## Cubical Sets And Their Site (2003)

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Venue: | Theory Appl. Categ |

Citations: | 15 - 3 self |

### BibTeX

@ARTICLE{Grandis03cubicalsets,

author = {Marco Grandis and Luca Mauri},

title = {Cubical Sets And Their Site},

journal = {Theory Appl. Categ},

year = {2003},

volume = {11},

pages = {185--211}

}

### OpenURL

### Abstract

Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid ; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K.

### Citations

924 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ... symmetric in general. References on cubical sets have been cited above; for simplicial sets see [30, 10, 13]. The characterisations of the category of finite ordinals can be found in Mac Lane's text =-=[27]-=-; finite cardinals, the site of (augmented) symmetric simplicial sets, have been similarly characterised in [17]. For monoidal categories, see [27] and Kelly's book [23]. Links with PRO's, PROP's, mon... |

379 | Basic Concepts of Enriched Category Theory
- Kelly
- 1982
(Show Context)
Citation Context ...an be found in Mac Lane's text [27]; finite cardinals, the site of (augmented) symmetric simplicial sets, have been similarly characterised in [17]. For monoidal categories, see [27] and Kelly's book =-=[23]-=-. Links with PRO's, PROP's, monoidal theories and rewrite systems will be given in the text. Outline. The classical notion of an abstract interval in a monoidal category (with two faces and a degenera... |

365 |
Confluent reductions: Abstract properties and applications to term rewriting systems
- Huet
- 1980
(Show Context)
Citation Context ...st, we want to mention a relevant information due to the Referee. An alternative proof to the present one can be based on the theory of rewrite systems, originated in the framework of #-calculus, cf. =-=[11, 19]-=-: one would reduce the argument to showing that all critical pairs (#, # # ) are joinable, for suitable pairs of composed connections. This new proof would be clearer and placed in a well-established ... |

266 | Simplicial Objects in Algebraic Topology - May - 1967 |

242 |
Braided tensor categories
- Joyal, Street
- 1983
(Show Context)
Citation Context ...ric cubical monoid is a cubical monoid A as in (13) with a symmetry (or interchange) # # : A# A # A# A, (25) under the following axioms, added to (14) (the second is a Yang-Baxter condition on #, see =-=[24]-=- and references therein) ## = 1, (## A)(A# #)(## A) = (A# #)(## A)(A# #), (## A)# = A# #, #(# ## A) = A# # # , # # # = # # , #(# ## A) = (A# # # )(## A)(A# #). (26) Higher interchanges are constructed... |

191 |
Homotopy invariant algebraic structures on topological spaces
- Boardman, Vogt
- 1973
(Show Context)
Citation Context ...he theories. The exposition is modelled on the case of algebraic theories in cartesian categories [9, 20, 25] with the necessary generalisations; some of the ideas behind the analysis can be found in =-=[2]-=- and [21]. For conciseness, we restrict here to the framework needed to discuss the cubical sites. Thus, the signatures are single sorted and the languages only allow weakening and exchange as structu... |

156 |
Generators and Relations for Discrete Groups
- Coxeter, Moser
- 1980
(Show Context)
Citation Context ...nsor power A n . (Recall that S n , the group of automorphisms of the set {1, ...n}, is generated by the main transpositions # i = (i, i + 1), for 1 # isn, under the relations (28); see Coxeter-Moser =-=[7]-=-, 6.2; or Johnson [22], Section 5, Thm. 3.) 196 MARCO GRANDIS AND LUCA MAURI 7. Interchanges and the extended cubical site Let K be the subcategory of Set consisting of the elementary cubes 2 n , toge... |

124 | On the algebra of cubes
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...ed 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 ... |

112 |
Categorical logic and type theory
- Jacobs
- 1999
(Show Context)
Citation Context ...ver the universal property of the cubical sites and to exhibit them as presentation-free versions of the theories. The exposition is modelled on the case of algebraic theories in cartesian categories =-=[9, 20, 25]-=- with the necessary generalisations; some of the ideas behind the analysis can be found in [2] and [21]. For conciseness, we restrict here to the framework needed to discuss the cubical sites. Thus, t... |

