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Yep yep. My experience is only with finite state spaces, where the Markov process is described by a transition matrix (see Wikipedia). In that case, you have a directed edge from state to state labeled by a transition probability .
Note: I should be more careful. Sometimes, I like to "explode" categories into category of elements. For example, if we are thinking about sets , and a function sometimes I like to "explode" this so we have one morphism for each and .
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