OK. Sorry. I should have assumed you'd already thought it through. On 6/24/20 8:55 AM, Jon Zingale wrote: > Gisin's yapping is yapping about sequences, as are morphisms into Lawvere's > (Y^Nat, β) object. The onto-property of morphisms from X gives the tipping > point where all sequences from the perspective of Y are covered even though > X may be doing more. That Lawvere is constructing his objects in a category > of dynamical systems, he is talking about evolution of state. One of the > best treatments of control theory from a categorical perspective is in Arbib > and Manes. There, they construct observability and realizability via > free/co-free dynamics and highlight the connection the two concepts share > via duality. Similar to the point I was making about Markov being a matter > of perspective (model), while dynamics are not static in one frame they are > in another. I hope that I am not being too obvious while missing your point. > There are graph-theoretic interpretations of randomness as complete graphs, > where everything is connected to everything. One interpretation is that any > structure imaginable arises as a sub-object. Another, perhaps by assigning > non-zero transition probabilities to all the edges, would be that any state > is reachable from any other. I am not sure I am responding appropriately to > your post.
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