R. Street, The Algebra of Oriented Simplexes, J. Pure Appl. Algebra 49(1987), 283-335. 52  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theory might evolve. The starting point must of course be the present state of the field: we have a dozen definitions of n categories, and a few comparisons. The various definitions are at uneven stages of internal development. The earliest, due to Street, was presented briefly and tentatively in [139, 140], 1987 88 papers that focused primarily on strict n categories. The explosion of interest in the area came almost a decade later. As already noted, the basic monograph [66] on 3 categories did not appear until 1995. A wave of definitions of n categories, some still available only in brief ....

....(n 1) categories are defined as n categories with suitable additional structure. In some cases, the recursive definition is not the original one, but rather is obtained by examining a more direct definition or by suitably specializing a definition of # categories. Street s original definition [139], which he has since developed much further, is motivated by the classical simplicial approach to algebraic topology. A complete analysis in low dimensions has been given by Duskin [45] The starting point is that any category can be viewed as a simplicial set via the nerve functor, and that one ....

Ross Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra 49 (1987), no. 3, 283--335.

....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, t n (a) s n (b) to A ( a, b) being carried over a n b) ....

....the nondecreasing monotone functions. One is used to distinguish in this category the morphisms # i : n 1] n] and # i : n 1] n] defined as follows for each n and i = 0, n : # i (j) j if j i j 1 if j # i , # i (j) j if j j 1 if j i It is well known ([16]) that the map [n] induces a functor # # from # to #Cat by setting # i and # i . # s ) of # # s ) is the only face of # as set of vertices ; # r ) of # # r ) is the only face of # as set of vertices. For a face # = with # 1 ....

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R. Street. The algebra of oriented simplexes. J. Pure Appl. Algebra, 49(3):283--335, 1987.

....evolve. The starting point must of course be the present state of the field: we have a dozen definitions of n categories, and a few comparisons. The various definitions are at uneven stages of internal development. The earliest definition, due to Street, was presented briefly and tentatively in [133, 134], 1987 88 papers that focused primarily on strict n categories. The explosion of interest in the area came almost a decade later. As already noted, the basic monograph [62] on 3 categories did not appear until 1995. A wave of definitions of n categories, some still available only in brief ....

....as n categories with additional structure, specified in various ways. In some cases, the recursive definition is not the original one, but rather is obtained 12 by examination of a more direct definition or by suitably specializing a definition of # categories. Street s original definition [133], which he has since developed much further, is motivated by the classical simplicial approach to algebraic topology. A complete analysis in low dimensions has been given by Duskin [38] The starting point is that any category can be viewed as a simplicial set via the nerve functor, and that one ....

Ross Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra 49 (1987), no. 3, 283--335.

....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. for all x; y 2 A, x y is defined as soon as tx = sy, 4. x (y z) x y) z as soon as the ....

....confluences and globes (or, computer scientifically, mutual exclusions) This is constructed through suitable nerve functors. This is made possible because of the choice of a nice category of cubical sets first, and also because simplicial sets can be represented as categories as well (see [62]) In some ways, the branching and confluence nerves describe simplicially all achronal cuts of HDA, as hinted at in Section 7.2. These constructions have a number of advantages over the ones of [25] ffl They are more discriminating (for instance, the room with three barriers example should ....

Street, R., The algebra of oriented simplexes, J. Pure Appl. Algebra (1987), pp. 283--335.

....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 from A to A and with a family of partially de ned 2ary operations ( n ) n 0 where for any n 0, n is a map from f(a; b) 2 A A; t n (a) s n (b)g to A ( a; b) being carried over a n b) ....

....the nondecreasing monotone functions. One is used to distinguishing in this category the morphisms i : n 1] n] and i : n 1] n] de ned as follows for each n and i = 0; n : i (j) j if j i j 1 if j i ; i (j) j if j 6 i j 1 if j i It is well known ([Str87]) that the map [n] 7 n induces a functor from to Cat by setting i 7 i and i 7 i where for any face ( 0 s ) of n 1 , i ( 0 s ) is the only face of n having i f 0 ; s g as set of vertices ; for any face ( 0 ....

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R. Street. The algebra of oriented simplexes. J. Pure Appl. Algebra, 49(3):283{ 335, 1987.

....to the category of pre sheaves Sets # op over a small category #. The corresponding category of precubical sets is isomorphic to the category of pre sheaves Sets # preop over a small category # pre . Theory and Applications of Categories, Vol. 8, No. 12 329 2.3. Definition. 5] [26] [24] A (globular) # category is a set A endowed with two families of maps (d n = s n ) n#0 and (d n = t 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, t n (a) s n (b) to A ( a, b) ....

