# The Unapologetic Mathematician

## An example of an enriched category

Sorry for the delay, but the cable setup took more than I’d expected (there will me more on this over at Yankee Freak-Out). Today, I’d like to run through an example of a monoidal category, and what sort of enriched categories it gives rise to.

The category I’m interested in is the ordinal $\mathbf{2}$. Remember that this consists of the objects ${0}$ and $1$, with one non-identity arrow $0\rightarrow1$. We can make this into a monoidal category by saying $a\otimes b=ab$. Then $1$ is the monoidal identity object.

So what is a category enriched over $\mathbf{2}$? Well, first it has a collection of objects. For each pair $(A,B)$ of objects we either have the hom-object $\hom_\mathcal{C}(A,B)=0$ or $\hom_\mathcal{C}(A,B)=1$.

To have “identity morphisms” means we need an arrow $1\rightarrow\hom_\mathcal{C}(C,C)$ for each object $C$. But the only such arrow in $\mathbf{2}$ is $1\rightarrow1$, so $\hom_\mathcal{C}(C,C)=1$. For composition, we need arrows $\hom_\mathcal{C}(B,C)\otimes\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(A,C)$. Thus if $\hom_\mathcal{C}(A,B)$ and $\hom_\mathcal{C}(B,C)$ are both $1$, then so must be $\hom_\mathcal{C}(A,C)$.

Now we can see that this is just a different way of talking about a preorder. The identity morphism corresponds to the reflexive axiom, and the composition morphism corresponds to the transitive action. In short: $A\preceq B$ if and only if $\hom_\mathcal{C}(A,B)=1$.

Another example I’ve seen bandied about uses the category $\mathbf{pSet}$ of “pointed sets”. This is just a set with an identified “point”. For example, $(\{1,2,3\},1)$ is a pointed set, and $(\{1,2,3\},2)$ is a different pointed set. The morphisms are “pointed functions”, which have to preserve the point — a morphism $f:(X,x_0)\rightarrow(Y,y_0)$ consists of a function $f:X\rightarrow Y$ with $f(x_0)=y_0$. This category has finite products, so it’s monoidal.

The usual statement is that categories enriched over $\mathbf{pSet}$ are the same as categories with “zero morphisms”. These are like categories with zero objects, but without needing an object to factor things through. Every hom-set has a special “zero” morphism, and the composition of a zero morphism with any other morphism is another zero morphism. The problem is, whenever I try to show this it doesn’t seem to work out. I think that there’s something askew or oversimplified with the statement somewhere.

Your mission, should you choose to accept it, is to figure out what the right statement is, and to prove it. Let me know by email (if you can’t find my email you aren’t trying very hard) and I’ll post it up for all to see, and for your own greater glory.

August 15, 2007 - Posted by | Category theory

1. John — it’s good to see you’re introducing enriched category theory in your blog. It’s always been one of my favorite chapters in the story of category theory.

But I’m surprised that so far as I can tell, you haven’t yet mentioned in this thread that some of the most potent examples for the development of enriched category are where the monoidal category V is enriched in itself! Were you going to say something about that?

For those interested: this is where V-category theory begins to become truly “autonomous”: where V begins to play the same fundamental role in V-Cat that Set plays in ordinary category theory. The world of V-categories, or V-Cat, becomes truly self-sufficient once the fundamental constructions of ordinary category theory, e.g., functor categories, are appropriately internalized within that world. Explorations of such autonomous worlds then reveal phenomena (e.g., the importance of *weighted limits and colimits*) which yield new insights even for ordinary categories!

I’ll (mostly) resist the temptation to explain the archetypal way in which this happens, since John may want to address this himself and tie it in with other things he’s discussed. But it can be summarized in just two words: “closed categories”. I’ll add that the full autonomous development of V-category theory becomes possible (only) if V is sufficiently nice — precisely, a complete, cocomplete, symmetric monoidal closed category — and in discussions of enriched category theory it’s usually a good exercise to check for such structure. Examples include the category 2 of the entry above, Ab, of course Set… How does it work out for pSet (pointed sets)? It’s a symmetric monoidal category under cartesian product; is pSet cartesian closed? (How does one check for that?) What more can one say about this situation?

(Possibly jumping the gun, this bears on the ‘zero morphism’ conundrum posed in the entry, but for the moment I’ll leave that thread dangling, in case John is awaiting more emails on this puzzle.)

Comment by Todd Trimble | August 17, 2007 | Reply

2. I may well come back to those sorts of things. I wanted to skim enriched categories in general before moving on to abelian categories rather than just giving the definition of an $\mathbf{Ab}$-category on its own as most authors do. Then there’s homological algebra, topology, analysis, geometry…

There’s so much material out there that I’ll let some of it slide at a first pass. I’m intending to work on this thing for a good long time, so I’m comfortable waiting and coming back to subjects later.

Comment by John Armstrong | August 17, 2007 | Reply

3. Okay. In that case allow me to mention, for readers who might like to pursue this now: the canonical reference for enriched category theory is the late Max Kelly’s book, Basic Concepts of Enriched Category Theory. It is now available online:

http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html

Comment by Todd Trimble | August 17, 2007 | Reply

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