Weakly globular \(\text{cat}^n\)-groups and Tamsamani's model
This long and well-written paper makes several important contributions to the study of algebraic models for connected homotopy \((n+1)\)-types, i.e., connected CW complexes with vanishing homotopy groups in degrees higher than \(n+1.\) The author introduces a subcategory \(\text{Cat}^n(\text{GP})_S\) of Loday's category of \(\text{cat}^n\)-groups [\textit{J.-L. Loday}, J. Pure Appl. Algebra 24, 179--202 (1982; Zbl 0491.55004)]. She shows that these ``special'' \(\text{cat}^n\)-groups model connected \((n+1)\)-types and provide natural algebraic formulations for weak-equivalences and Postnikov decompositions. Special \(\text{cat}^n\)-groups are then compared to Tamsamani's model of weak \((n+1)\)-groupoids [\textit{Z. Tamsamani}, K-Theory 16, No.~1, 51--99 (1999; Zbl 0934.18008)]. The author constructs a functor from the category of \(\text{cat}^n\)-groups to Tamsamani's category \(\mathcal{T}_{n+1}\) and shows this functor factors through \(\text{Cat}^n(\text{GP})_S\). This comparison functor preserves homotopy type. A consequence of this fact is a ``semistrictification'' result for \(\mathcal{T}_{n+1}\). The author identifies a full subcategory \(\mathcal{S}_{n+1}\) of \(\mathcal{T}_{n+1}\) and proves that every \(\mathcal{G} \in \mathcal{T}_{n+1}\) with \(B\mathcal{G}\) connected is linked by a zig-zag of \((n+1)\)-equivalences to an object of \(\mathcal{S}_{n+1}\). The category \(\mathcal{S}_{n+1}\) is semistrict in the sense it cannot be strictified while preserving homotopy type. The details of these constructions and results are extensive. The author does an excellent job of motivating and providing background making the paper relatively self-contained.
scientific article; zbMATH DE number 5597915
scientific article