  We provide a new approach to categorical graph and hypergraph theory by using
categorical syntax and semantics. For each monoid $M$ and action on a set $X$,
there is an associated presheaf topos of $(X,M)$-graphs where each object can
be interpreted as a generalized uniform hypergraph where each edge has
cardinality $\#X$ incident vertices (including multiplicity) and where the
monoid informs what type of cohesivity the edges possess. One distinguishing
feature of $(X,M)$-graphs is the presence of unfixed edges. We prove that
unfixed edges are a necessary feature of a category of graphs or uniform
hypergraphs if one wants exponentials and effective equivalence relations to
exist in the category. The main advantage of separating syntax (the
$(X,M)$-graph theories) from semantics (the categories of $(X,M)$-graphs) is
the ability to interpret the theory in any cocomplete category. This
interpetation functor then yields a nerve-realization adjunction and allows us
to transfer structure between the category of $(X,M)$-graphs and the receptive
cocomplete category.
