nLab
unit enriched category

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

From a 22-categorical point of view, the unit enriched category \mathcal{I} satisfies the property that [,][\mathcal{I},\mathcal{I}] (the category of 11-endomorphisms of \mathcal{I}) is a category with one object, and the 22-functor [,][\mathcal{I},-] is the functor returning the underlying category of an enriched category.

In other words, the VV-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 VV-category 𝒜\mathcal{A} can be identified with VV-functors 𝒜\mathcal{I}\to\mathcal{A}.

Note that in general the unit VV-category is not a terminal object of VCatV Cat, just as II is not usually terminal in VV. We can see in particular that when VV is closed and II is not terminal, \mathcal{I} is not terminal since VV embeds in VCatV Cat.

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