<html><head><meta http-equiv="Content-Type" content="text/html; charset=us-ascii"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Jon, thank you so much for all of these, and also for the pointers to where to look for connections.<div class=""><br class=""></div><div class=""><div>This would be a cool route to expressing dualities, if I am asking the right question.</div><div><br class=""></div><div>Eric</div><div><br class=""></div><div><br class=""><blockquote type="cite" class=""><div class="">On Oct 30, 2021, at 3:00 PM, Jon Zingale <<a href="mailto:jonzingale@gmail.com" class="">jonzingale@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" class=""><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">It some searching, but I found the paper where I first got a sense about<br class="">operads. I hope that you also find it useful:<br class=""><a href="https://arxiv.org/pdf/1806.08304.pdf" class="">https://arxiv.org/pdf/1806.08304.pdf</a><br class=""><br class="">In particular, at the top of page 4 are a collection of diagrams<br class="">depicting composition in categories, monoidal categories, traced monoidal<br class="">categories and finally in hypergraph categories. Each, as far as I<br class="">understand the concept (which is new to me) has a "governing" operad.<br class="">For instance, the category of 1-dimensional oriented Cobordisms *governs*<br class="">the traced monoidal category and the category of Cospan algebras does<br class="">the same for hypergraphs. I figure you probably know the last relation<br class="">because you are familiar with the bipartite graph representation of<br class="">hypergraphs.<br class=""><br class="">I suspect there are a number of nice duality properties in the Cospan<br class="">category, I am not sure if they are the kind you were looking for (a<br class="">certain duality between states and events).<br class=""></div></div>
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