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 John Armstrong | Category theory

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

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