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Hi,
Ahman, Chapman and Uustalu have given some prescriptions as to when a container is a comonad and when it is a monad.
The prototypical data structure that is a monad is List, as described here.
In their paper on Containers and comonads, Ahman et alias give the example of Trees as a comonad, where the comultipication is trees of trees. Each node gets labelled with the sub-tree that extends from that node.
The next step in this reasearch is to find the container that is a Frobenius Monad. One could also find the container that is a bimonad, but I am most interested in the Frobenius monad.
I am a physicist and my intuitions come from physics. I am fairly convinced that spacetimes can be seen as containers.
I have written a paper that gives a model of experiments and how theories relate to experiments. I give an example of how experiments can be modeled with the List monad. It needs to be improved to include the idea from General relativity that there is no universal background clock. I believe I need Frobenius monads for that.
Panangaden and Martin showed that spacetimes are equivalent to interval domains.
My conjecture is that something very much like a domain of intervals is a Frobenius container. I was toying around with the idea and thought about a domain where each node contained a total order. Then the multiplication could take a domain of total orders of total orders to a domain of total orders by just concatenating.
I am interested in any example of a Frobenius container, so if you can think of one, please post it.
I havenâ€™t considered this question yet, but let me point out that we do have an article on containers at the nLab.
Also let me point out that if you set the format to Markdown+Itex, then you can easily link to your articles and dropbox. For example, see the source of this comment for creating this (irrelevant) link.
Incidentally, I just added to Frobenius monad a proof that every such comes from an ambidextrous adjunction.
Updated the links. Thank you Todd!!
I would like to point out that if you like the work I am doing and would like to join me, I am badly in need of collaborators. I would like to publish my current paper and the next one in a major journal if possible.
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