Width of poset of posets

Question

Let $P$ be the set of all labelled posets on $[n]$ ordered naturally by edge extension. What is the cardinality of the largest antichain in $P$?

Answer 1

Nice question. This is a long-comment to effect that: "okay, here's a natural family of generalizations of your question." So in particular, I make no attempt at answering your (very interesting) question.

Definition. Given a functor $\mathbf{f} : \mathbf{D} \leftarrow \mathbf{C}$ and an object $Y$ of $\mathbf{D}$, lets define that $\mathbf{f}^*(Y)$ is the subcategory of $\mathbf{C}$ given as follows:

• objects: those objects $X$ of $\mathbf{C}$ satisfying $\mathbf{f}(X)=Y.$

• arrows: those arrows $\varphi : X_1 \leftarrow X_0$ of $\mathbf{C}$ satisfying $\mathbf{f}(\varphi) = \mathrm{id}_Y.$

It is readily checked that:

Proposition. If $\mathbf{f} : \mathbf{D} \leftarrow \mathbf{C}$ is a faithful functor, then $\mathbf{f}^*(Y)$ is always a thin category, for each object $Y$ of $\mathbf{D}$.

In other words, if $\mathbf{f}$ is faithful, then $\mathbf{f}^*(Y)$ is a preordered set.

Now given a preordered set $P$, write $\sim_P$ for the binary relation defined on $P$ as follows:

$$x \sim_P y \iff x \lesssim_P y \wedge y \lesssim_P x$$

It follows that $P/\sim_P$ is a partially ordered set. Define that $P/\sim$ is shorthand for $P/\sim_P.$

What this all means is the following. Suppose $\mathbf{C}$ is a concrete category and write $\mathbf{u}_\mathbf{C}$ for the corresponding forgetful functor. Then we get a functor $$\mathbf{p}_\mathbf{C} : \mathbf{Pos} \leftarrow \mathrm{core}(\mathbf{Set})$$ given on sets as follows:

$$\mathbf{p}_\mathbf{C}(Y) = (\mathbf{u}^*_\mathbf{C}(Y))/\sim$$

As a general rule, if $\mathbf{C}$ is a concrete category, then $\mathbf{p}_\mathbf{C}$ deserves to be studied! One thing we can ask about $\mathbf{p}_\mathbf{C}$ is: given a set $Y$, what is the largest cardinality of an antichain in $\mathbf{p}_\mathbf{C}(Y)$?

Your question is the special case where $\mathbf{C} = \mathbf{Pos}$. But we can replace $\mathbf{Pos}$ with $\mathbf{Top}$, or the category of digraphs, or whatever, and we'll get different answers.