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Finite sets and symmetric simplicial sets

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Marco Grandis

The category of finite cardinals (or, equivalently, of finite sets)
is the symmetric analogue of the category of finite ordinals, and the ground
category of a relevant category of presheaves, the * augmented symmetric
simplicial sets.* We prove here that this ground category has
characterisations similar to the classical ones for the category of finite
ordinals, by the existence of a universal symmetric monoid, or by generators
and relations. The latter provides a definition of symmetric simplicial sets
by faces, degeneracies and transpositions, under suitable relations.

Keywords: Simplicial sets, monoidal categories, generators and relations.

2000 MSC: 18G30, 55U10, 18D10, 20F05.

*Theory and Applications of Categories*, Vol. 8, 2001, No. 8, pp 244-252.

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