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I entered one characterisation of Markov chain as a coalgebra.
I was doing some research on Lawvere-Giry monad with Roland Friedrich few years ago, and I am a bit disappointed, that this formulation seems just to scratch the surface of the stochastic theory. I read some of Doberkat's papers on the subject, they are written very formally, lots of notation and text for my taste and much less content. I did not know he published the book in the meantime. I stll hope to resume and finish the work with Roland, we had some generalization of Giry monad related to projective measures and coherent states in physics. This is conceptually nice, and one needs to know a little bit of spectral theory to do it properly, but it is not very deep so far. Roland was prompted to think about this starting after he heard a talk of Voevodsky on this approach, who wanted to sue the formalism in applciations in mathematical biology. Gromov was in the audience and complained that it is just a reformulation and Voevodsky agreed but said that it fist his way of concentrating to put it into categorical framework. I don't find thinking about stochastic processes via Giry monad simpler but more difficult. It was however our hope that in some continuous limits where the study of various spaces in probability things are nontrivial, some variant of Giry monad could help with nontrivial insight. Also I was expecting some generalizations of Giry monad for a class of abstract lattices...and I think that in the meantime some paper roughly in similar direction appeared by other authors.
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