R. Brown and P. J. Higgins, The algebra of cubes, J. Pure Appl. Algebra , 21 (1981) 233--260.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The higher groupoid approach to homotopy theory gives an example of the kind of technical interaction that this project should foster. In formulating and proving a generalized Van Kampen theorem in the form of colimit theorems for relative homotopy modules or crossed modules, Brown and Higgins [26, 27] were led to extend cubical techniques (which have generally been discarded in the U.S. They introduced extra cubical degeneracies, called connections because of an analogy with connections in di#erential topology, that obey an appropriate transport law. Now, much later, this insight is playing ....

Ronald Brown and Philip J. Higgins, On the algebra of cubes. J. Pure Appl. Algebra 21 (1981), 233--260.

....information, however, is stable under bipointed dhomotopy (3.3.2) with one point in that triangle and the second at b. 29 4. Complements Directed homotopy of categories is briefly considered and related with d spaces. We end by discussing directed geometric realisation of cubical sets (cf. [4]) and directed metrisability of d spaces. 4.1. Directed homotopy for categories. Classically, the homotopical invariance of the fundamental groupoid functor P 1 : Top = Gpd (small groupoids) means that it turns homotopy equivalence of spaces into ordinary equivalence of groupoids; the latter can ....

R. Brown - P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233-260.

....) The set valued functor X n , is produced by the congruence n defined by the real extension rD n Top(R ) formed of real n dimensional delays. In particular, P (X) X and P (X) PX. For a space X, these functors produce a cubical set P X with connections (g i ) [BH1, BS] and symmetries (the interchanges s i and the involutions r i ) 2) X = P (X) P . P . only faces are displayed) the structural maps (for 1 i n; a = 3) e i : P s i : P , r i : P are induced by the following natural ....

.... P X X = colim rc X rG n , and, within the latter, the fundamental w groupoid, defined as in (7) 9) P w (X) P X, Again, P X and P X contain the same information , since cubical w groupoids with connections are equivalent to (globular) w groupoids, a fact already known from [BH1, BH2]. A congruence class of an n cube a: i 1 , j 1 ] i n , j n ] X in P n X (resp. in P n X) will be written as a . resp. a . one can always use a representative defined on a cube [i, j] 2.7. Fundamental n groupoids. Finally, the fundamental n groupoid of a space X (1) P n ....

R. Brown - P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233-260.

.... to be covered with 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 ....

....# # # i 1 # j , j i # i # i , j = i i # j 1 , j i # # # # # # # i 1 # , j i 1, i; # = # i # i , j = i 1, i; # j 1 , j i. 16) The dual relations have appeared quite recently, in [1] Section 3; but a partial version with one connection can be found in [4], p. 235) Let J be the subcategory of Set consisting of the elementary cubes 2 , together with the mappings generated by all faces, degeneracies and connections (# i : 2 ) Note, again, that J is a PRO. We prove now that every J map has a unique canonical factorisation, as in the ....

R. Brown, P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233--260.

....2000 Mathematics 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 ....

....semicubical set) and most of the theorems known so far are expressible in this wider framework, hence the importance of the latter notion. In this paper as well, most of the results will be stated in the wider framework of non contracting # categories. A precubical set is a cubical set as in [2] but without degeneracy maps of any kind. It is easy to view such an object as a contravariant functor from some small category # the category of sets. The objects of are the nonnegative integers. The small category is then the quotient of the free category generated by the arrows # i : ....

R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21(3):233--260, 1981.

....geometric realization functor (which should be homotopically equivalent to the former one composed with the realization of precubical sets as local po spaces of [FGR99] in some sense) transforming a precubical set into a globular CW complex. We rst need a few (classical) remarks. De nition 3.6. [BH81] [KP97] A precubical set (or HDA) consists of a family of sets (M n ) n 0 and of a family of face maps M n i ## M n 1 for 2 f0; 1g and 1 6 i 6 n which satis es the following axiom (called sometime the cube axiom) i j = j 1 i for all 1 6 i j 6 n and ; 2 f0; 1g. ....

R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21(3):233-260, 1981.