....gives exactly the branching homology of C (in degree greater than or equal to 1) The map N induces a functor from #Cat 1 to the category Sets # op of simplicial sets. The globular # category # n . Now let us recall the construction of the # category called by Street the n th oriental [26]. We use actually the construction appearing in [17] Let O n be the set of strictly increasing sequences of elements of 0, 1, n . A sequence of length p 1 will be of dimension p. If # = # 0 . # p is a p cell of O n , then we set # j # = # 0 . # # j . ....

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R. Street. The algebra of oriented simplexes. J. Pure Appl. Algebra, 49(3):283--335, 1987.

....categorical structures, which however need to have strict identities. The main step in the proof is an elegant Eckmann Hilton type argument. 1. Introduction In 1987 Street, following Roberts, suggested that that n cohomology should be developed using (weak) n categories as coecient objects [19, 18]. Part of this development will be by clarifying the notion of n stack [12, 14, 3, 4] In 1995, Gordon, Power and Street made signi cant progress by proving a coherence theorem for tricategories, showing that they are triequivalent to Gray categories [13] For higher dimensions, I introduced teisi ....

R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), 283-335.

....3 Definition 6 Write dim X for the greatest i such that there exists an x 2 X which is i dimensional, and say that X is dim X dimensional. The above four equations are very basic to the theory and will be referred to as the st equations. They were first used in the definition of categories in [7]. The letters s and t are chosen as mnemonics for source and target. Each element x of an multigraph will have for each i an i dimensional source s i fxg and an i dimensional target t i fxg. If x is n dimensional we will draw it as oriented from its (n Gamma 1) dimensional source to its (n ....

R.H. Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra, 49 (1987) 283--335. 7

....maximal chains in B(n;k) by certain elementary equivalences. The posets B(n;k) have rank functions and a unique minimal and a unique maximal element. Another interpretation of higher Bruhat orders was given by Kapranov and Voevodsky [15] in terms of pasting schemes [13] and strict n categories [20]. The combinatorial n cube L n has a (more or less canonical) structure of a well formed loop free pasting scheme [7, Section 3 3] denote the free strict n category on L n by I n . Because I n is the free strict n category on a well formed loop free pasting scheme, it is very structured: the ....

R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), 283--335.

....categorical structures, which however need to have strict identities. The main step in the proof is an elegant Eckmann Hilton type argument. 1 Introduction In 1987 Street, following Roberts, suggested that that n cohomology should be developed using (weak) n categories as coecient objects [19, 18]. Part of this development will be by clarifying the notion of n stack [12, 14, 3, 4] In 1995, Gordon, Power and Street made signi cant progress by proving a coherence theorem for tricategories, showing that they are triequivalent to Gray categories [13] For higher dimensions, I introduced teisi ....

R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), 283-335.

....set. And their associated set of execution paths and homotopies between them have a natural structure of globular category. All these ideas already appear in [Pra91] Pratt uses the term of n complex which is in fact nothing else but a small n category. We use the notations of [Ste91] and [Str87] for the following definition. 5 ff fi A B u v Figure 2: A 3 dimensional hole ff ff A BB ff B fi Figure 3: Composition of two 2 morphims 6 The following definition already appears in [BH81a] Definition 2.2. An category is a set A endowed with two families of maps (s n = d ....

.... g . m m m m m m m m m m m m m m m m h 66 Figure 12: A pasting scheme We only want here to recall the construction of the free category I n generated by the faces of the n cube. For more details see [Cra95] for an analogous construction for simplices see [Str87], and for some explicit calculations on I n see [Ait86] Set n = f1; ng and let n be the set of maps from n to f Gamma; 0; g. We say that an element x of n is of dimension p if x Gamma1 (0) is a set of p elements. We can identify the elements of n with the words of length n in ....

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Ross Street. The algebra of oriented simplexes. Journal of Pure and Applied Algebra, 49:283--335, 1987.

....a simplicial complex as a category with the objects the 0 faces and the morphisms the 1 faces. If the simplicial complex involves simplexes of dimension higher than one, it has actually the structure of a higher category by attaching n morphisms to the n faces. The way this can be done is given in [20] . Here, for technical reasons, an orientation of the faces is assumed but since this is not an inherent structure of the complex, we assume that the morphisms (of all levels) are invertible, thereby dispensing of it again. According to the universality postulate for quantum mechanics of Part I, ....

....of a simplicial complex is encoded in its cohomology. Therefore, we have to consider not simply the representations in a higher quantized vector space but simplicial cohomology theory with values in a higher quantized vector space. Categorical cohomology theory is considered in ( 19] and [20] ) where n th cohomology takes values in an n category. This is just what appears here. Let us shortly explain the idea (for the details, see the above cited literature) In abelian first cohomology (i.e. values in an abelian group) we have the 1 cocycle condition stating that a Gamma b c = 0 ....

