The Unapologetic Mathematician

Mathematics for the interested outsider


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 Ffrom 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.

September 14, 2007 Posted by | Category theory | 1 Comment