various works Work supported by MIUR Research Projects Received by the editors 2002 03 08 and, in revised form, 2003 05 12. Transmitted by Ronald Brown. Published on 2003 05 15. 2000 Mathematics ....

P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, Math. Struct. in Comp. Science 10 (2000), 481--524.

....and pullbacks, mapping cones and homotopy fibres, suspensions and loops, cofibre and fibre sequences. Introduction Directed Algebraic Topology is a recent subject, arising from domains where privileged directions appear, like concurrent processes, tra#c networks, space time models, etc. cf. [2, 3, 4, 5, 9, 10]) Its domain should be distinguished from classical Algebraic Topology by the principle that directed spaces have privileged directions, and directed paths therein need not be reversible. Its homotopical tools will also be non reversible : directed homotopies and fundamental category instead of ....

P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, in: Geometry and concurrency, Math. Struct. in Comp. Science 10 (2000), no. 4, pp. 481-524.

....similarly to the case of the category of precubical sets, the category of cubical sets is an elementary topos, which is complete and cocomplete. We do not talk about cubical sets with connections and compositions here [3] but they have a great interest for our purposes, see for instance [12]. 4.3 Some useful functors There again, we need two interesting (and quite classical in spirit) functors. Let Upsilon n be the category of Upsilon , whose objects are the n dimensional cubical sets, i.e. the cubical sets M with M k = for all k n . This category can be seen as the presheaf ....

P. Gaucher. Homotopy invariants of higher-dimensional categories and concurrency in computer science (i). Mathematical Structures in Computer Science, to appear, 2000.

....coordinate gives a copy of X in GlobX as an achronal cut , a subspace on which the partial order reduces to equality. The exact relationships among homotopy theory, dihomotopy theory, and higher category theory are not yet understood. An idea of the state of the subject is given in the papers [56, 57, 59, 61, 63] of Gaucher and Goubault. Practical problems in this area are currently requiring strict ncategories and have stimulated the development of that theory. However, it should in due course interact with the weak theory. This area also suggests interesting and di#cult computational questions. 6. ....

Philippe Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science. Mathematical Structure in Computer Science, 10(2000), 481--524.

....00 0 0 0 0Figure 18: The category representing a 2 transition. As V. Pratt already noticed in [49] the axioms of categories encode the composition properties of dipaths and of dihomotopies in an HDA. The interested reader can find the exploitation of these ideas in [19] 18] and [21]. We will give the formal definition of an category in three steps (see for instance [7] 62] and [61] for more details) A 1 category is a pair (A; s; t) satisfying the following properties: 1. A is a set, 2. s and t are fonctions from A to A called source and target respectively, 3. ....

....the automaton generated by a unique 2 transition is represented as the category of Figure 18 18 . 18 The double arrow is a 2 morphism in the corresponding category. This is more generally due to a result in [1] Homology, Homotopy and Applications, vol. No. 2001 25 P. Gaucher in [21] and [20] uses the category generated by a cubical set to construct three homological theories, corresponding respectively to branchings, confluences and globes (or, computer scientifically, mutual exclusions) This is constructed through suitable nerve functors. This is made possible because of ....

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Gaucher, P., Homotopy invariants of higher dimensional categories and concurrency in computer science, Mathematical Structures in Computer Science 10(4) (2000).

....to di erent ones of them do not represent the same component although leading to the same empty extension set. This shows that the extension sets alone carry less information than the components. The component category thus seems to capture the inherent branchings in a (local) po space. P. Gaucher [12] has another categorical approach to branching and merging. 3 It is not more dicult to handle the situation with a set of leaves X 1 . TOPOLOGICAL ANTIDOTES TO THE STATE SPACE EXPLOSION PROBLEM 9 In suciently simple situations, the 1 connected components can be derived by an easy geometric ....

....f L from L to T . The component category 0 ( 1 (X) 2 ) is of the form B [ R [ L ## ## f L ## ## ## f R ## T : The component category thus seems to keep track of the merging situations in a state space. For an alternative treatment, we refer again to the work of P. Gaucher [12] 4.3. State space up to potential. A ner analysis of state spaces arises when one tries to encompass both branching and merging situations: De nition 4.10. A morphism s 2 Mor(x; y) belongs to 3 if and only if s # : P(x) P(y) is the identity. Along a weakly invertible morphism s 2 3 (x; ....

P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, Math. Structures Comput. Sci. 10 (2000), no. 4, 481-524.

....dicylinders. The respective posets will describe the merging of execution trajectories, possible pasts of the systems; and, in particular, the unreachable configurations. This duality corresponds conceptually to the negative and positive corner homologies studied by Gaucher in [2] and [3]. In the one dimensional case, it has also been discussed in [1] See also Sec. 5.1 and Sec. 5.3 below. n Dihomotopy in X is a continuous mapping h : I (C n X) compact open topology in C n X) such that 2 From directed cylinder a reference to monotonicity. 3 Is there a better ....