....Americans. 7 The higher groupoid approach to homotopy theory gives an example of the kind of interaction that this project should foster. In formulating and proving a generalized Van Kampen theorem in the form of colimit theorems for relative homotopy modules or crossed modules, Brown and Higgins [22, 23] were led to extend cubical techniques which had previously been almost universally discarded in the USA. They introduced extra cubical degeneracies, called connections because of an analogy with connections in di#erential topology, including a crucial transport law. Now, twenty years later, this ....

Ronald Brown and Philip J. Higgins, On the algebra of cubes. J. Pure Appl. Algebra 21 (1981), 233--260.

.... cut 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 ....

....# # # # # # 0 00 # # # # # # ;C # # # # # # Figure 1: Part of the 2 source of the 3 cube Notice that (R( 00) and (R(00 ) become composable in P C, although they are not composable in the initial category C. Now let us recall the notion of strict globular groupoid : De nition 4.4. [BH81b] Let C be a strict globular category. Then C is a strict globular groupoid if and only if for any p morphism A of C with p 1 and any r 0, then there exists A 0 (a priori depending on r) such that A r A 0 = s r A = t r A 0 and A 0 r A = s r A 0 = t r A. Theorem 4.5. Let C be ....

R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21(3):233-260, 1981. 24

....realization functor (which should be homotopically equivalent to the former one composed with the realization of precubical sets as local po spaces of [FGR99] in some sense) transforming a precubical set into a globular CW complex. We first need a few (classical) remarks. Definition 3.6. [BH81] [KP97] A precubical set (or HDA) consists of a family of sets (M n ) n 0 and of a family of face maps M n ff i M n Gamma1 for ff 2 f0; 1g and 1 6 i 6 n which satisfies the following axiom (called sometime the cube axiom) ff i fi j = fi j Gamma1 ff i for all 1 6 i j 6 n ....

R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21(3):233--260, 1981.

....326 cubes which are necessary to treat the branching case. The first kind is well known in cubical set theory : this is for example B = # 1 v or # 1 A = # 1 # 1 v. The second kind is for example A which will be denoted by # 1 u and which corresponds to extra degeneracy maps as defined in [6]. To take into account the symmetric problem of merging areas of execution paths, a third family # i of degeneracy maps will be necessary. In this paper, we will only treat the case of branchings. The case of mergings is similar and easy to deduce from the branching case. The solution ....

....and Applications of Categories, Vol. 8, No. 12 327 2 = u v u v v v X u v u u A B A B Figure 3: Identifying u 1 v and u 1. the extra structure of connections # on cubical sets, which allow extra degenerate elements in which adjacent faces coincide. This structure was first introduced in [6]. 2. the notion of folding operator. This was introduced in the groupoid context in [6] to fold down a cube to an element in a crossed complex, and in the category context in [1] to fold down a cube to an element in a globular category. Properties of this folding operator are further developed in ....

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R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21(3):233--260, 1981.

....HDA. We would like to explain here the construction of these functors, some of their known properties and some perspectives. Many explanations are given in a very informal way. We refer to the bibliography for more details. In [21] and [14] HDA are formalized using cubical sets in the sense of [6]. The link between cubical sets and categories will be described at the end of Section 3. There are two equivalent approaches of the notion of category : a cubical one and a globular one [3] The globular approach will be used everywhere in this paper except in Figure 6 where cubical ....

....fact that close to a corner, the intersection 9 Gaucher Fig. 5. model of 2 simplex in the branching nerve F E G H K L I J A B C D Fig. 6. A 2 dimensional branching area of an n cube by an hyperplane is an (n 1) simplex. Considering the maps i 1 and i 1 is not new (cf. the operations i 1 in [6] and i 1 ; 0 i 1 in [2] Our notations are adapted to the simplicial structure of De nition 4.3 noticed in [13] for the rst time. Figure 6 represents a 2 dimensional branching area. It corresponds to the homology class of the cycle (A) F ) I) One can prove that the cycles (A; B; C; D) E; ....

Brown, R. and P. J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), pp. 233-260.

....z 2 Cat(I n Gamma1 ; C) Then fx(ff: Gamma] i : ff) x(ff: 0] i : ff) x(ff: i : ff)g is a singleton so x(ff: 0] i : ff) is 0 dimensional. Hence the sufficiency of the condition. 5.2. Combinatorial structure on the cubical nerve of a multiple category Following the example of (Brown and Higgins, 1981) for the cubical nerve of a topological space, we are going to equip the cubical nerve of a multiple category with a combinatorial structure. First of all some connections and in a second part operations j . Let us start with a definition. Definition 5.2.1. An oriented cubical complex ....

