in the ncatlab https://ncatlab.org/nlab/show/hypergraph they defined the category of hypergraphs: SimpHGrph has objects consisting of a pair of sets (V,H) equipped with a relation R⊆V×H, and morphisms (R⊆V×H)→(R′⊆V′×H′) consisting of pairs of functions (f:V→v′,g:E→E′) which preserve the relation, i.e., such that for all v∈V,e∈H, if (v,e)∈R then (fv,ge)∈R′.
my question is what does it mean by relation here? what type of relation would be in this? would it be the pair; the vertex singletons set and the edge containing those vertexes? for example the pair ({{v1},{v2},{v3}}, {v1,v2,v3}). Pardon me, if this is too stupid to ask.
Relation $R \subset V\times H$ literally means a subset in $V\times H$. It describes which vertices and edges are connected. For instance, the edge $e\in H$ is connected to a vertex $v\in V$ if and only if $(v,e) \in R$.
It generalises ordinary unoriented graphs in the sense that an edge $e$ might connect more than two vertices, e.g. if $(v,e),(v',e),(v'',e) \in R$ for three different vertices $v,v',v'' \in V$.