Non-Simplicial Nerves for Two-Dimensional Categorical Structures
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 several combinatorial categories. 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 (infinity, 0)-categories, i.e. E-infinity spaces.