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.

From a $2$-categorical point of view, the unit enriched category $\mathcal{I}$ satisfies the property that $[\mathcal{I},\mathcal{I}]$ (the category of $1$-endomorphisms of $\mathcal{I}$) is a category with one object, and the $2$-functor $[\mathcal{I},-]$ is the functor returning the underlying category of an enriched category.

In other words, the $V$-category $\mathcal{I}$ plays a role in enriched category theory 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 in general the unit $V$-category is *not* a terminal object of $V Cat$, just as $I$ is not usually terminal in $V$. We can see in particular that when $V$ is closed and $I$ is not terminal, $\mathcal{I}$ is not terminal since $V$ embeds in $V Cat$.

Revised on April 15, 2014 03:50:41
by Toby Bartels
(98.16.173.18)