Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: community: our work

Topic: Eric Dolores

Eric Dolores (Oct 11 2026 at 21:31):

Hello. I work with algebras over the operad of posets. I will describe here my previous work and, hopefully soon, ill talk about new preprints that we are still working on.

So I work with sets $Z$ together with operadic maps $End_{Poset}\rightarrow End_{Z}.$ And I use the language of category theory to bring properties of posets into what ever field $Z$ belongs to.

Eric Dolores (Oct 11 2026 at 21:41):

Today Ill talk about a paper on Combinatorics that studies the action of the Operad of series parallel posets on order polynomials.

What is category theory doing here? I first need to define Hasse diagrams and the lexicographic sum.

Hasse diagrams: This are graphs where every point of the poset is a node, and we draw edges between an element and their successors. The successors go above the elements.
Then a linear order is a vertical line with $n$ nodes. Here in the attached image from Wikipedia user ed g2s, we can find the Hasse diagram of the poset of divisors of 60.

[1/4]

Eric Dolores (Oct 11 2026 at 21:48):

Then the lexicographic sum is defined graphically as follows:
Let $P$ be a poset with $n$ points (an $$n-$$ary operation) and let $Q_1,\dots,Q_n$ be finite posets.
For every $i$ we take the Hasse diagram of $P$ and on the $i$ node we put the Hasse diagram of $Q_i$, then if there was an edge $p_i$ to $p_j$ in the Hasse diagram of $P$, this means the successor of $p_i$ is $p_j$. Then in the lexicographic sum we connect all the maximum points of $Q_i$ to the minimum points of $Q_j$.

In the case where $P=\{x_1<x_2\}$ and $Q_1=\{q_1,q_2\}, Q_2=\{q_3\}$ what we obtain is the poset $\{q_1<q_3,q_2<q_3\}$ with Hasse diagram two nodes at the same level, both of them connected to a third node above them.
[2/4]

Eric Dolores (Oct 11 2026 at 21:56):

Here come the operad part:

I work with the operad of finite posets. The operad has as $n$-ary operations posets with $n$ points, and the operadic composition
$o(P;Q_1,\dots,Q_n)=P(Q_1,\dots,Q_n)$
is the lexicographic sum.

Now in combinatorics people study the following polynomial introduced by Richard Stanley: $\Omega(P,n)$ which counts the number of "continuous maps/order preserving maps" from $P$ to the lineal order with $n$ points. Example, $\Omega(m,x)$ will be the number of maps from a linear order on $m$ points to a linear order on $x$ points. That is ${x\choose m}$ a polynomial on the variable $x$.

And my idea was to consider the set of generating series $Z(P)=\sum \Omega(P,n) x^n$ and define the action of the operad of poset on that set as:

$P(Z(Q_1),\dots,Z(Q_n))=Z(P(Q_1,\dots,Q_n))$ where $P,Q_1,\dots,Q_n$ are posets.

And then, at least for the posets that are linear orders, or disjoint union of points, or lexicographic sum of this two families, describe explicitly the action.
[3/4]

Eric Dolores (Oct 11 2026 at 22:13):

This sounds abstract and clean, just how category theorist like it, but in combinatorics we prefer explicit calculations. So note that for the poset on two points and no relation between those points $\{x,y\}$ we have
$\{x,y\}(Z(Q_1),Z(Q_2))=Z(\{x,y\}(Q_1,\dots,Q_n))$, now the lexicographic sum with $\{x,y\}$ is just the disjoint union of posets so we obtain $\{x,y\}(Z(Q_1),Z(Q_2))=Z(Q_1\cup Q_2)$ and by the definition of the polynomial, the number of maps from the disjoint union of posets to a line is the product of the number of maps from each poset to a line. So $\{x,y\}(Z(Q_1),Z(Q_2))$ is the Hadamard product of the generating series. It takes more effort to describe the action of $\{x_1<x_2\}$ but you can find it in the paper.

Why people in combinatorics may be interested in the operadic structure? the existence of this structure implies several combinatorial identities and new proofs of old combinatorial identities.

The other thing is, there were several results in combinatorics that were previously isolated results, and we show they are all part of a coherent story, which I think is a more interesting contribution. Now that I remember, there were already papers trying to study operads without saying it.

This is our paper An algebra over the operad of posets and structural binomial identities.

Interacting in another field does require some politics. For example, a common reply by reviewers is that we are just packing definitions. Which is true, but this brings functoriality. So one has to spend time convincing them that (sometimes) this is worth doing.

[4/4]

Eric Dolores (Oct 12 2026 at 21:01):

Today I will talk about the tensor product of operads.

Here I will follow the book Simplicial and dendroidal homotopy theory by Gijs-Heuts and Ieke Moerdijk.

We consider colored tree operads. That is, you take a tree, then the n-ary operations are the n-ary vertices of the tree. The compositions that are non zero are those compositions that generate a subtree.

For example consider the image (taken from the book Simplicial and dendroidal homotopy theory)

Then $k(l,m)$ is well defined but $k(k,k)$ should be zero. This is why this is a colored operad.

In particular there are linear trees, you can take $n$ unary operations and stack one into the other.
[1/n]