# The Unapologetic Mathematician

## 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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