# 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.

### Like this:

Posted by John Armstrong | Category theory

## 1 Comment	»

1. [...] of Ab-Categories There are a number of things we can say right off about the -categories we defined last time. As is common practice, we’ll blur the distinction between an abelian group and its [...]

Pingback by | September 17, 2007 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 54, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9094445705413818, "perplexity_flag": "head"}