Picard Infinity Groupoids with Underlying Globular Sets

The Stable Homotopy Hypothesis, or SHH for short, states that the homotopy category of Picard $n$-groupoids equipped with categorical equivalences is equivalent to the homotopy category of stable homotopy $n$-types equipped with stable homotopy equivalences for all $n\ge 0$ (see [3]). In essence, the SHH is a statement about how two different categories, both of whose objects are obtained by weakening the point-set equality and axioms for an abelian group, have equivalent homotopy theories. Moreover, the SHH is motivated by the Homotopy Hypothesis, the Baez-Dolan Stablization Hypothesis [2], the Freudenthal Suspension Theorem (see Section 11.2 of [6]), and Thomason’s Theorem [9].

On one end of the conjectured equivalence in the SHH, we have the stable homotopy types, i.e. connective spectra and their $n$-truncated models for all $n\ge 0$. Connective spectra are well-studied and several models for connective spectra exist, such as the infinite loop spaces of [4] and the Segal spaces of [8]. On the other end, we have Picard $\mathrm{\infty }$-groupoids and their $n$-truncated models for all $n\ge 0$. There exist simplicial and topological models where the Stable Homotopy Hypothesis is known to be true, such as group-like $E_{\mathrm{\infty }}$-spaces [5] and the Picard–Tamsamani model [7]. On the other hand, we would like a model for Picard $n$-groupoid with an underlying globular set where the SHH is true; however, there is no notion of Picard $n$-groupoid past dimension 2 in the literature with an underlying globular set.

In this talk, we provide the first globular model for Picard $\mathrm{\infty }$-groupoids, which have an underlying Grothendieck $\mathrm{\infty }$-groupoid (see [1]), as models over a specially constructed limit sketch. Since these Picard $\mathrm{\infty }$-groupoids have an underlying Grothendieck $\mathrm{\infty }$-groupoid, they have an underlying globular set. Then Picard $n$-groupoids are merely defined to be Picard $\mathrm{\infty }$-groupoids whose underlying Grothendieck $\mathrm{\infty }$-groupoids are $n$-truncated. Our main goal in this talk is to provide the tools and construction strategy used to define the limit sketch whose models are defined to be Picard $\mathrm{\infty }$-groupoids.