The Unapologetic Mathematician

Mathematics for the interested outsider

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 V(C)=\hom_{\mathcal{V}_0}(\mathbf{1},C). 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 \mathcal{V}\mathbf{-Cat}.

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 \mathcal{V}-category \mathcal{I} that has a single object I and \hom_\mathcal{I}(I,I)=\mathbf{1}. This behaves sort of like a “unit \mathcal{V}-category”, and we define (\underline{\hphantom{X}})_0:\hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{I},\underline{\hphantom{X}}). This is a 2-functor from \mathcal{V}\mathbf{-Cat} to \mathbf{Cat}, and it assigns to an enriched category the “underlying” ordinary category. Let’s look at this a bit more closely.

A \mathcal{V}-functor F:\mathcal{I}\rightarrow\mathcal{C} picks out an object F(I)\in\mathcal{C}, while a \mathcal{V}-natural transformation \eta:F\rightarrow G consists of the single component \eta_I:\mathbf{1}\rightarrow\hom_\mathcal{C}(F(I),G(I)) — an element of V(\hom_\mathcal{C}(F(I),G(I))). Thus the underlying category \mathcal{C}_0 has the same objects as \mathcal{C}, while \hom_{\mathcal{C}_0}(A,B) is the “underlying set” of \hom_\mathcal{C}(A,B).

Given a \mathcal{V}-functor T:\mathcal{C}\rightarrow\mathcal{D} we get a regular functor T_0:\mathcal{C}_0\rightarrow\mathcal{D}_0. It sends the object F:\mathcal{I}\rightarrow\mathcal{C} of \mathcal{C}_0 to the object T\circ F:\mathcal{I}\rightarrow\mathcal{D} of \mathcal{D}_0. Its action on arrows of \mathcal{C}_0 (natural transformations of functors from \mathcal{I} to \mathcal{C} shouldn’t be too hard to work out.

Given a \mathcal{V}-natural transformation \eta:S\rightarrow T of \mathcal{V}-functors we get a natural transformation \eta_0:S_0\rightarrow T_0. Its component \eta_{0A}:S(A)\rightarrow T(A) in \mathcal{D}_0 is an element of an “underlying hom-set” — an arrow from \mathbf{1} to the appropriate hom-object. But this is just the same as the component \eta_A of the \mathcal{V}-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 \mathcal{V}-naturality condition for a transformation between \mathcal{V}-functors, for example. In general if I start talking about \mathcal{V}-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 \mathcal{V}-categories \mathcal{C} and \mathcal{D}, then if I talk about a functor F:\mathcal{C}\rightarrow\mathcal{D} I automatically mean a \mathcal{V}-functor. If I mean to talk about a regular functor F:\mathcal{C}_0\rightarrow\mathcal{D}_0 I’ll say as much. Similarly, if I assert a natural transformation \eta:S\rightarrow T I must mean \mathcal{V}-natural, or I would have said \eta:S_0\rightarrow T_0.

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August 20, 2007 - Posted by | Category theory

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