....to yield isomorphic groups; if this is indeed the case then its shortcomings are isomorphic too. Besides, both proposals have a strong flavor of homologies rather than homotopies. On the other hand, higher dimensional directed homologies have been studied by Goubault [6] and by Gaucher [2] and [3]. The latter author defines three homology theories: globular H gl , negative corner H Gamma and positive corner H . Negative corner homology is based on similar intuitions as the dicylinder classes presented here. Very informally, the cycles correspond to dicones (dicylinders with ....

P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. To appear in Mathematical Structure in Computer Science, 2000. Electronically available at http://www-irma.u-strasbg.fr/~gaucher/homotopiecat.ps.gz. REFERENCES 25

....Subject Classification: 55U10, 18G35, 68Q85. Key words and phrases: cubical set, thin element, Kan complex, branching, higher dimensional automata, concurrency, homology theory. c Philippe Gaucher, 2003. Permission to copy for private use granted. 75 of topological spaces as in [2] The papers [6, 8] demonstrate that the formalism of strict globular # categories (see Definition 2.1) freely generated by precubical sets (see below) provides a suitable framework for the introduction of new algebraic tools devoted to the study of deformations of HDA. In particular, three augmented simplicial ....

....nerve is called the branching semi globular homology and is denoted by H (C) the simplicial homology shifted by one of the merging semi globular nerve is called the merging semi globular homology and is denoted by H (C) 2. Preliminaries The reader who is familiar with papers [6, 7, 8] may want to skip this section. 2.1. Definition. 1, 16, 14] An # category is a set A endowed with two families of maps (s n = d n ) n#0 and (t n = d n ) n#0 from A to A and with a family of partially defined 2 ary operations (# n ) n#0 where for any n 0, n is a map from (a, b) A A, ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Math. Structures Comput. Sci., 10:481--524, 2000.

....this model and the new topological model we introduce here ( globular CW complexes ) in Section 3. 2, the relation between local po spaces and cubical sets can be found in [FGR99] There are other formulations of the same problems using homological methods [Gou95] strict globular categories [Gau00c]. An important motivation in these pieces of work is that of reducing the complexity of the semantics (given by a local po space for instance) by considering deformation retracts. The classi cation of the possible concurrent semantics (and behaviours) should then be the result nding the right ....

....execution path of X, f must not only be an execution path (f must preserve partial order) but also f must be non constant as well : we say that f must be non contracting. The condition of non contractibility is very analogous to the notion of non contracting functors appearing in [Gau00c]. Notice also that the attaching maps in the de nition of a globular CW complex are morphisms in glCW. This non contractibility condition will be justi ed in Section 6. A non constant execution path of a globular CW complex X induces a morphism of globular CW complexes from I to X. ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Math. Structures Comput. Sci., 10:481-524, 2000.

.... 20 7 Comparison with the branching semi cubical nerve 21 1 1 Introduction An categorical model for higher dimensional automata has been rst proposed in [Pra91] followed by [Gou95] for a rst homological approach using these ideas and cubical models of topological spaces as in [BH81b] Since [Gau00b, Gau01a], it is clear that non contracting categories are a good framework to introduce new algebraic tools (in particular new homology theories) to study deformations of higher dimensional automata (HDA) Non contracting categories freely generated by precubical sets encode the algebraic properties ....

....homology shifted by one of the branching semi globular nerve is called the branching semi globular homology and the simplicial homology shifted by one of the merging semi globular nerve is called the merging semi globular homology. 2 Preliminaries The reader who would be familiar with papers [Gau00b, Gau01b, Gau01a] may want to skip this section. Only the recall of Steiner s formula for the n source and the n target maps in an complex is of importance for the sequel. De nition 2.1. BH81a, Str87, Ste91] An category is a set A endowed with two families of maps (s n = d n ) n 0 and (t n = d n ) n 0 ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Math. Structures Comput. Sci., 10:481-524, 2000.

....strict globular categories 1 Even if the limited required number of pages for this paper too entails to make some shortcuts. 2 are supposed to encode the algebraic structure of the possible compositions of execution paths and homotopies between them, initiated by [Pra91] and continued in [Gau00] where connections with homological ideas of [Gou95] were made. 2) The topological approach which consists, loosely speaking, to locally endow a topological space with a closed partial ordering which is supposed to represent the time : this is the notion of local po space developed for example in ....

....execution path of X, f must not only be an execution path (f must preserve partial order) but also f must be non constant as well : we say that f must be non contracting. The condition of non contractibility is very analogous to the notion of non contracting functors appearing in [Gau00], and is necessary for similar reasons. In particular, if the constant paths are not removed from P X (see Section 3.2 for the de nition) then this latter spaces are homotopy equivalent to the discrete set X 0 (the 0 skeleton of X ) And the removing of the constant paths from P X ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Math. Structures Comput. Sci., 10:481-524, 2000.

....this model and the new topological model we introduce here ( globular CW complexes ) in Section 3. 3, the relation between local po spaces and cubical sets can be found in [FGR99] There are other formulations of the same problems using homological methods [Gou95] strict globular categories [Gau00c]. An important motivation in these pieces of work is that of reducing the complexity of the semantics (given by a local po space for instance) by considering deformation retracts. The classification of the possible concurrent semantics (and behaviours) should then be the result finding the right ....

