Hopf Algebras from Posets
Posted by John Baez
As usual, visiting my friend Bill Schmitt in DC made me think about combinatorics. A long time ago, based on ideas of his advisor Gian-Carlo Rota, he developed some nice ways to get Hopf algebras from collections of partially ordered sets, or ‘posets’:
- Bill Schmitt, Incidence Hopf algebras, Journal of Pure and Applied Algebra 96 (1994), 299–330.
The last example in this paper later became known as the ‘Connes–Kreimer’ Hopf algebra. Why? Because those guys rediscovered it and applied it to renormalization in quantum field theory!
I would like to understand Bill’s constructions in a more category-theoretic way. But this raises some questions about posets of subobjects… and I’m hoping some of you can help me out!
Bill can construct a Hopf algebra from any collection of posets satisfying certain conditions. In examples, these collections of posets are often formed by taking a category and looking at all its ‘posets of subobjects’.
Any object $c$ in any category $C$ has a poset of subobjects, which I’ll call $Sub(c)$. For example, the subobjects of a set are just its subsets. The subobjects of a vector space are just its linear subspaces. And so on.
The two examples I just mentioned give interesting Hopf algebras… at least if we work with finite sets and finite-dimensional vector spaces. But I want to know why. In other words: I want to know which categories give collections of posets — namely, posets of subobjects — that make Bill’s construction work.
Here’s the condition that’s puzzling me most. I think some of you category theory experts can help me out.
Question: What conditions on a category make the following condition true? Given any object $c$ and any subobject $x$ of $c$, there’s an object $c/x$ such that $Sub(c/x)$ is isomorphic to the poset of all subobjects of $c$ that are greater than or equal to $x$.
For example, the category of sets has this property, where $c/x$ is just the set $c$ minus the subset $x$. Why? Because subsets $a$ with $x \subseteq a \subseteq c$ are in natural one-to-one correspondence with subsets of $c/x$… and this correspondence is order-preserving.
Also, the category of vector spaces has this property, where $c/x$ is just the vector space $c$ modulo the subspace $x$. Why? Because subspaces $a$ with $x \subseteq a \subseteq c$ are in natural one-to-one correspondence with subspace of $c/x$… and this correspondence is order-preserving.
Now, in both these cases I know what’s going on. The category in question has coproducts, and for any subobject $x$ of $c$, there’s an object $y$ such that $x + y \cong c$. We can then take $c/x = y$.
In other words, every subobject $x$ has a ‘complement’ $y$, such that $c$ is the sum of $x$ and $y$.
But this is a very drastic condition. It’s not true in the category of modules of a typical ring. Nonetheless, in any such category we can still get things to work: just $c/x$ to be the usual quotient. In fact, we can do this in any abelian category.
So: what condition most naturally handles both abelian categories and categories like $Set$, which are not abelian?
I’ll restate my question more formally, just for people who like lots of precision.
Given two elements $x,y$ of a poset $P$, the interval $[x,y]$ is the set of all $a \in P$ lying between $x$ and $y$:
$x \le a \le y$
This interval becomes a poset in its own right.
Question: What conditions on a category $C$ make the following true? Given any object $c$ and any subobject $x \in Sub(c)$, there is an object I’ll call $c/x \in C$ such that
$[x,c] \cong Sub(c/x)$
as posets. (Here note that $c$ is a subobject of itself — the biggest one.)
Re: Hopf Algebras from Posets
Incomplete as it is, maybe the $n$Lab entry on subobject ($\to$ $n$Lab) is already better than the Wikipedia entry.