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:
infinite: $\forall_{i} \exists_{j \geq i} \exists_{u \in U_j} \ u \in H$
directed: $\forall_{u,v \in H} \exists_{i,j,k \in I} \exists_{w \in H} \ u R_{i,k} w \ \wedge \ v R_{j,k} w$
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]