....the same as the one of connections on the cubical nerve of a topological space. Thus there is nothing to verify in the axioms of oriented cubical complex except the relations mixing the two families of degeneracies Gamma i and Gamma Gamma i : all other axioms are already verified in (Brown and Higgins, 1981). So it remains to verify that Gamma Sigma i Gamma Upsilon j = Gamma Upsilon j 1 Gamma Sigma i if i j, and Gamma Sigma i Gamma Upsilon j = Gamma Upsilon j Gamma Sigma i Gamma1 if i j 1, that can be done quickly. The only remainding point we have to verify is ....

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Brown, R. and Higgins, P. (1981). On the algebra of cubes. Journal of Pure and Applied Algebra, pages 233--260.

....to the 2 groupoid G. Of course the usual semidirect product construction allows one to rebuild G from (M;N; We should mention that 2 groupoids resp. cat 1 groupoids are also equivalent to double groupoids with thin structure and crossed modules over groupoids ( BS, Spe, SW, BH, P 1, Br] 2. Tools from homotopical algebra In this section we recall the necessary tools from 1 and 2 dimensional homotopical algebra. We shall make use of the algebra of the homotopy bigroupoid of a topological space developed in [HKK 2] If X is a topological space and x and y points of ....

R. Brown and P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233 -- 260.

.... categorical definition is that a simplicial set is a functor Delta op Sets, where Delta is the category of finite ordered sets and order preserving maps between them [12, 18] For cubical sets, there are two analoga of the first description, one without and one with so called connections [8, 35]. Perhaps for this reason there seems to be no categorical description of cubical sets available. I intend to fill part of this gap, by defining a category Gamma which is to cubical sets, without connections, what Delta is to simplicial sets. In fact, I define a cubical set as a functor Gamma ....

....of cubical sets available. I intend to fill part of this gap, by defining a category Gamma which is to cubical sets, without connections, what Delta is to simplicial sets. In fact, I define a cubical set as a functor Gamma Sets, and I show that this definition coincides with the usual one [8]. Analogous to the simplicial case the objects of Gamma are called the standard cubes. 2.1 Cubes combinatorially Aichison has given an extensive account on cubes [1] from which I will use the following combinatorial definition of the n dimensional cube. Let n be the ordered set f1; ng ....

R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21:233--260, 1981.

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R. Brown and P. J. Higgins, The algebra of cubes, J. Pure Appl. Algebra , 21 (1981) 233--260.

....in the related 2 groupoid enrichment. 1. The singular cubical set of a topological space We shall be concerned with the low dimensional part (up to dimension 3) of the singular cubical set R # (X) R # n (X) # i , # i , # i ) of a topological space X. We recall the definition (cf. BH 2,BH 1] For n # 0 let R # n (X) Top(I n , X) denote the set of singular n cubes in X, i.e. continuous maps I n # X, where I = 0, 1] is the unit interval of real numbers. We shall identify R # 0 (X) with the set of points of X. For n = 1, 2, 3 a singular n cube will be called a ....

..... s n 1 ) a(s 1 , s i 1 , 1, s i 1 , s n 1 ) resp. # i (b) s 1 , s n ) b(s 1 , s i 1 , s i 1 , s n ) where a : I n # X, resp. b : I n 1 # X, are continuous maps. The face and degeneracy maps satisfy the usual cubical relations (cf. 1. 1) of [BH 2] 5.1) of [KP] If a # R # n (X) then #(a) # 1 (a) # 1 (a) # n (a) # n (a) A HOMOTOPY DOUBLE GROUPOID OF A HAUSDORFF SPACE 73 will be called the boundary of a. A path a # R # 1 (X) will sometimes be denoted a s by abuse of language and will be called a ....

R. Brown, P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233--260.

....Double category, 2 category, thin structure, connection. AMS 1991 CLASSIFICATION: 18D05. y email: r.brown bangor.ac. uk z email: kallami hotmail.com 1 2 There is an equivalence between cubical 1 groupoids with connections and globular 1 groupoids this follows from the main results of [6, 7], which give equivalences with crossed complexes. In the category case, the above equivalence for 2 categories has been extended to 3 categories in the thesis of Al Agl [1] It is conjectured that there is an equivalence of categories between globular 1 categories and cubical 1 categories with ....

