The traditional denotation of a structured set object is something like

$(1)\qquad(X,\mathcal S_1,\dots\mathcal S_n)$

for some "structures" $\mathcal S_1,\dots\mathcal S_n$ on a set X.

The modern definition of a category with sets as objects and functions as morphisms includes a faithful functor $F:\mathbf C\to\mathbf{Set}$, that is, a functor such that $\alpha\ne \beta\implies F(\alpha)\ne F(\beta)$ for all morphisms $\alpha,\beta\in Mor(\mathbf C)$.

Now there is a faithful functor from Rel to Set, $\mathcal P(\cdot)$ the covariant power set functor, and my question is:

is it possible to express this category as a category of structured sets as in $(1)$ with structure preserving functions as morphisms?

up vote 2 down vote accepted

Well, $\mathbf{Rel}$ is a full subcategory of $\mathbf{CJS}$, the category of complete join-semilattices (and join-preserving maps). The embedding sends a set $X$ to the powerset $\mathscr{P} X$ and a relation $R : X \not\to Y$ to the unique join-preserving map $r : \mathscr{P} X \to \mathscr{P} Y$ where $r \{ x \} = \{ y \in Y : x \mathrel{R} y \}$. Thus you can realise $\mathbf{Rel}$ as a full subcategory of a category of (infinitary) algebraic structures.

  • I almost expected something like that. But it would be cool to see an explicit description. – Lehs Jan 5 '15 at 14:59
  • This is an explicit description. – Zhen Lin Jan 5 '15 at 15:13
  • Do you mean that $(\mathscr PX,\vee)$ are the objects I asked for? – Lehs Jan 5 '15 at 18:28

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.