Cubical sets and their site

Marco Grandis and Luca Mauri

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.

Keywords: Simplicial sets, cubical sets, monoidal categories, algebraic theories, generators and relations, word problem, classifying categories

2000 MSC: 18G30, 55U10, 18D10, 18C10, 20F05, 20F10

Theory and Applications of Categories , Vol. 11, 2003, No. 8, pp 185-211.

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