The Clowder Project

¹ $\webleft (-1\webright )$-categories should be thought of as being “categories enriched in $\webleft (-2\webright )$-categories”, having a collection of objects and, for each pair of objects, a $\textup{Hom}$-object $\textup{Hom}\webleft (x,y\webright )$ that is a $\webleft (-2\webright )$-category (i.e. trivial).

Therefore, a $\webleft (-1\webright )$-category $\mathcal{C}$ is either:²

1. Empty, having no objects.
2. Contractible, having a collection of objects $\webleft\{ a,b,c,\ldots \webright\} $, but with $\textup{Hom}_{\mathcal{C}}\webleft (a,b\webright )$ being a $\webleft (-2\webright )$-category (i.e. trivial) for all $a,b\in \text{Obj}\webleft (\mathcal{C}\webright )$, forcing all objects of $\mathcal{C}$ to be uniquely isomorphic to each other.

As such, there are only two $\webleft (-1\webright )$-categories, up to equivalence:

1. The $\webleft (-1\webright )$-category $\mathsf{false}$ (the empty one);
2. The $\webleft (-1\webright )$-category $\mathsf{true}$ (the contractible one).