# The Unapologetic Mathematician

## Ab-Categories

Now that we’ve done a whole lot about enriched categories in the abstract, let’s look at the very useful special case of categories enriched over $\mathbf{Ab}$ — the category of abelian groups.

We know that $\mathbf{Ab}$ is a monoidal category, with the tensor product of abelian groups as its monoidal structure and the free abelian group $\mathbb{Z}$ as the monoidal identity. Even better, it’s symmetric, and even closed. That is, for any two abelian groups $A$ and $B$ we have an isomorphism $A\otimes B\cong B\otimes A$, and there is a natural abelian group structure on the set of homomorphisms $B^A=\hom_\mathbf{Ab}(A,B)$ satisfying the adjunction $\hom_\mathbf{Ab}(A\otimes B,C)\cong\hom_\mathbf{Ab}(A,C^B)$.

Further, $\mathbf{Ab}$ is complete and cocomplete. All together, this means it’s a great candidate as a base category on which to build enriched categories. Of course, these will be called $\mathbf{Ab}$-categories.

So let’s read the definitions. An $\mathbf{Ab}$-category $\mathcal{C}$ has a collection of objects, and between objects $A$ and $B$ there is an abelian hom-group $\hom_\mathcal{C}(A,B)$.

For each object $C$ we have a homomorphism of abelian groups $\mathbb{Z}\rightarrow\hom_\mathcal{C}(C,C)$ which picks out the “identity morphism” from $C$ to itself at the level of the underlying sets. Remember that we’re no longer thinking of an abelian group as having elements — only its underlying set has elements anymore, and the underlying set of an abelian group $X$ is the set of abelian group homomorphisms $\mathbb{Z}\rightarrow X$.

Given three objects $A,B,C\in\mathcal{C}$ we have a “composition” arrow in $\mathbf{Ab}$: $\circ:\hom_\mathcal{C}(B,C)\otimes\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(A,C)$. This is associative and the identity morphism acts as an identity in the sense that the appropriate diagrams commute. Of course, since the composition arrows are morphisms in $\mathbf{Ab}$ they are linear functions in each input.

An $\mathbf{Ab}$-functor $F$ between $\mathbf{Ab}$-categories $\mathcal{C}$ and $\mathcal{D}$ is defined by a function $F$from the objects of $\mathcal{C}$ to the objects of $\mathcal{D}$, and for each pair of objects $C,C'\in\mathcal{C}$ a homomorphism of abelian groups $F_{C,C'}:\hom_\mathcal{C}(C,C')\rightarrow\hom_\mathcal{D}(F(C),F(C'))$. Two diagrams are required to commute, saying that these linear functions preserve the composition and identity functions.

An $\mathbf{Ab}$-natural transformation is one of two forms. In one we’re given two $\mathbf{Ab}$-functors $F$ and $G$. Then a natural $\eta:F\rightarrow G$ is a collection of linear functions $\eta_C:\mathbb{Z}\rightarrow\hom_\mathcal{D}(F(C),G(C))$ making one diagram commute. In the other we’re given an object $K\in\mathcal{D}$ and a bifunctor $T:\mathcal{C}^\mathrm{op}\otimes\mathcal{C}\rightarrow\mathcal{D}$. Then $\eta:K\rightarrow T$ is a collection of linear functions $\eta:K\rightarrow T(C,C)$ making another diagram commute.

Together, $\mathbf{Ab}$-categories, $\mathbf{Ab}$-functors between them, and $\mathbf{Ab}$-natural transformations (of the first kind) form a 2-category. We can pair off $\mathbf{Ab}$-categories $\mathcal{C}$ and $\mathcal{D}$ to get the product category $\mathcal{C}\otimes\mathcal{D}$ (in fact we already did once above) and we can take the opposite category $\mathcal{C}^\mathrm{op}$. Thus $\mathbf{Ab}$-categories form a symmetric monoidal 2-category with a duality involution.

There’s a whole lot of structure here, but ultimately it boils down to “the hom-sets all have the structure of abelian groups, and everything in sight is $\mathbb{Z}$-linear”. And that’s the usual definition given, that I decided to forgo back when I started in on enriched categories.