Semi-strictification of $(\mathrm{\infty },n)$-categories

In this talk, I will present my joint work with Clémence Chanavat [1] providing the first (to our knowledge) equivalence between a fully weak, non-algebraic model and a semi-strict algebraic model of $(\mathrm{\infty },n)$-categories. Since the models satisfy the homotopy hypothesis in the case $n=0$, this also exhibits the first semi-strict model of homotopy types with algebraic units and composition.

Since the semi-strict model satisfies a strict form of associativity and interchange, while unitality laws only hold up to coherent equivalence, this is a semi-strictification result in the spirit of Carlos Simpson’s “weak units” conjecture [2], although it does not map neatly to a classification in terms of globular algebra: the model separates the formation of pasting diagrams (which uses globular pasting operations) from composition, which is restricted to the subclass of round diagrams, similar to Simon Henry’s notion of composition for regular $\omega$-categories [3]. Globular composition operations are derived from a combination of “padding up pasting diagrams with units” and round diagram composition, as in the following picture:

Both the definition of the semi-strict model and the semi-strictification result are founded on a “category of diagram shapes” that plays the same role that Joyal’s $\mathrm{\Theta }$ category plays in the theory of strict $\omega$-categories, but in which—unlike in $\mathrm{\Theta }$—morphisms that classify face, unit, and composition operations are neatly separated into a ternary factorisation system. This category is based on the combinatorial theory of regular directed complexes [4] and a further study of classes of morphisms between regular directed complexes. Suitably limit-preserving presheaves on the restriction to (face, unit)-morphisms support weak (partially non-algebraic) models of $(\mathrm{\infty },n)$-categories, and freely extending with composition-morphisms realises semi-strictification, following a proof strategy first sketched in [5]. More technically, semi-strictification is exhibited by acyclic cofibrations constituting the derived unit components of a Quillen equivalence between weak model categories [6] whose fibrant objects are, respectively, the weak $(\mathrm{\infty },n)$-categories and (up to an acyclic fibration) the semi-strict ones; weak functors lift to functors which strictly preserve composition, but only weakly preserve units. The constructions are explicit and combinatorial, in the spirit of Mac Lane’s strictification of bicategories.