[FRIAM] (no subject)

[Jon Zingale]

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 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 the
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 domains 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 woman 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, I simply needed to read a little further in Spivak and Fong
;)
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