The Unapologetic Mathematician

Mathematics for the interested outsider

Free Enriched Categories

Now we’re going to assume that our monoidal category \mathcal{V} is also cocomplete. In particular, we’ll assume that it has all small coproducts.

This is enough to ensure that the “underlying set” functor \hom_{\mathcal{V}_0}(\mathbf{1},\underline{\hphantom{X}}) has a left adjoint (\underline{\hphantom{X}})\cdot\mathbf{1} that sends a set E to the coproduct of a bunch of copies of \mathbf{1} indexed by E. The adjunction says that \hom_{\mathcal{V}_0}(\coprod\limits_E\mathbf{1},X)\cong\prod_E\hom_{\mathcal{V}_0}(\mathbf{1},X) is naturally isomorphic to \hom_\mathbf{Set}(E,\hom_{\mathcal{V}_0}(\mathbf{1},X)). That is, a function from E to the underlying set of X is the same as an E-indexed collection of elements of the underlying set of X.

It’s straightforward from here to verify that this adjunction interchanges the cartesian product on \mathbf{Set} and the monoidal structure on \mathcal{V}. That is, (E\times F)\cdot\mathbf{1}\cong(E\cdot\mathbf{1})\otimes(F\cdot\mathbf{1}) and {*}\cdot\mathbf{1}\cong\mathbf{1}.

And now the 2-functor (\underline{\hphantom{X}})_0:\mathcal{V}\mathbf{-Cat}\rightarrow\mathbf{Cat} has a left 2-adjoint (\underline{\hphantom{X}})_\mathcal{V}. Starting with an ordinary category \mathcal{C} (with hom-sets) we get the “free \mathcal{V}-category” \mathcal{C}_\mathcal{V} with the same objects as \mathcal{C}, and with the hom-objects given by \hom_{\mathcal{C}_\mathcal{V}}(C,C')=\hom_\mathcal{C}(C,C')\cdot\mathbf{1}. Compositions and identities for this \mathcal{V}-category are induced by the above exchange of cartesian and monoidal structures. The actions of this 2-functor on functors and natural transformations are straightforwardly defined.

For example, when \mathcal{V}=\mathbf{Ab}, we just replace each hom-set \hom_\mathcal{C}(C,C') by the free abelian group on \hom_\mathcal{C}(C,C'). We extend the composition and identity maps from the ordinary category \mathcal{C} by linearity. In particular, if \mathcal{C} has only one object — if it’s a monoid M — then \mathcal{C}_\mathbf{Ab} is the free ring \mathbb{Z}[M] on M.

To finish off, let \mathcal{C} be a small category, so \mathcal{C}_\mathcal{V} is a small \mathcal{V}-category. Then we can define functor categories and functor \mathcal{V}-categories. Verify that \left(\mathcal{B}^{\mathcal{C}_\mathcal{V}}\right)_0\cong\left(\mathcal{B}_0\right)^\mathcal{C} by the above 2-adjunction.

September 13, 2007 - Posted by | Category theory

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