The most natural notion of a simplicial nerve for a (weak) bicategory wasgiven by Duskin, who showed that a simplicial set is isomorphic to the nerve ofa $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if andonly if it is a quasicategory which has unique fillers for inner horns ofdimension $3$ and greater. Using Duskin's technique, we show how his nerveapplies to $(2,1)$-category functors, making it a fully faithful inclusion of$(2,1)$-categories into simplicial sets. Then we consider analogues of thisextension of Duskin's result for several different two-dimensional categoricalstructures, defining and analysing nerves valued in presheaf categories basedon $\Delta^2$, 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 uniquenesscondition, we get proposed models for several different kindshigher-categorical structures, with each of these models closely analogous toquasicategories. 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 theUniversity of California, Berkeley. Minor corrections and changes are included.
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