The Catalan simplicial set and uniform classification of monoidal-type categories

Abstract

Many monoidal-type objects are known to be classified by maps from the Catalan simplicial set $\mathbb{C}$ to various nerves of categories and higher categories. There are, for example, three different nerves of the 2-category of categories $\mathsf{Cat}$ with the property that maps from $\mathbb{C}$ into them are in one to one correspondence with strict monoidal, monoidal, and skew monoidal categories. In this last case, the classification result can be understood as justifying the definition of skew monoidal categories intrinsically in terms of the structure of $\mathbb{C}$. In this paper, we consider a single nerve -- the homotopy coherent nerve of $\mathsf{Cat}$ viewed as a one object simplicially enriched category -- and verify that maps from $\mathbb{C}$ to it classify the three monoidal-type categories mentioned above, as well as lax monoidal categories and $\Sigma$-monoidal categories. Furthermore, we identify a new monoidal-type category corresponding to the most general such maps and present them in the usual way: a collection of data subject to explicit coherence requirements. These results suggest that the space of maps from $\mathbb{C}$ to the coherent nerve of $\mathsf{Cat}$ provides a uniform framework for the definition and study of many species of monoidal-type category, old or new.