The Unapologetic Mathematician

Functor Categories as Exponentials

The notation we used for the enriched category of functors between two enriched categories gives away the game a bit: this will be the exponential between the two categories.

First off, the arrows $E_C$ that we used to read off the $C$-component of a natural transformation turn out to fit into a more general structure. As we might hope there’s an “evaluation” functor $E:\mathcal{D}^\mathcal{C}\otimes\mathcal{C}\rightarrow\mathcal{D}$ that takes a $\mathcal{V}$-functor from $\mathcal{C}$ to $\mathcal{D}$ and evaluates it on an object of $\mathcal{C}$ to give an object of $\mathcal{D}$. The partial functors are $E(\underline{\hphantom{X}},C)=E_C:\mathcal{D}^\mathcal{C}\rightarrow\mathcal{D}$ and $E(F,\underline{\hphantom{X}})=F:\mathcal{C}\rightarrow\mathcal{D}$. That this is actually a $\mathcal{V}$-functor follows yet again from that litany of naturalities we keep referring to.

Now the $\mathcal{V}$-functor $E$ induces an ordinary functor $\hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{C},\mathcal{B}^\mathcal{A})\rightarrow\hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{C}\otimes\mathcal{A},\mathcal{B})$. Remember here that $\mathcal{V}\mathbf{-Cat}$ is a 2-category — a category enriched over ordinary categories — so each hom-object is a category, and an arrow from one hom-object to another is a functor. In particular, given a functor $F:\mathcal{C}\rightarrow\mathcal{B}^\mathcal{A}$ we get a functor $F\otimes1_\mathcal{A}:\mathcal{C}\otimes\mathcal{A}\rightarrow\mathcal{B}^\mathcal{A}\otimes\mathcal{A}$. And then we compose with the evaluation functor to get a functor $\mathcal{C}\otimes\mathcal{A}\rightarrow\mathcal{B}$. It turns out that this functor is an isomorphism of categories.

This means that every functor $P:\mathcal{C}\otimes\mathcal{A}\rightarrow\mathcal{B}$ is of the above form for some unique $F:\mathcal{C}\rightarrow\mathcal{B}^\mathcal{A}$. Let’s look at the partial functors of $P$ and $E\circ(F\otimes1_\mathcal{A})$.

• $P(C,\underline{\hphantom{X}})=[F(C)](\underline{\hphantom{X}})$
• $P(\underline{\hphantom{X}},A)=E_A\circ F$

The first of these equations completely determines the functor in $\mathcal{B}^\mathcal{A}$ that $F$ must assign to an object of $\mathcal{C}$. The second uniquely determines the action of $F$ on hom-objects because $E_A$ is the counit of an end, and it comes with a universal property. Thus the above map is bijective on objects. But since it’s a functor we also need it to be bijective on morphisms — natural transformations in this case.

So, if we’ve got functors $F$ and $F'$ from $\mathcal{C}$ to $\mathcal{B}^\mathcal{A}$, with images $P$ and $P'$ from $\mathcal{C}\otimes\mathcal{A}\rightarrow\mathcal{B}$. Now we need to check that every $\mathcal{V}$-natural $\alpha:P\rightarrow P'$ is of the form $1_E\circ(\beta\otimes1_{1_\mathcal{A}})$ for some unique $\mathcal{V}$-natural $\beta:F\rightarrow F'$. But by the equation above between the second partial functors, this says that $\alpha_{C,A}=E_A\circ\beta_C=(\beta_C)_A$. Thus $\beta:F\rightarrow F'$ is the $\mathcal{V}$-natural transformation $\alpha_{C,\underline{\hphantom{X}}}:P(C,\underline{\hphantom{X}})\rightarrow P'(C,\underline{\hphantom{X}})$.

We now have a 2-natural isomorphism (natural isomorphism enriched over $\mathbf{Cat}$): $\hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{C},\mathcal{B}^\mathcal{A})\cong\hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{C}\otimes\mathcal{A},\mathcal{B})$. Equivalently, this means that $E:\mathcal{B}^\mathcal{A}\otimes\mathcal{A}\rightarrow\mathcal{B}$ is 2-natural in each variable. Using this naturality it’s straightforward to show that given $M:\mathcal{A}'\rightarrow\mathcal{A}$ and $N:\mathcal{B}\rightarrow\mathcal{B}'$ we get a $\mathcal{V}$-functor $N^M:\mathcal{B}^\mathcal{A}\rightarrow\mathcal{B}'^{\mathcal{A}'}$ by composing functors.

All of this fits together to say that the 2-functor $\underline{\hphantom{X}}\otimes\mathcal{C}:\mathcal{V}\mathbf{-Cat}\rightarrow\mathcal{V}\mathbf{-Cat}$ has a right adjoint $(\underline{\hphantom{X}})^\mathcal{C}$ when $\mathcal{D}^\mathcal{C}$ exists for all $\mathcal{D}$. At this point, the existence tends to hinge on a lot of smallness technicalities. The 2-category of all $\mathcal{V}$-categories is thus “partially closed”, in a similar way to the 2-category of all ordinary categories. However, if we restrict to small $\mathcal{V}$-categories we actually do have a symmetric, monoidal, closed 2-category.

In either case, partial closedness together with the Fubini theorem is enough for us to get the standard isomorphisms $\mathcal{B}^\mathcal{I}\cong\mathcal{B}$ and $\mathcal{B}^{\mathcal{C}\otimes\mathcal{A}}\cong(\mathcal{B}^\mathcal{A})^\mathcal{C}$. This latter holds in the sense that either side exists if the other one does.