The empty category is the category $\emptyset _{\mathsf{cat}}$ where
- Objects. We have
\[ \text{Obj}\webleft (\emptyset _{\mathsf{cat}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\emptyset . \]
- Morphisms. We have
\[ \textup{Mor}\webleft (\emptyset _{\mathsf{cat}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\emptyset . \]
- Identities and Composition. Having no objects, $\emptyset _{\mathsf{cat}}$ has no unit nor composition maps.
