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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.

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

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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.

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