112 |
An Introduction to the Theory of Groups
- Rotman
- 1984
(Show Context)
Citation Context ...ITE 191 4.3. Remark. (a) Our results, Lemma 4.1 and Theorem 4.2, not only give a reduced form for the maps of I, but solve the word problem for I, as presented above, by generators and relations (cf. =-=[31, 3]). In-=- fact we have proved that any (categorically well formed) word in faces and degeneracies can be rewritten in a unique canonical form, by applying finitely many times our relations (5), as "rewrit... |

80 |
Categories for Types
- Crole
- 1993
(Show Context)
Citation Context ...[j # , j ## ] where all points are j-indices with the same #-weight, except possibly j ## which need not be a j-index; (b) the remaining singletons. Thus, in case (32), we have j = (1, 2, 4, 5, 8) in =-=[1, 9]-=-, with the following weights # and decomposition D 9 (j, #) 1 2 3 4 5 6 7 8 9 - + + + - # # # # # # # # # # D 9 (j, #) (36) the corresponding S 9 (j, #) is precisely the subgroup of S 9 considered abo... |

77 | Colimit Theorems for Relative Homotopy Groups
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...ed 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 ... |

60 |
Simplicial homotopy theory
- Curtis
- 1971
(Show Context)
Citation Context ...noidal functor I # : K # Cat(C, C), where Cat(C, C) is monoidal with respect to composition, though not symmetric in general. References on cubical sets have been cited above; for simplicial sets see =-=[30, 10, 13]-=-. The characterisations of the category of finite ordinals can be found in Mac Lane's text [27]; finite cardinals, the site of (augmented) symmetric simplicial sets, have been similarly characterised ... |

49 | Homotopy invariants of higher dimensional categories and concurrency in computer science
- Gaucher
- 2000
(Show Context)
Citation Context ..., (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 rev... |

46 | Multiple categories: the equivalence of a globular and a cubical approach
- Al-Agl, Brown, et al.
(Show Context)
Citation Context ..., (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 rev... |

43 | Tensor products and homotopies for ω-groupoids and crossed complexes
- Brown, Higgins
- 1987
(Show Context)
Citation Context ...ed 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 ... |

36 |
First-order Categorical Logic
- Makkai, Reyes
- 1977
(Show Context)
Citation Context ...ver the universal property of the cubical sites and to exhibit them as presentation-free versions of the theories. The exposition is modelled on the case of algebraic theories in cartesian categories =-=[9, 20, 25]-=- with the necessary generalisations; some of the ideas behind the analysis can be found in [2] and [21]. For conciseness, we restrict here to the framework needed to discuss the cubical sites. Thus, t... |

29 |
Simplicial Homotopy Theory, Birkhäuser
- Goerss, Jardine
- 1999
(Show Context)
Citation Context ...noidal functor I # : K # Cat(C, C), where Cat(C, C) is monoidal with respect to composition, though not symmetric in general. References on cubical sets have been cited above; for simplicial sets see =-=[30, 10, 13]-=-. The characterisations of the category of finite ordinals can be found in Mac Lane's text [27]; finite cardinals, the site of (augmented) symmetric simplicial sets, have been similarly characterised ... |

28 |
Singular Homology Theory
- Massey
- 1980
(Show Context)
Citation Context ...cubes are closed under products, while products of tetrahedra have to be "covered" with tetrahedra; this advantage appears clearly when studying singular homology based on cubical chains, (c=-=f. 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 th... |

28 |
The petit topos of globular sets
- Street
(Show Context)
Citation Context ...ar case, one can use the integral traces of the standard discs, D n # Z n = {e 1 , . . . , e n } (coinciding with the traces of the standard octahedra); but this will not be treated here (one can see =-=[32]-=-). 3. The pointwise embedding of a discrete site Let C be a small category with a terminal object 1. A point (or global element, or global section) of a C-object C is a map x : 1 # C; the set of such ... |

