Weak vertical composition (Q6593825)

It is well-known that every weak 2-category is equivalent to a strict one [\url{https://www2.irb.hr/korisnici/ibakovic/groth2fib.pdf}], while the analogous result for weak 3-categories does not hold [\textit{R. Gordon} et al., Coherence for tricategories. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0836.18001)] in which it was shown that every tricategory is equivalent to one where everything is strict except interchange. \textit{C. Simpson} [``Homotopy types of strict 3-groupoids'', Preprint, \url{arXiv:math/9810059}] conjectured that weak units would be enough, which was established for the case \(n=3\)\ by \textit{A. Joyal} and \textit{J. Kock} [Contemp. Math. 431, 257--276 (2007; Zbl 1137.18004)]. This paper addresses semi-strict tricategories in which the only weakness lies in vertical composition.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 1] defines the type of semi-strict tricategories, which are categories enriched in the category of bicategories and strict functors. The authors characterize the doubly-degenerate ones as certain categories with two monoidal structures, one weak and one strict, abiding by strict interchange, and demonstrate that each one has an associated braided monoidal category. These categories with two monoidal structures are a particular strict form of 2-monoidal category [\textit{M. Aguiar} and \textit{S. Mahajan}, Monoidal functors, species and Hopf algebras. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1209.18002)].\N\N\item[\S 2] gives some background on cliques.\N\N\item[\S 3] introduces the labelled configuration spaces of points, together with their braided monoidal structure.\N\N\item[\S 4] gives the main construction starting with any braided monoidal category \(B\)\ and producing a doubly-degenerate \(\boldsymbol{Bicat}_{s} \)-category \(\Sigma B\)\ from it.\N\N\item[\S 5] establishes the main theorem claiming that the braided monoidal category associated with \(\Sigma B\)\ is braided monoidal equivalent to \(B\). It is shown that all braided monoidal categories arise from doubly-degenerate \(\boldsymbol{Bicat}_{s}\)-categories.\N\N\item[\S 6] gives a brief account of future work.\N\end{itemize}