The Underlying Category
In the setup for an enriched category, we have a locally-small monoidal category, which we equip with an “underlying set” functor . This lets us turn a hom-object into a hom-set, and now we want to extend this “underlying” theme to the entire 2-category .
Okay, we could start by finding “underlying” analogues for each piece of the whole structure, but there’s a better way. We just take the setup of the “underlying set” from our monoidal categories and port it over to our 2-categories of enriched categories.
In particular, there’s a -category that has a single object and . This behaves sort of like a “unit -category”, and we define . This is a 2-functor from to , and it assigns to an enriched category the “underlying” ordinary category. Let’s look at this a bit more closely.
A -functor picks out an object , while a -natural transformation consists of the single component — an element of . Thus the underlying category has the same objects as , while is the “underlying set” of .
Given a -functor we get a regular functor . It sends the object of to the object of . Its action on arrows of (natural transformations of functors from to shouldn’t be too hard to work out.
Given a -natural transformation of -functors we get a natural transformation . Its component in is an element of an “underlying hom-set” — an arrow from to the appropriate hom-object. But this is just the same as the component of the -natural transformation we started with, so we don’t really need to distinguish them.
At this point, some of these conditions tend to diverge. The ordinary naturality condition for a transformation between functors acting on the underlying categories turns out to be weaker than the -naturality condition for a transformation between -functors, for example. In general if I start talking about -categories then everything associated to them will be similarly enriched. If I mean a regular functor between the underlying categories I’ll try to say so. That is, once I lay out -categories and , then if I talk about a functor I automatically mean a -functor. If I mean to talk about a regular functor I’ll say as much. Similarly, if I assert a natural transformation I must mean -natural, or I would have said .
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