# The Unapologetic Mathematician

## 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.