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


  1. […] now what about comonoid objects? We’ve got diagrams to talk about them […]

    Pingback by Diagrammatics for Monoid Objects « The Unapologetic Mathematician | July 25, 2007 | Reply

  2. […] spaces over . That’s all well and good, but what’s a comonoid object? We’ve mentioned them before, but let’s be more explicit this time […]

    Pingback by Coalgebras « The Unapologetic Mathematician | November 5, 2008 | Reply

  3. […] object in the category of vector space over . Then a compatible coalgebra structure makes it a comonoid object in the category of algebras over . Or in the other order, we have a monoid object in the category […]

    Pingback by Bialgebras « The Unapologetic Mathematician | November 6, 2008 | Reply

  4. […] what’s a comonoid object in this 2-category? When we defined comonoid objects we used a model category . Now let’s augment it to a 2-category in the easiest way possible: […]

    Pingback by The Category of Representations of a Hopf Algebra « The Unapologetic Mathematician | November 18, 2008 | Reply

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