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The 'Probabilistic compilation as functor' job description has enlarged: 'probabilistic compilation as functor' is only one instance of studying the properties of 'kleisli database instances', as defined by Spivak here: https://arxiv.org/abs/1209.1011 . The enlarged description is to establish existence of limits/colimits, kan extensions/lifts, for particular monads.
Quick update - we have identified one of the categories of particular interest for this project: https://arxiv.org/abs/1701.02547