....path OE of X, f ffi OE must not only be an execution path (f must preserve partial order) but also f ffi OE must be non constant as well : we say that f must be non contracting. The condition of non contractibility is very analogous to the notion of non contracting functors appearing in [Gau00c]. Notice also that the attaching maps in the definition of a globular CW complex are morphisms in glCW. This non contractibility condition will be justified in Section 6. A non constant execution path of a globular CW complex X induces a morphism of globular CW complexes from Gamma I to X . ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Math. Structures Comput. Sci., 10:481--524, 2000.

.... Concurrent Pascal into a text file coding a precubical set is presented in [10] At this step, one does not yet consider cubical sets with or without connections since the degenerate elements have no meaning at all from the point of view of computer scientific modeling (even if in the beginning of [12], the notion of cubical sets is directly introduced by intellectual reflex) In [14] the following fundamental observation is made : given a precubical set (K n ) n#0 together with its two families of face maps (# # i ) for # # , then both chain complexes (ZK # , # # ) where ZX ....

....have explained above the situation in dimension 1. The 2 dimensional case is depicted in Figure 8. Additional explanations are available at the end of Section 9. The branching homology (or negative corner homology) and the merging homology (or positive corner homology) were already introduced in [12]. This invariance with respect to the cubifications of the underlying HDA was already suspected for other reasons. The branching and merging homology theories are the solution to overcome the drawback of Goubault s constructions. There are three key concepts in this paper which are not so common ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. Math. Structures Comput. Sci., 10(4):481--524, 2000. Geometry and concurrency.

....Descartes 67084 Strasbourg Cedex France gaucher irma.u strasbg.fr March 2000 Abstract We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new de nition of the globular homology. With this new de nition, the drawbacks noticed with the construction of [Gau99] disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets. Contents 1 Introduction 2 2 Conventions and notations 2 3 Cut of categories 4 4 The globular cut 8 5 Informal ....

....Proofs 13 6.1 Proof of Theorem 4.1 . 13 6.2 Proof of Theorem 4.3 . 15 1 Introduction The question of correcting the de nition of the globular homology was already mentioned in [Gau99]. An analysis of the weaknesses of the old de nition led us to adding new relations inside the globular chain complex. This goal is concretely implemented by the introduction of a new simplicial nerve of categories called the globular nerve whose simplicial homology yields the solution. After a ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. to be published in Math. Structures Comput. Sci., 1999. math.CT/9902151. 18

.... to appear From concurrency to algebraic topology Philippe Gaucher Institut de Recherche Math ematique Avanc ee, ULP et CNRS, 7 rue Ren e Descartes, 67084 Strasbourg Cedex, France, gaucher math.u strasbg.fr Abstract This paper is a survey of the new notions and results scattered in [13], 11] and [12] Starting from a formalization of higher dimensional automata (HDA) by strict globular categories, the construction of a diagram of simplicial sets over the threeobject small category gl is exposed. Some of the properties discovered so far on the corresponding simplicial ....

....threeobject small category gl is exposed. Some of the properties discovered so far on the corresponding simplicial homology theories are explained, in particular their links with geometric problems coming from concurrency theory in computer science. 1 Introduction We have already argued in [13] for modeling higher dimensional automata (HDA) using strict globular categories. To our knowledge, the link between globular categories and concurrent automata was rst noticed in [21] Papers [13] 11] and [12] show that this way of formalizing HDA is very well adapted to getting interesting ....

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Gaucher, P., Homotopy invariants of higher dimensional categories and concurrency in computer science, to be published in Math. Structures Comput. Sci. (2000).

.... 8 Comparison of x and Phi Gamma n (x) in the reduced corner complex 34 9 Folding operations and composition maps 36 10 Folding operations and differential map 43 11 Some consequences for the reduced corner homology 46 12 Acknowledgment 50 1 Introduction Two homology theories, introduced in [Gau00], formalize respectively the branching areas of execution paths (the negative corner homology) and the merging areas of execution paths (the positive corner homology) in higher dimensional automata (HDA) Pra91] FGR98] These homology theories improve the idea of [Gou95] which consisted of ....

....In [BS99] it is proved that the category of cubical categories and the category of globular categories are equivalent. If S is a set, the free abelian group generated by S is denoted by ZS. By definition, an element of ZS is a formal linear combination of elements of S. Definition 2.4. [Gau00] Let C be an category. Let C n be the set of n dimensional morphisms of C. Two n morphisms x and y are homotopic if there exists z 2 ZC n 1 such that s n z Gamma t n z = x Gamma y. This property is denoted by x y. Definition 2.5. Gau00] Let f be an functor from C to D. The morphism f is ....

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P. Gaucher. Homotopy invariants of higher dimensional categories and concurrency in computer science. to be published in Math. Structures Comput. Sci., 2000. available at math.CT/9902151.

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P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, Math. Struct. in Comp. Science 10 (2000), 481-524.

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