R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49, 283-335 (1987).

....satisfy various laws. As with 2 categories, we can try to impose these laws either strictly or weakly. Strict n categories have been understood for quite some time now [23, 28] but more interesting for us are the weak ones. Various definitions of weak n category are currently under active study [5, 10, 36, 57, 58, 61, 62, 63], and we discuss our own in Section 5. Here, however, we wish to sketch the main challenges any theory of weak n categories must face, and some of the richness inherent in the notion of weak n category. Nota bene: Throughout the rest of this paper, n category will mean weak n category unless ....

....quadruple of 1 cells, one obtains the pentagon coherence law by considering a suitable 4 simplex. In fact, all the 16 higher coherence laws for associativity, which Stasheff [59] organized into polyhedra called associahedra , have been obtained from higher dimensional simplices by Street [61] in his simplicial approach to categories. If we take the liberty of calling Kan complexes groupoids , we can set up a correspondence between groupoids and homotopy types as follows. Given a topological space X, we can form an groupoid Pi(X) whose j cells are all the continuous maps ....

R. Street, The algebra of oriented simplexes, Jour. Pure Appl. Alg. 49 (1987), 283-335.

....positive dimension and p 1. Then s 1 s p x = s 1 x therefore s p x cannot be 0 dimensional. If x p y then s 1 (x p y) s 1 x if p = 1 and if p 1 for two di erent reasons. Therefore x p y cannot be 0 dimensional. Here are the conventions of notations for the sequel (see [May67] BH81] [Str87] [Ste91] Gau99] Gau00] 1. Cat : category of globular categories 2. Cat 1 : category of non 1 contracting globular categories with non 1 contracting functors 3. Sets : category of sets 4. Sets op : category of simplicial sets 5. Sets op : category of cubical sets (without ....

R. Street. The algebra of oriented simplexes. Journal of Pure and Applied Algebra, 49:283-335, 1987. 19

....the composition of higher dimensional homotopies (cf. Figure 3(a) As already noticed in [21] the axioms of globular categories encode the geometric properties of compositions of execution paths and homotopies between them. Let us recall the de nition of category in three steps (see [5] [26] [24] for more details) De nition 3.1 A 1 category is a pair (A; s; t) satisfying the following axioms : i) A is a set (ii) s and t are set maps from A to A respectively called the source map and the target map (iii) for x; y 2 A, x y is de ned as soon as tx = sy (iv) x (y z) x ....

.... the category I n associated to the n dimensional cube and that of the n dimensional simplex (this latter is denoted by n ) For the cube, the older attempt of constructing a structure of category on the set of faces of the n cube is maybe in [1] As for the n simplex, the seminal work is [26]. Since then, many constructions have been proposed. Both families of categories can be characterized in the same way. The rst step consists of labelling all faces of the n cube and of the n simplex. For the n cube, this consists of considering all words of length n in the alphabet f; 0; g, ....

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Street, R., The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), pp. 283-335.

.... g 4. ff i ffl j = ffl j ff i Gamma1 for i j 6 n and ff 2 f Gamma; g 5. ff i ffl i = Id The corresponding category of cubical sets, with an obvious definition of its morphisms, is isomorphic to the category of presheaves Sets op over a small category . Definition 2.2. BH81a] [Str87] [Ste91] A (globular) category is a set A endowed with two families of maps (d Gamma n = s n ) n 0 and (d n = t 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 f(a; b) 2 A Theta A; t n (a) s n (b)g to A ( a; ....

....by C[1] the unique category such that its set of n morphisms is exactly the set of (n 1) morphisms of C for any n 0 with an obvious definition of the source and target maps and of the composition laws. Now let us recall the construction of the category called by Street the n th oriental [Str87]. We use actually the construction appearing in [KV91] Let O n be the set of strictly increasing sequences of elements of f0; 1; ng. A sequence of length p 1 will be of dimension p. If oe = foe 0 : oe p g is a p cell of O n , then we set j oe = foe 0 : b oe j : ....

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R. Street. The algebra of oriented simplexes. Journal of Pure and Applied Algebra, 49:283--335, 1987.

....a cell, and they all become explicit in their simplicial forms if one makes composition into an operation of the traditional sort by arbitrarily choosing an extension of every horn. It is tempting, therefore, to develop a simplicial approach to weak n categories. This was done by Street [29], who actually dealt with weak categories. Like Kan complexes, these are simplicial sets. However, only certain admissible horns, having the correct sort of orientation, are required to have extensions. For example, we do not require the horn shown in Figure 2 to have an extension, since the ....