27 |
On combinatorial models for higher dimensional homotopies
- Crans
- 1995
(Show Context)
Citation Context ...results will be proved, much less trivially, for wider cubical sites --- J and K--- in the next sections. (b) A di#erent global description of I, as embedded in Set op , can be found in Crans' thesis =-=[8]-=-, Section 3.2. In fact, an I-map f : 2 m # 2 n can be represented by a mapping f # : n # m # {-, +} (where n = {1, . . . , n}) which reflects the order of m, as in the following example f : 2 5 # 2 7 ... |

25 |
Rewrite systems," in Handbook of Theoretical Computer Science
- Dershowitz, Jouannaud
- 1989
(Show Context)
Citation Context ...st, we want to mention a relevant information due to the Referee. An alternative proof to the present one can be based on the theory of rewrite systems, originated in the framework of #-calculus, cf. =-=[11, 19]-=-: one would reduce the argument to showing that all critical pairs (#, # # ) are joinable, for suitable pairs of composed connections. This new proof would be clearer and placed in a well-established ... |

16 |
Tensor products and homotopies for � -groupoids and crossed complexes
- Brown, Higgins
- 1987
(Show Context)
Citation Context |

15 |
Categorically algebraic foundations for homotopical algebra
- Grandis
- 1997
(Show Context)
Citation Context ...to copy for private use granted. 185 186 MARCO GRANDIS AND LUCA MAURI of the first named author on homotopy theory, based on a cylinder (or path) functor and its structure of cubical (co)monad (e.g., =-=[14, 15, 16]-=-). All these maps have their origin in the standard topological interval I = [0, 1] and its structure as an involutive lattice (cf. (12)). Here, we give characterisations, similar to (a)--(c) above, f... |

13 |
Cubical monads and their symmetries
- Grandis
- 1993
(Show Context)
Citation Context ...to copy for private use granted. 185 186 MARCO GRANDIS AND LUCA MAURI of the first named author on homotopy theory, based on a cylinder (or path) functor and its structure of cubical (co)monad (e.g., =-=[14, 15, 16]-=-). All these maps have their origin in the standard topological interval I = [0, 1] and its structure as an involutive lattice (cf. (12)). Here, we give characterisations, similar to (a)--(c) above, f... |

11 | Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
- Grandis
(Show Context)
Citation Context ...= lim #- (X : C op # Set), (2) give the global section functor # = #y of C, and it is easy to prove that # is faithful if and only if all the representable presheaves on C are simple (in the sense of =-=[18]-=-, 1.3). The simplicial sites have pointwise embedding, the ordinary one. We prove below that this is also true for the cubical sites I, J, K, which will thus be embedded in Set with objects 2 n (since... |

11 |
Languages for monoidal categories
- Jay
- 1989
(Show Context)
Citation Context ... strict monoidal category M with a faithful strict monoidal functor #S n # M, bijective on objects; the category #S n is the disjoint union of the groups S n , with the obvious monoidal structure; cf =-=[26, 21]-=-.) Observe that the object 2 itself with the obvious operations is a symmetric cubical monoid in K, which will be called the generic symmetric cubical monoid. To determine a canonical form for K-maps,... |

9 |
Using rewriting systems to compute left kan extensions and induced actions of categories
- Brown, Heyworth
- 2000
(Show Context)
Citation Context ...ITE 191 4.3. Remark. (a) Our results, Lemma 4.1 and Theorem 4.2, not only give a reduced form for the maps of I, but solve the word problem for I, as presented above, by generators and relations (cf. =-=[31, 3]). In-=- fact we have proved that any (categorically well formed) word in faces and degeneracies can be rewritten in a unique canonical form, by applying finitely many times our relations (5), as "rewrit... |

6 |
Topics in the theory of presentation of groups
- Johnson
- 1980
(Show Context)
Citation Context ...all that S n , the group of automorphisms of the set {1, ...n}, is generated by the main transpositions # i = (i, i + 1), for 1 # isn, under the relations (28); see Coxeter-Moser [7], 6.2; or Johnson =-=[22]-=-, Section 5, Thm. 3.) 196 MARCO GRANDIS AND LUCA MAURI 7. Interchanges and the extended cubical site Let K be the subcategory of Set consisting of the elementary cubes 2 n , together with the maps gen... |

