6
$\begingroup$

Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids?

This category has (at least) two different descriptions, namely it is also the category of cocontinuous functors $\mathbf{Mon} \to \mathbf{Mon}$ (given by tensoring with a comonoid), but also the opposite of the category of continuous functors $\mathbf{Mon} \to \mathbf{Mon}$ (the functors represented by comonoids). From the description of cocontinous functors $\mathbf{Mon} \to \mathbf{Mon}$ we also get a structure of a cocomplete monoidal category (with $\otimes=\circ$ being cocontinous in each variable) with a zero object.

So far I have found three basic comonoids: $\langle x \rangle$ with $\nabla(x) = x_1 x_2$, $\langle x^{\pm 1} \rangle$ with $\nabla(x)=x_1 x_2$, and $\langle x \rangle$ with $\nabla(x)=x_2 x_1$. The continuous functors $\mathbf{Mon} \to \mathbf{Mon}$ represented by them are the identity functor, the group of units functor (which factors over $\mathbf{Grp}$), and the opposite monoid functor. Their left adjoints are the identity $\mathbf{1}$, the group completion $K$ and the opposite monoid functor $D$. Hence, every coproduct of them is also an example. Apart from the unit $\eta : \mathbf{1} \to K$, I think that all other morphisms between them are zero, and it seems that we have $D \circ K \cong K \circ D \cong K \circ K \cong K$.

In case it turns out that this monoidal category is too complicated, what about the category of monoids in it, i.e. the category of Tall-Wraith monoids in $\mathbf{Mon}$? Can we classify them? For $\mathbf{Grp}$ it is well-known by a result of Freyd.

$\endgroup$
7
  • 1
    $\begingroup$ George Bergman characterizes representable endofunctors of monoids in 10.6 of math.berkeley.edu/~gbergman/245/3.2.pdf $\endgroup$ Dec 15 '19 at 16:08
  • $\begingroup$ Thanks! You can make this an answer, right? It is spelled out in great detail in Section 10.6. The comonoids are classified by E-systems. $\endgroup$
    – HeinrichD
    Dec 15 '19 at 20:50
  • $\begingroup$ Ok so another specific example is $\langle x,y : xy = 1 \rangle$ with $\nabla(x)=x_1 x_2$ and $\nabla(y)=y_2 y_1$. $\endgroup$
    – HeinrichD
    Dec 15 '19 at 23:11
  • $\begingroup$ your example co-represents the functor taking a monoid $M$ to the submonoid of $R\times L$ (where $R$ is the submonoid of right invertible elements and $L$ is the submonoid of left invertible elements) choosing all pairs $(x,y)$ with $xy=1$. $\endgroup$ Dec 16 '19 at 19:21
  • $\begingroup$ It's just a submonoid $M \times M^{op}$, also in alignment with Bergman's notation. $\endgroup$
    – HeinrichD
    Dec 18 '19 at 8:42
4
$\begingroup$

George Bergman characterizes representable endofunctors of monoids in 10.6 of https://math.berkeley.edu/~gbergman/245/3.2.pdf by classifying the comonoids in the category of monoids.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.