The universal property of the multitude of trees
In about 1975, Grothendieck put forward an idea for extending the link between covering spaces and \(G\)-sets, for \(G\) the fundamental group of the base space being considered. The attack he proposed involved (truncated) algebraic models of homotopy types and stacks of such. The models he proposed were globular sets and the corresponding globular \(\infty\)-categories. No details were worked out and even in his long notes on Pursuing Stacks, which emerged nine years later, that particular part of his idea did not get worked on in any detail. At about the same time, mid 1970s, various other types of \(\infty\)- or \(\omega\)-categories were coming to light in work by Brown and Higgins, Roberts and others. Understanding how to extend the well known theory of bicategories to higher-dimensional weak \(\omega\)-categories looked more and more important. The search took nearly twenty years. In the mid 1990s various formulations of the concept became available. One due to the first author of this paper explicitly uses the globular approach. Globular sets are somewhat akin to simplicial sets, but the standard \(n\)-dimensional object is an \(n\)-ball with one top dimensional cell, \(t\), \((n-1)\)-dimensional faces which share two \((n-2)\)-dimensional ones, and so on. The problem of encoding the information in a globular set was solved by using a description in terms of trees. The power of the usual simplicial category \(\Delta\) lies in its rich combinatorial structure, which follows from its structure in terms of words; in fact \textit{J. Benabou} showed [Lect. Notes Math. 1488, 20-28 (1991; Zbl 0759.18003)] that the nerve of \(\Delta\) is the standard resolution of the terminal monoid, relative to the free-forget comonad from \({\mathcal M}on\) to \({\mathcal S}ets\). The aim of this paper is to study the extent that a similar description can explain the role of trees in the context of (weak) \(\omega\)-categories. If \({\mathcal O}mcat\) denotes the category of \(\omega\)-categories, and \({\mathcal G}lob\) that of globular sets, there is a category \(\Omega\), whose nerve is the standard resolution of the terminal \(\omega\)-category relative to the free-forget comonad from \({\mathcal O}mcat\) to \({\mathcal G}lob\). This \(\Omega\) has a concrete description in terms of trees.
scientific article; zbMATH DE number 1538796