#
Free precategories as presheaf categories

##
Simon Forest and Samuel Mimram

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.

http://www.tac.mta.ca/tac/volumes/41/24/41-24.pdf

TAC Home