....an 0 ary morphism riding A(g 0 ) which we call A(G) a(g 1 ) a(g k ) In item 3 we require this to equal a(g 0 ) ut 5 Conclusions In addition to our approach to weak n categories, there are a number of others. We have already mentioned Street s original simplicial approach [29]. After a sketch of our definition appeared [5] Makkai has begun studying it, and a modified version has been developed by Makkai, Hermida, and Power, but the details of this have not yet been published. Independently, Tamsamani [30] developed an approach using multisimplicial sets: simplicial ....

R. Street, The algebra of oriented simplexes, Jour. Pure Appl. Alg. 49 (1987), 283-335.

.... on the relation between the local and global theory can be found in [16] The category of 2 groupoids enjoys a closed model structure, as shown by Moerdijk and Svensson in [28] It is related to the closed model structure on the category of simplicial sets via the 2 categorical nerve functor (see [32]) and its left adjoint. This last relation induces an adjointness at the homotopy level, which gives a categorical description of homotopy 2 types: every topological space with trivial homotopy groups in dimensions 3 is homotopy equivalent to the classifying space of a 2 groupoid. Thus, ....

....necessary to describe the closed model structure on 2Grpd in full detail for two reasons. First, because full detail is already supplied by [28] and second because I will only use a very specific part. There is an adjoint functor pair sSets W 2Grpd N where N is the 2 categorical nerve (see [32]) N(G) G 0 G 1 G 2 Theta G 1 (G 1 Theta G 0 G 1 ) Delta Delta Delta ; 1) and W is the Whitehead 2 groupoid described in [28] W (X) X 0 F (X 1 ) F (X 2 ) X 3 (2) where F describes a free construction, i.e. F (X i ) consists of finite formal composites of i cells of X and their formal ....

R. Street. The algebra of oriented simplexes. J. Pure Appl. Algebra, 49:283-- 335, 1987.

.... of an category 57 11.1 Generators : 57 11.2 Relations : 58 11.3 The category (G C ; R C ) 59 References 62 1 Introduction In [19], Street, following Roberts [15] suggests that n cohomology should be developed using n categories as coefficient objects. As already suggested there, weaker versions would eventually do a better job at this. Still, n categories are worth studying, partly as an exercise for the weaker case, but ....

....computer science [14] Because of the combinatorics involved, it is quite difficult to say precisely what a composable diagram in an n category should be. Closely related to this is the problem of describing free n categories. Street defined the free n category on an n simplex: his n th oriental [19]. Motivated by this, Johnson defined pasting schemes, and proved a pasting theorem for them [8] Other suggested structures are Power s pasting schemes [12, 13] defined geometrically, Street s parity complexes [20] Steiner s directed complexes [17] and Street s inductively defined n computads ....

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R. Street. The algebra of oriented simplexes. J. Pure Appl. Algebra, 49:283-- 335, 1987.

.... following universal property: For every functor M : E S and natural transformation ff: X Delta MK, ff factors through via ff = Delta Koe for a unique oe: L Delta M , 3, 6] A left Kan extension, when it exists, gives answers to a variety of questions of presentation and design, [2, 4, 6]. There is no general guarantee that the left Kan extension does exist, but if it does it is unique up to equivalence. When S = Finset, the category of finite sets and functions between them, there is a procedure to compute the left Kan extension due to Sean Carmody and R.F.C. Walters and sketched ....

....between the terms are arrows between the arrows, or 2 cells. The theory of such structures is the theory of 2 categories. We give the definition shortly, but as we are going to generalize, we first give a definition of an ordinary category as a 1 category. The next four definitions are from [4]. definition. A 1 category (A; s; t; is a set A of arrows structured by a source function s: A A, a target function t: A A, and a composition defined on the set of composable pairs f(a; b) 2 A Theta A j t(a) s(b)g with this structuring data satisfying ss = ts = s; tt = st = t; s(a ....

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R. Street, The algebra of oriented simplexes, J. Pure Appl. Alg. 49(1987) 283--330.

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R. Street, The Algebra of Oriented Simplexes, J. Pure Appl. Algebra 49(1987), 283-335. 52

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R. Street, The Algebra of Oriented Simplexes, J. Pure Appl. Algebra 49(1987.

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R. Street, The Algebra of Oriented Simplexes, J. Pure Appl. Algebra 49(1987.

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R. Street. The algebra of oriented simplexes. J. Pure Appl. Algebra, 49(3):283{ 335, 1987.

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R.H. Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra, 49 (1987) 283--335.

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