Free precategories as presheaf categories (Q6575458)

The notion of \textit{polygraph}, also known as \textit{computad}, was introduced by Street and Burroni as a generalization of the notion of presentation for strict \(n\)-categories. From an algebraic viewpoint, they constitute the right notion of free \(n\)-category in the sense that they have been established as being the cofibrant objects in the folk model structure on the category of \(n\)-categories, based on the construction of resolutions or cofibrant replacements of categories of interest.\N\NFrom a categorical viewpoint, strict polygraphs are adapted to strict \(n\)-categories without being equivalent to weak \(n\)-categories, which are the real object of interest. Besides, strict polygraphs do not form a presheaf category. From a standpoint of rewriting, polygraphs lack the fundamental property for rewriting that a finite rewriting system has a finite number of critical branchings [\textit{Y. Lafont}, J. Pure Appl. Algebra 184, No. 2--3, 257--310 (2003; Zbl 1037.94015); \textit{Y. Guiraud} and \textit{P. Malbos}, Theory Appl. Categ. 22, 420--478 (2009; Zbl 1190.18002)]. For these reasons, it seems natural to investigate the framework of \(n\)-precategories, which correspond to \textit{R. Street}'s sesquicategories [in: Handbook of algebra. Volume 1. Amsterdam: North-Holland. 529--577 (1996; Zbl 0854.18001)] in dimension 2. The authors [\textit{S. Forest} and \textit{S. Mimram}, LIPIcs -- Leibniz Int. Proc. Inform. 108, Article 15, 16 p. (2018; Zbl 1462.68095); \textit{S. Forest} and \textit{S. Mimram}, Math. Struct. Comput. Sci. 32, No. 5, 574--647 (2022; Zbl 1517.68162)] have defined an associated notion of polygraphs, developing a theory of rewriting in this setting.\N\NThis paper further investigates the category of \(n\)-categories for \(n\)-precategories, showing that they form a presheaf category. The authors' proof is based on the characterization of concrete presheaf categories given by \textit{M. Makkai} [\url{https://www.math.mcgill.ca/makkai/Foundations\%20seminar/Item2\_IntroWordProble.pdf}]. Simultaneously and independently, another proof of this result has been given by \textit{M. Araújo} [Cah. Topol. Géom. Différ. Catég. 65, No. 2, 111--149 (2024; Zbl 1543.18017)]. It should be mentioned that a notion of polygraphs for weak categories has been developed and shown to be a presheaf category [\textit{C. J. Dean} et al., Adv. Math. 450, Article ID 109739, 70 p. (2024; Zbl 1541.18030)].\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 1] introduces precategories and associated polygraphs.\N\N\item[\S 2] shows that functors between precategories induced by polygraphs have the significant property of being Conduché.\N\N\item[\S 3] recalls Makkai's theorem characterizing presheaf categories [\url{https://www.math.mcgill.ca/makkai/Foundations\%20seminar/Item2\_IntroWordProble.pdf}].\N\N\item[\S 4] considers the support of a cell in a precategory, which allows of retrieving some properties of a morphism of polygraphs \(F\)\ from the associated free functor \(F^{\prime}\).\N\N\item[\S 5] defines and studies polyplexes, which are shapes parametrizing compositions in precategories.\N\N\item[\S 6] shows the main theorem of this paper that polygraphs form a presheaf category.\N\N\item[\S 7] derives adjunction together with an associated generic-free factorization for precategories, which gives a more conceptual view of the good syntactical properties of precategories.\N\N\item[\S 8] presents two open questions on homotopical aspects of polygraphs of precategories:\N\N\begin{itemize}\N\item[(1)] whether polygraphs are the cofibrant objects for a reasonable model structure on precategories, and\N\N\item[(2)] whether the presheaf category of polygraphs is able to model homotopy types. \N\end{itemize}\N\N\end{itemize}