I'm looking for the directed (or equivalently projective) limit of a directed family of relations in the category of sets with relations between them.

Consider a family $\{ R_{ij} \subseteq U_i \times U_j \mid i,j \in I \}$ of relations, with $\langle I, \leq \rangle$ a directed order, and $R_{ik} = R_{ij} \cdot R_{jk}$ for any $i \leq j \leq k$. I think the directed limit and the projective limit in the category of sets and relations coincide and the resulting set given by a subset of $\mathcal{P}(\uplus_{i \in I} \, U_i)$ consisting of exactly those sets $H \subseteq \uplus_{i \in I} \, U_i$ that are:

  1. infinite: $\forall_{i} \exists_{j \geq i} \exists_{u \in U_j} \ u \in H$

  2. directed: $\forall_{u,v \in H} \exists_{i,j,k \in I} \exists_{w \in H} \ u R_{i,k} w \ \wedge \ v R_{j,k} w$

  3. projecting: $\forall_{i \leq j} \forall_{u \in U_i} \forall_{v \in U_j} (u R_{i,j} v \ \wedge \ v \in H) \ \Rightarrow \ u \in H$

Can anyone confirm this (and perhaps even provide a reference?)

The question pops up because I am studying a particular kind of ordering as a category and would like to justify my notion of limit there. The best way seems to be to show that it is actually the category-theoretic notion of limit, and because I only want one notion (not a notion and a co-notion) I'm turning to the category of relations for inspiration. The definitions I use are based on [Adámek-Herrlich-Strecker] and [Mac Lane]

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    $\begingroup$ Hi Pieter and welcome to Math.SE ! What is the "I" in the formula n.2? In your first line you say "directed family of relations" what do you mean by this? On the other hand, in your 3rd line you say that this family only has the property of being "closed under concatenation". So which is which? Also "concatenation" means "composition in Rel", right? So this family could consist of just one arrow $R_{12}$? If yes, why do you call formula 1 infinite? Can you please give a reference where you found this problem? $\endgroup$ – magma Mar 21 '13 at 13:24
  • $\begingroup$ Editted the question to clarify... The problem comes from my own research. I'm not a category theorist, but I'd like to use it as a compass... $\endgroup$ – Pieter Cuijpers Mar 22 '13 at 11:38
  • $\begingroup$ Indeed, a directed order can just be {1,2}. The idea, however, is that it is something infinite. For the definition this is not necessary, but the limit trivially becomes `the maximum' if the index set is not... Maybe infinite is not a good name for 1), but I don't have a better one. $\endgroup$ – Pieter Cuijpers Mar 22 '13 at 12:40

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