Geodesic
Cubical sets and their site
Theory and applications of categories, Tome 11 (2003), pp. 185-211 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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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.

Classification : 18G30, 55U10, 18D10, 18C10, 20F05, 20F10
Keywords: Simplicial sets, cubical sets, monoidal categories, algebraic theories, generators and relations, word problem, classifying categories 
@article{TAC_2003_11_a7,
     author = {Marco Grandis and Luca Mauri},
     title = {Cubical sets and their site},
     journal = {Theory and applications of categories},
     pages = {185--211},
     year = {2003},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2003_11_a7/}
}
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Marco Grandis; Luca Mauri. Cubical sets and their site. Theory and applications of categories, Tome 11 (2003), pp. 185-211. http://geodesic.mathdoc.fr/item/TAC_2003_11_a7/