....these matters elsewhere. We will develop in [9] the equivalence given in [13] between edge symmetric double algebroids with thin structure and certain crossed modules of algebroids this is an analogue of the equivalence between edge symmetric double groupoids with connection and crossed modules [10, 6]. For more discussion on matters of higher dimensional group theory see the web article [4] We would like to thank Rafael Sivera for help with the layout of section 4. 2 Double categories A double category is a category object internal to the category of small categories. It may also be ....

Brown, R. and Higgins, P.J., `On the algebra of cubes', J. Pure Appl. Algebra 21 (1981) 233-260.

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Brown, R., Higgins, P.J. (1981). On the algebra of cubes. J. Pure and Appl. Algebra 21, 233--260.

....nicely with our earlier results, the techniques will be used in other situations elsewhere, and it should be useful to workers in higher dimensional algebra. There is an equivalence between cubical 1 groupoids with connections and globular 1 groupoids this follows from the main results of [6, 7], which give equivalences with crossed complexes. In the category case, the above equivalence for 2 categories has been extended to 3 categories in the thesis of Al Agl [1] It is conjectured that there is an equivalence of categories between globular 1 categories and cubical 1 categories with ....

....these matters elsewhere. We will develop in [9] the equivalence given in [15] between edge symmetric double algebroids with thin structure and certain crossed modules of algebroids this is an analogue of the equivalence between edge symmetric double groupoids with connection and crossed modules [10, 6]. For more discussion on matters of higher dimensional group theory see the web article [4] We would like to thank Rafael Sivera for help with the layout of section 4 and the referee for helpful comments. 2. Double categories A double category is a category object internal to the category of ....

Brown, R. and Higgins, P.J., `On the algebra of cubes', J. Pure Appl. Algebra 21 (1981) 233-260.

....and of double groupoids with connection [11] The case of these equivalences when P is a singleton is due to Brown and Spencer [19, 20] Each of these forms of double groupoids have their own particular value, and circumstances when they are most appropriate. Crossed modules over groupoids [10] These objects are more obviously related to classical tools, namely (i) groupoids, ii) modules over groups, iii) second relative homotopy groups, iv) chain complexes [13] For these reasons, it is natural to attempt to compute with these objects rather than with the other forms. There is a ....

....over groups, iii) second relative homotopy groups, iv) chain complexes [13] For these reasons, it is natural to attempt to compute with these objects rather than with the other forms. There is a useful monoidal closed structure on this category, see [14] Double groupoids with connection [19, 10] These are an essential tool in one proof of the Van Kampen Theorem for the fundamental crossed module, because they nicely handle subdivision and the homotopy addition lemma [9] They also have a monoidal closed structure, related to a notion of homotopy, and which may be derived from that for ....

Brown, R. and Higgins, P.J., `The algebra of cubes', J. Pure Appl. Algebra, 21 (1981) 233-260.

....crossed modules had arisen much earlier in the work of J. H. C. Whitehead on 2 dimensional homotopy. This result is easily extended to give an equivalence between arbitrary special double groupoids with special connection and crossed modules over groupoids; this is included in the results of [3]. We recall these results in more detail in x3 below. Special double groupoids with special connection and a singleton double base are also equivalent to the cat 1 groups of Loday. Cat n groups, for any positive integer n, were introduced in [12] as algebraic models of homotopy (n ....

.... 3 SPLIT DOUBLE GROUPOIDS By split double groupoids we mean the differentiable analogue of the special double groupoids with special connection which were introduced by Brown and Spencer [6] and whose generalization to arbitrary dimensions has been extensively developed by Brown and Higgins ([3], 4] and elsewhere) for proving Generalized Van Kampen Theorems. Special double groupoids with special connection differ from general double groupoids in that their side groupoids are identical and in that they admit special connections . These special connections encode aspects of the ....

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R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21:233--260, 1981.

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R.Brown, P.J.Higgins, , `The algebra of cubes', J. Pure Appl. Algebra, 21, 233-260, 1981a.

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R. Brown and P. J. Higgins. On the algebra of cubes. J. Pure Appl. Algebra, 21(3):233-260, 1981.

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