Let $\mathcal C = (\mathcal C_0,\mathcal C_1)$ be a (small) strict monoidal category. Pick a field $\mathbb K$, and let $\mathbb K[\mathcal C_1]$ be the vector space with basis the morphism of $\mathcal C$. It is an associative unital algebra under tensor product $\otimes$ (the identity morphism on the $\otimes$ unit is the algebra unit).
I will now define a coassociative comultiplication on $\mathbb K[\mathcal C_1]$, although without restriction on $\mathcal C$ the comultiplication will not converge. I'll give two descriptions:
- $\mathbb K[\mathcal C_1]$ is an associative algebra not only under $\otimes$, but also under composition: if $a,b \in \mathcal C_1$, then $ab = a\circ b$ if that composition is defined in $\mathcal C_1$, and $0$ otherwise. But $\mathbb K[\mathcal C_1]$ has a distinguished basis (namely $\mathcal C_1$), and hence a distinguished map $\mathbb K[\mathcal C_1] \to (\mathbb K[\mathcal C_1])^\*$; using this map, turn the composition multiplication into a comultiplication.
- For each morphism $c \in \mathcal C_1$, there is some set $\{(a,b)\in \mathcal C_1 \times \mathcal C_1 \text{ s.t. } a\circ b = c \}$ of ways to factorize $c$. Define $\Delta(c) = \sum_\{ a\circ b = c \} a\otimes b$; where here the $\otimes$ is the exterior one (not the other multiplication on $\mathbb K[\mathcal C_1]$.
From either description, it's clear that the comultiplication isn't really defined: in general that sum diverges. So let's suppose that $\mathcal C$ has the property that any morphism has only finitely many factorizations. Clearly this requirement is evil.
Question 0: Is there a less evil way to talk about this comultiplication? Actually, even the requirement that $\mathcal C$ be strict is evil, but without it $\mathbb K[\mathcal C_1]$ is not associative. Is there a less evil fix for this?
The comultiplication is co-unital. The counit on $\mathbb K[\mathcal C_1]$ sends identity morphisms to $1\in \mathbb K$ and non-identity morphisms to $0$. (A less-evilization might want to send, say, isomorphisms to $1$, or something.)
So, I have a vector space $\mathbb K[C_1]$ with a multiplication (coming from the monoidal structure on $\mathcal C$) and a comultiplication (coming from the composition structure on $\mathcal C$).
Question 1: Are there simple general conditions that assure that this structure is a bialgebra?
In the categories I am most interested in, $\mathbb K[\mathcal C_1]$ is a bialgebra. My intuition is that when $\mathcal C$ is sufficiently free, everything works. Here's an example. The category of braided graphs has objects the non-negative integers, thought of as distinguished subsets of $\mathbb R$. A morphism between $m$ and $n$ is: a graph $G$ with $m$ univalent vertices marked "in" and $n$ univalent vertices marked "out", along with a smooth embedding $G \to \mathbb R^2 \times [0,1]$ so that $G \cap \mathbb R^2 \times\{0\}$ consists of precisely the $m$ "in" vertices, spaced out on the integers $\{1,\dots,m\} \times \{0\} \times \{0\}$, and similarly for the out vertices, and such that every edge of $G$ is never horizontal. Two morphisms are identified if they are isotopic rel boundary among embedded graphs with non-horizonal edges. Composition are the monoidal structure are obvious. Equivalently, the category of braided graphs is the free braided monoidal category generated by a single basic object $V$ and a basic morphism in each $\hom (V^{\otimes m}, V^{\otimes n})$.
In any case, once you have a bialgebra, you are lead inexorably to the following question:
Question 2: When is $\mathbb K[\mathcal C_1]$ Hopf?
For very free categories, it is Hopf: a free category is graded, by setting the generators to have grading $1$; the degree-zero part is $\mathbb K[\text{identity maps}]$, and these themselves are graded by the number of objects; the degree-zero part of this is $\mathbb K$, generated by the identity map on the monoidal unit; then bootstrap back up. Probably this works for less-free things too, using filtrations rather than gradings (i.e. filtered quotients of free monoidal categories).
