The Unapologetic Mathematician

Mathematics for the interested outsider

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.

September 11, 2007 Posted by | Category theory | Leave a comment