(reposted for readability) EricS,
I am sorry to say that with the disruption of the nabble Friam server, and with my head buried in work, I managed to miss your response to my queries about your approach to Fisher's Theorem. Thankfully, RogerF brought your response to my attention. In the last few months since we engaged this thread, I have steeped myself in some of the hypergraph literature[⍼]. What follows is an attempt to state what I think I understand and to see how closely it comes to your own understanding. My question (at the time) regarding whether hypergraphs make up a topology was in part my wondering whether we get an algebra from hypergraph parts, which it seems is how *cospan algebras* enter the picture. The Wolfram podcast you sent, and to which I am presently listening, is giving me insight into how and why hypergraphs came to be considered by those interested in generative/constructor physics. His metaphysical starting point: that space is *material*, that it has a kind of *local logic of connectivity*, and that in the Kelvin and Tait tradition we can interpret all that we see as a manifestation of self-interaction in an ether that we call space. This monadic/monist description is quite general and so it is becoming a *Pauling point* in many complexity-oriented theories. A brief survey includes Petri nets, object recognition, protein interactions, open Markov networks, chemical reactions, GUI design, automata theory, and anywhere that one can imagine Conway's madman sitting at an infinitely sprawling synthesizer patch bay. One striking feature of these models is that whether or not anything in the universe actually *happens simultaneously*, we can witness that some of the details at one timescale are found to be indistinguishable at another. This suggests that it can be useful to write an algebra of *boxed* interactions, where we may know nothing about the implementation except for which nodes to use as inputs and which to use as outputs (though possibly stronger types). At an extreme, as with the *operad* formalization, one can simply specify ports. What seems to make the hypergraph formalization so useful is that we now have semantics for these things, whether *decorated* or *structured* cospans. In effect, this means that we get functors that not only *name* ports through the typical unit adjunction (giving rise to all the familiar play of adjoint relationships and the tracing of natural equivalences) but also whose domain can be dynamical. This is especially wonderful if you want the *names for things* and the *things you name* to be made of the *same stuff* and without concern for whether or not things are simply points "way down there" somewhere. This week, through work, I met a researcher whose recent work concerns long-timescale dynamics (was it milliseconds?). She studies protein self-interactions at various stages of denaturing. I left the conversation with the sense that *behavioral classification* of proteins is important and that being able to identify some small *generative germ* of probable interactions is key. It seems to me that this is another place where it might be interesting to investigate via a hypergraph approach. There, we see that the *names* are effectively the internal dynamics of these core interactions, that *composability* of interactions corresponds to an algebra of names, and that all of the above can be situated harmoniously enough in a computational context. So far, what is written above is about as much as I understand. Thank you for the Springer link. Unfortunately, the pay-wall around that work is too rich for my blood. I would love it if someone with access can gift me a copy off-list. Cheers, Jon [⍼] For interested lurkers, I have compiled a list of papers that I found helpful in getting up to speed with hypergraphs: https://arxiv.org/pdf/1911.04630.pdf https://arxiv.org/pdf/1704.02051.pdf https://arxiv.org/pdf/1812.03601.pdf https://arxiv.org/pdf/1806.08304.pdf Embarrassingly enough, I simply needed to read a little further in Spivak and Fong ;)
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