Non-Simplicial Nerves for Two-Dimensional Categorical Structures

Abstract

The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it is a quasicategory which has unique fillers for inner horns of dimension $3$ and greater. Using Duskin's technique, we show how his nerve applies to $(2,1)$-category functors, making it a fully faithful inclusion of $(2,1)$-categories into simplicial sets. Then we consider analogues of this extension of Duskin's result for several different two-dimensional categorical structures, defining and analysing nerves valued in presheaf categories based on $\Delta^2$, on Segal's category $\Gamma$, and Joyal's category $\Theta_2$. In each case, our nerves yield exactly those presheaves meeting a certain "horn-filling" condition, with unique fillers for high-dimensional horns. Generalizing our definitions to higher dimensions and relaxing this uniqueness condition, we get proposed models for several different kinds higher-categorical structures, with each of these models closely analogous to quasicategories. Of particular interest, we conjecture that our "inner-Kan $\Gamma$-sets'' are a combinatorial model for symmetric monoidal $(\infty,0)$-categories, i.e. $E_\infty$-spaces. This is a version of the author's Ph.D. dissertation, completed 2013 at the University of California, Berkeley. Minor corrections and changes are included.