# nLab unit enriched category

If $V$ is a monoidal category, the unit $V$-category is the $V$-enriched category $\mathcal{I}$ having one object, say $\star$, with $\mathcal{I}(\star,\star)=I$, the monoidal unit object of $V$, and the composition and identity-assigning morphisms being the canonical coherence isomorphism $I\otimes I\cong I$ and the identity arrow $id_I$, respectively.

The $V$-category $\mathcal{I}$ plays a role in enriched category theory that is similar to the role played by the terminal category in ordinary unenriched category theory. (In fact, the terminal category is the unit Set-category.) For instance, objects of a $V$-category $\mathcal{A}$ can be identified with $V$-functors $\mathcal{I}\to\mathcal{A}$. Note that the unit $V$-category is not a terminal object of $V Cat$, in general, just as $I$ is not usually terminal in $V$.

Revised on November 4, 2009 23:45:28 by Toby Bartels (173.51.68.54)