Acyclicity conditions on pasting diagrams (Q6631561)

A category is a fundamental object in modern mathematics with a wide range of applications in algebra, geometry, and topology. By definition a category consists of objects and morphisms between them, and is equipped with an associative and unital composition. In particular, to every category we can associate a \textit{directed graph}, obtained by ``forgetting'' the composition. On the other side, to each directed graph one can associate a category with compositions freely generated by the edges in the graph, known as the \textit{free category}. One important class of directed graphs are the \textit{acyclic graphs}, which are essentially graphs without cycles. Indeed, we can characterize the structure of a composition in a category via acyclic graphs. For example, the directed graph with \(n+1\) vertices \(\{0,\dots, n\}\) and \(n\) edges going from \(i\) to \(i+1\) results in the free category \([n]\), and a functor out of \([n]\) is precisely the data of \(n\) composable morphisms.\N\NCategory theory has been generalized to higher categories, and specifically \textit{strict \(n\)-categories} (including the infinite case \(n = \omega\)), which beyond objects and morphisms includes higher morphisms, usually called \(k\)-morphisms (for \(1 \leq k \leq n\)), which are suitably composable. Similar to the \(1\)-categorical situation, every higher category comes with an underlying generalized directed graph, called a \textit{polygraph}. Similar to the \(1\)-categorical situation, every polygraph generates an \(n\)-category, called an \textit{\(n\)-computad}. Similar to above, certain classes of such polygraphs, sometimes called \textit{acyclic polygraphs} or \textit{pasting diagrams}, play a particular role, as they can help us understand the nature of compositions in a strict \(n\)-category. This has resulted in a variety of \textit{pasting theorems}, which prove that given an appropriate pasting diagram, there is an essentially unique way to construct a composition. Such results can be found throughout the literature, from figures such as \textit{R. Street} [Cah. Topologie Géom. Différ. Catégoriques 32, No. 4, 315--343 (1991; Zbl 0760.18011)] and \textit{A. J. Power} [Lect. Notes Math. 1488, 326--358 (1991; Zbl 0736.18004)], up until more recent results [\textit{P. Hackney} et al., Trans. Am. Math. Soc. 376, No. 1, 555--597 (2023; Zbl 1505.18031)]. While these results are very powerful, their focus on pasting diagrams excludes some examples of interest (for example diagrams relevant in the study of adjunctions).\N\NIn this paper the authors consider several generalizations of the standard notion of acyclicity. In particular, they introduce the notion of \textit{frame-acyclicity} and \textit{(strong) dimension-wise acyclicity}. They prove that these more general notions still exhibit many desirable properties of more traditional acyclic diagrams. Moreover, in \textit{Section 4}, and particularly \textit{Theorem 4.29}, they study how these notions relate to more common notions of acyclicity and polygraphs, and particularly work of \textit{R. Steiner} [Appl. Categ. Struct. 1, No. 3, 247--284 (1993; Zbl 0804.18005)]. Finally, in \textit{Section 6} the authors exploit the more general definition to prove that, in contrast to usual acyclicity, these generalized notions are stable under many common categorical constructions, such as joins and Gray tensor products (\textit{Proposition 6.12}).