The Unapologetic Mathematician

Mathematics for the interested outsider

Examples of Monoid Objects

It’s all well and good to define monoid objects, but it’s better to see that they subsume a lot of useful concepts. The basic case is, of course, that a monoid object in \mathbf{Set} is a monoid.

Another example we’ve seen already is that a ring with unit is a monoid object in \mathbf{Ab} — the category of abelian groups with the tensor product of abelian groups as the monoidal structure. Similarly, given a commutative ring K, a monoid object in the category K\mathbf{-mod} with tensor product of K-modules as its monoidal structure is a K-algebra with unit. For extra credit, how would we get rings and K-algebras without units?

Here’s one we haven’t seen (and which I’ll talk more about later): given any category \mathcal{C}, the category of “endofunctors” \mathcal{C}^\mathcal{C} has a monoidal structure given by composition of functors from \mathcal{C} to itself. This is the one I was thinking of that doesn’t have a symmetry, by the way. A monoid object in this category consists of a functor T:\mathcal{C}\rightarrow\mathcal{C} along with natural transformations \mu:T\circ T\rightarrow T and \eta:1_\mathcal{C}\rightarrow T. These turn out to be all sorts of useful in homology theory, and also in theoretical computer science. In fact, the programming language Haskell makes extensive and explicit use of them.

And now for a really interesting class of examples. Let’s say we start with a monoidal category \mathcal{C} with monoidal structure \otimes. We immediately get a monoidal structure \otimes^\mathrm{op} on the opposite category \mathcal{C}^\mathrm{op}. Just define A\otimes^\mathrm{op}B=A\otimes B for objects. For morphisms we take f:A\rightarrow C and g:B\rightarrow D (which are in \hom_{\mathcal{C}^\mathrm{op}}(C,A) and \hom_{\mathcal{C}^\mathrm{op}}(D,B), respectively), and define f\otimes^\mathrm{op}g=f\otimes g, which is in \hom_{\mathcal{C}^\mathrm{op}}(C\otimes^\mathrm{op}D,A\otimes^\mathrm{op}B).

So what’s a monoid object in \mathcal{C}^\mathrm{op}? It’s a contravariant functor from \mathrm{Th}(\mathbf{Mon}) to \mathcal{C}. Equivalently, we can write it as a covariant functor from \mathrm{Th}(\mathbf{Mon})^\mathrm{op} to \mathcal{C}. It will be easier to just write down explicitly what this opposite category is.

So we need to take \mathrm{Th}(\mathbf{Mon}) and reverse all the arrows. It’s enough to just reverse the arrows we threw in to generate the category, and their composites will be reversed as well. We’ll also have to dualize the relations we imposed to make everything work out right. So we’ll have an arrow \Delta:M\rightarrow M\otimes M called comultiplication and another arrow \epsilon:M\otimes\mathbf{1} called the counit. These we require to satisfy the coassociative condition (\Delta\otimes1_M)\circ\Delta=(1_M\otimes\Delta)\circ\Delta and the left and right coidentity conditions (\epsilon\otimes1_M)\circ\Delta=1_M=(1_M\otimes\epsilon)\circ\Delta.

Now a functor from this category to another monoidal category picks out an object C\in\mathcal{C} and arrows (reusing the names) \epsilon:C\rightarrow\mathbf{1} and \Delta:C\rightarrow C\otimes C satisfying coassociativity and coidentity conditions. We call such an object with extra structure a “comonoid object” in \mathcal{C}. In \mathbf{Set} we call them “comonoids”. In \mathbf{Ab} we call them “corings” (with counit), in K\mathbf{-mod} we call them “coalgebras” (with counit), and in \mathcal{C}^\mathcal{C} we call them “comonads”. In general, we call this new model category \mathrm{Th}(\mathbf{CoMon}) — the “theory of comonoids”.

July 24, 2007 Posted by | Category theory | 4 Comments

Tell it to Einstein!

I’m adding another new link, this time to God Plays Dice. This one is run by a mysterious and shadowy figure known only as “The Probabilist”. I don’t know why, though. There’s a lot of great stuff here, very accessible to a general audience. In fact, it’s rather like another direction I could have gone with this site six months ago, but I think “The Probabilist” does a better job of it than I would have.

So let this also be a call for “The Probabilist” to unmask and accept credit for this work! I’ve already figured out the secret, and I imagine others have as well, so we’re all just waiting for the other shoe to drop. However, I will respect “The Probabilist”‘s pseudonymity, however little I understand it.

July 24, 2007 Posted by | Uncategorized | 1 Comment