# Simple examples of comonoids

In textbooks on Category Theory, monoids pop-up all over the place, we several easy examples such as integers, lists and so on. I was then wondering about comonoids. What are some "simple" examples of comonoids appearing in mathematics? I've tried searching, but most texts either only mentions the concept of a comonoid, or provide some convoluted example.

I'm trying to get a better feel for what a comonoid is, thus I'm searching for easy to grasp examples.

• The diagonal map makes any object of a cartesian monoidal category uniquely a comonoid. Nov 28, 2023 at 15:08

The notion of a comonoid is defined in every monoidal category.

If the monoidal category is cartesian, i.e. $$\otimes$$ is the product and $$1$$ is the terminal object, then comonoids are very easy to describe. In fact, every object $$X$$ has a unique comonoid structure: The counit is the unique morphism $$X \to 1$$, and the comultiplication is the diagonal morphism $$X \to X \times X$$, $$x \mapsto (x,x)$$. For a specific example, take $$(\mathbf{Set},\times,\{\star\})$$, here every set admits a unique comonoid structure.

A more interesing case is the monoidal category $$(\mathbf{Mod}_k,\otimes_k,k)$$ of $$k$$-modules, where $$k$$ is a commutative ring. Here, comonoids are called coalgebras over $$k$$, and there are plenty of examples. A particular important subclass are the Hopf algebras.

For a specific example, take the polynomial ring $$k[X]$$ with the counit $$k[X] \to k,\quad X \mapsto 1$$ and the comultiplication $$k[X] \to k[X] \otimes k[X],\quad X \mapsto X \otimes X$$ More generally, if $$G$$ is any set, then the free $$k$$-module $$k[G]$$ on the set $$G$$ carries the structure of a coalgebra over $$k$$ (the same formulas apply), and if $$G$$ is a group this is actually a Hopf algebra (with the multiplication and antipode induced by $$G$$).

Another interesting example (used in combinatorics) is the incidence coalgebra of a finite poset $$P$$. We take the free module $$C := k[\{(x,y) : x,y \in P, \, x \leq y\}]$$ on the set of non-empty intervals in $$P$$ and define the comultiplication by $$C \to C \otimes C, \quad (x,y) \mapsto \sum_{u \in P,\, x \leq u \leq y} (x,u) \otimes (u,y).$$ We only need that $$P$$ is locally finite (:=every interval is finite) to render this sum well-defined, so this also applies for example to the poset $$(\mathbb{N},\leq)$$.

Under the duality between commutative $$k$$-algebras and affine schemes over $$k$$, commutative Hopf algebras correspond to the affine group schemes over $$k$$. This means that Hopf algebras (and hence, coalgebras) also appear in algebraic geometry. For a simple example, the affine group scheme $$\mathrm{GL}_n$$ is represented by the commutative $$k$$-algebra $$R = k[(X_{ij})_{1 \leq i,j \leq n}][{\det}^{-1}],$$ where $$\det$$ is a polynomial in the variables, defined by the Leibniz formula. The counit is defined by $$R \to k,~ X_{ij} \mapsto \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}$$ The comultiplication is defined by $$R \to R \otimes R, ~ X_{ij} \mapsto \sum_{s=1}^{n} X_{is} \otimes X_{sj}.$$

It was mentioned before that comonoids are not that interesting when $$\otimes$$ is the product. It is quite the opposite when $$\otimes$$ is the coproduct. For example, the sphere $$S^n$$ admits a map $$S^n \to S^n \vee S^n$$ making it a comonoid in $$(\mathbf{Top}_*,\vee,\{\star\})$$ "almost": associativity only holds up to homotopy. Therefore, we get a comonoid in $$(\mathbf{hTop}_*,\vee,\{\star\})$$, and in fact it is a cogroup. It follows that for every $$X \in \mathbf{Top}_*$$ the set $$\hom(S^n,X)$$ carries the structure of a group: the multiplication is $$\hom(S^1,X) \times \hom(S^1,X) \to \hom(S^1 \vee S^1,X) \to \hom(S^1,X)$$. That is a highly conceptual construction of the $$n$$-th homotopy group of $$X$$. Put differently, the group structure on these sets exactly comes from a cogroup structure on $$S^n$$.

Monads are monoids in the monoidal category of endofunctors of a category, where $$\otimes = \circ$$. Dually, comonads are comonoids in the category of endofunctors of a category. Thus, a comonad structure on a functor $$T : \mathcal{C} \to \mathcal{C}$$ consists of two natural transformations $$T \to \mathrm{id}_{\mathcal{C}},\quad T \to T \circ T$$ satisfying three equations. Perhaps the simplest non-trivial example is the endofunctor $$T(X) := X \times W$$ of $$\mathbf{Set}$$, where $$W$$ is any set. (The same construction works over every cartesian category.) The natural transformations $$X \times W \to X, ~ (x,w) \mapsto x$$ $$X \times W \to (X \times W) \times W,~ (x,w) \mapsto ((x,w),w)$$ constitute a comonad structure on $$T$$. Many more examples of comonads appear from the observation that if $$L$$ is left adjoint to $$R$$, then $$L \circ R$$ carries the structure of a comonad (which is dual to the observation that $$R \circ L$$ carries the structure of a monad). For example, there is a comonad structure on the endofunctor of $$\mathbf{Grp}$$ that sends a group to the free group on its underlying set.