The Unapologetic Mathematician

Mathematics for the interested outsider

Functor Categories

Let’s consider two categories enriched over a monoidal category \mathcal{V} — \mathcal{C} and \mathcal{D} — and assume that \mathcal{C} is equivalent to a small category. We’ll build a \mathcal{V}-category \mathcal{D}^\mathcal{C} of functors between them.

Of course the objects will be functors F:\mathcal{C}\rightarrow\mathcal{D}. Now for functors F and G we need a \mathcal{V}-object of natural transformations between them. For this, we will use an end:
\hom_{\mathcal{D}^\mathcal{C}}(F,G)=\int_{A\in\mathcal{C}}\hom_\mathcal{D}(F(C),G(C))
This end is sure to exist because of the smallness assumption on \mathcal{C}. Its counit will be written E_C=E_{C,F,G}:\hom_{\mathcal{D}^\mathcal{C}}(F,G)\rightarrow\hom_\mathcal{D}(F(C),G(C)).

An “element” of this \mathcal{V} object is an arrow \eta:\mathbf{1}\rightarrow\hom_{\mathcal{D}^\mathcal{C}}(F,G). Such arrows correspond uniquely to \mathcal{V}-natural families of arrows \eta_C=E_C\circ\eta:\mathbf{1}\rightarrow\hom_\mathcal{D}(F(C),G(C)), which we know is the same as a \mathcal{V}-natural transformation from F to G. We also see that at this level of elements, the counit E_C takes a \mathcal{V}-natural transformation and “evaluates” it at the object C.

Now we need to define composition morphisms for these hom-objects. This composition will be inherited from the target category \mathcal{B}. Basically, the idea is that at each object a natural transformation gives a component morphism in the target category, and we compose transformations by composing their components. Of course, we have to finesse this a bit because we don’t have sets and elements anymore.

So how do we get a component morphism? We use the counit map E_C! We have the arrow
E_C\otimes E_C:\hom_{\mathcal{D}^\mathcal{C}}(G,H)\otimes\hom_{\mathcal{D}^\mathcal{C}}(F,G)\rightarrow\hom_\mathcal{D}(G(C),H(C))\otimes\hom_\mathcal{D}(F(C),G(C))
which we can then hit with the composition \circ:\hom_\mathcal{D}(G(C),H(C))\otimes\hom_\mathcal{D}(F(C),G(C))\rightarrow\hom_\mathcal{D}(F(C),H(C)). This is a \mathcal{V}-natural family indexed by C\in\mathcal{C}, so by the universal property of the end we have a unique arrow
\circ:\hom_{\mathcal{D}^\mathcal{C}}(G,H)\otimes\hom_{\mathcal{D}^\mathcal{C}}(F,G)\rightarrow\hom_{\mathcal{D}^\mathcal{C}}(F,H)

Similarly we get the arrow picking out identity morphism on T as the unique one satisfying E_C\circ i_T=i_{T(A)}

So \mathcal{D}^\mathcal{C} is a \mathcal{V}-category whose underlying ordinary category is that of \mathcal{V}-functors and \mathcal{V}-natural transformations between the \mathcal{V}-categories \mathcal{C} and \mathcal{D}. That is, it’s the hom-category \hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{C},\mathcal{D}) in the 2-category of \mathcal{V}-categories.

September 10, 2007 Posted by | Category theory | 1 Comment