It some searching, but I found the paper where I first got a sense about operads. I hope that you also find it useful: https://arxiv.org/pdf/1806.08304.pdf
In particular, at the top of page 4 are a collection of diagrams depicting composition in categories, monoidal categories, traced monoidal categories and finally in hypergraph categories. Each, as far as I understand the concept (which is new to me) has a "governing" operad. For instance, the category of 1-dimensional oriented Cobordisms *governs* the traced monoidal category and the category of Cospan algebras does the same for hypergraphs. I figure you probably know the last relation because you are familiar with the bipartite graph representation of hypergraphs. I suspect there are a number of nice duality properties in the Cospan category, I am not sure if they are the kind you were looking for (a certain duality between states and events).
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