4 | Finite sets and symmetric simplicial sets
- Grandis
(Show Context)
Citation Context ...he characterisations of the category of finite ordinals can be found in Mac Lane's text [27]; finite cardinals, the site of (augmented) symmetric simplicial sets, have been similarly characterised in =-=[17]-=-. For monoidal categories, see [27] and Kelly's book [23]. Links with PRO's, PROP's, monoidal theories and rewrite systems will be given in the text. Outline. The classical notion of an abstract inter... |

3 |
On the homotopy structure of strongly homotopy associative algebras
- Grandis
- 1999
(Show Context)
Citation Context ...to copy for private use granted. 185 186 MARCO GRANDIS AND LUCA MAURI of the first named author on homotopy theory, based on a cylinder (or path) functor and its structure of cubical (co)monad (e.g., =-=[14, 15, 16]-=-). All these maps have their origin in the standard topological interval I = [0, 1] and its structure as an involutive lattice (cf. (12)). Here, we give characterisations, similar to (a)--(c) above, f... |

3 |
Categorical Logic and Type Theory North Holland
- Jacobs
- 1999
(Show Context)
Citation Context ...ver the universal property of the cubical sites and to exhibit them as presentation-free versions of the theories. The exposition is modelled on the case of algebraic theories in cartesian categories =-=[9, 20, 25]-=- with the necessary generalisations; some of the ideas behind the analysis can be found in [2] and [21]. For conciseness, we restrict here to the framework needed to discuss the cubical sites. Thus, t... |

3 |
Categories for the working mathematician
- Soc, Providence
- 1994
(Show Context)
Citation Context ...roduct in the subcategory (exponents denote tensor powers). (Note that I is a PRO, i.e. a strict monoidal category whose monoid of objects is isomorphic to the additive monoid of natural numbers; cf. =-=[26, 2]-=-.) The object 2 is a bipointed object (both in (Set, *) and (I, \Lambda )), with (basic) faces ffiff and degeneracy " ffiff : 1 ! 2, " : 2 ! 1, "ffiff = 1 (ff = +-). (3) Higher faces and degeneracies ... |

2 |
Cubical groups which are
- Tonks
- 1992
(Show Context)
Citation Context ..., (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 rev... |

1 |
Algebraic theories in monoidal categories
- Mauri
(Show Context)
Citation Context ... Section 10 cannot be applied to prove that !K is the classifying category of !K. Nevertheless, a classifying category of !K can be constructed syntactically and will be proved to be equivalent to !K =-=[29]-=-. It follows that !K, defined above as a subcategory of Set, is the category generated by faces, degeneracies, connections, interchanges and reversions under 202 MARCO GRANDIS AND LUCA MAURI the relat... |

1 |
Categories for types, Cambridge University Press 1993. [10] E.B. Curtis, Simplicial homotopy theory, Adv
- Crole
- 1971
(Show Context)
Citation Context ...ype [j0, j00] where all points are j-indices with the same ff-weight, except possibly j00 which need not be a j-index; (b) the remaining singletons. Thus, in case (32), we have j = (1, 2, 4, 5, 8) in =-=[1, 9]-=-, with the following weights ff and decomposition D9(j, ff) 1 2 3 4 5 6 7 8 9 - + + + - ffffi ffi ffi ffi ffi ffi ffi ffi ffi D9(j, ff) (36) the corresponding S9(j, ff) is precisely the subgroup of S9... |

1 |
Simplicial homotopy theory, Birkh"auser 1999. [14] M. Grandis, Cubical monads and their symmetries, in
- Goerss, Jardine
- 1993
(Show Context)
Citation Context ...noidal functor I * : K ! Cat(C, C), where Cat(C, C) is monoidal with respect to composition, though not symmetric in general. References on cubical sets have been cited above; for simplicial sets see =-=[30, 10, 13]-=-. The characterisations of the category of finite ordinals can be found in Mac Lane's text [27]; finite cardinals, the site of (augmented) symmetric simplicial sets, have been similarly characterised ... |