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Purity of the Batanin monad. II. (Pureté de la monade de Batanin. II.) (French. English summary) Zbl 1472.18003

This is the sequel to [J. Penon, Cah. Topol. Géom. Différ. Catég. 61, No. 1, 57–110 (2020; Zbl 1456.18005)], where the notion of purity of a monad, and syntactic monads, were introduced. The aim of this second part is to utilise the machinery of the first part toward the study of Batanin’s monad \(\mathbb B\). In the first part, the monads were on the category \(\mathbf {Set}\), but for the purposes of the second part it is necessary to pass to the category \(\mathbf {Glob}\) of globular sets. The first steps are taken immediately as this part starts, obviously relying on the language of part I.
The language of syntactic monads from part I is new and unfamiliar. The author uses Section 2 to present the familiar monad \(\omega\) of strict \(\infty \)-categories, in terms of the language of part I, in part as a way to exemplify the machinery, as well as towards the weakened version of this monad. The free strict \(\infty \)-category is described in quite some detail.
Section 3 introduces the monad \(\mathbb P\). This is a variant of the author’s monad presented in [J. Penon, Cah. Topologie Géom. Différ. Catégoriques 40, No. 1, 31–80 (1999; Zbl 0918.18006)], following a technique of the author’s of weakening algebraic structures (commented on by M. A. Batanin [J. Pure Appl. Algebra 172, No. 1, 1–23 (2002; Zbl 1003.18010)]). The monad \(\mathbb P\) described in this section is shown to be syntactic and cartesian, but not pure.
In a sense, the failure of purity necessitates the passage, carried out in Section 4, to Batanin’s monad \(\mathbb B\). This monad is described in this section in terms of weak trees and using the formalism of the first part of the work. The description as given does not make it clear at all that this is indeed Batanin’s monad. That is the aim of Section 5. The presentation given makes it quite straightforward that \(\mathbb B\) is pure. The rest of the section, and of the work, is making the relation between this definition of \(\mathbb B\) and the monad arising from free strict \(\omega\)-categories, to arrive at Theorem 5.46, at the identification of \(\mathbb B\) as Batanin’s monad. It is thus pure.

MSC:

18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18N15 2-dimensional monad theory
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