Categorial description of the subposet of $\prod_{i \in I}P_i$ of all $x \in \prod_{i \in I}P_i$ with $\{i \in I \mid x_i = \bot_{P_i}\}$ cofinite

Question

By a grounded poset, I mean a poset $P$ with a least element $\bot_P$.

Definition. Whenever $P$ is a family of grounded posets, write $\bigotimes_{i \in I}P_i$ for the subposet of $\prod_{i \in I}P_i$ consisting of all $x \in \prod_{i \in I}P_i$ such that $\{i \in I \mid x_i = \bot_{P_i}\}$ is cofinite.

Example. Let

1. $\mathbf{N}$ denote the set $\{0,1,2,\ldots\}$ of natural numbers
2. $\mathbb{N}$ denote the poset of natural numbers equipped with the usual order
3. $\mathbb{N}^\times$ denote the poset of non-zero natural numbers equipped with the divisibility order

Then there is an order isomorphism $f : \bigotimes_{n \in \mathbf{N}}\mathbb{N} \rightarrow \mathbb{N}^\times$ given as follows.

$$x \mapsto \prod_{i \in \mathbf{N}}p_i^{x_i}$$

(where $p_i$ is the $i$th prime number.)

Question. Is there a sleek category-theoretic description of $\bigotimes_{i \in I}P_i$?

Answer 1

Your $\bigoplus_{i \in I} P_i$ is the colimit of all finite products of some of the $P_i$; in symbols, $\bigoplus_{i \in I} P_i =\mathop{\mathrm{colim}}_{F \subset I, |F|<\infty} \prod_{i \in F} P_i$.