Precategories generalize both the notions of strict n-category and sesquicategory: their definition is essentially the same as the one of strict n-categories, excepting that the various interchange laws are not required to hold. Those have been proposed as a framework in which one can express semi-strict definitions of weak higher categories. In particular, in dimension 3, Gray categories are particular 3-precategories which have been shown to be equivalent to tricategories. In this article, we are mostly interested in free precategories. Those can be presented by generators and relations, using an appropriate variation on the notion of polygraph (aka computad), and earlier works have shown that the theory of rewriting can be generalized to this setting, enjoying most of the fundamental constructions and properties which can be found in the traditional theory: with respect to this, polygraphs for precategories are much better behaved than their counterpart for strict categories. We further study here why this is the case, by providing several results which show that precategories and their associated polygraphs bear properties which ensure that we have a good syntax for those. In particular, we show that the category of polygraphs for precategories form a presheaf category.
Keywords: precategory, polygraph, Conduché functor, polyplex, parametric adjunction
2020 MSC: 18N99
Theory and Applications of Categories, Vol. 41, 2024, No. 24, pp 785-824.
Published 2024-07-17.
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