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This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references about a categorical treatment of probability theory I'm interested in a probabilistic treatment of category theory. I wonder whether concepts such as random morphisms or functors have been considered so far, for example in a "dynamical graph theoretic" perspective. The idea is to build categories depending probabilistically on objects and morphisms of other classical, "deterministic" categories. My goal is to apply this kind of considerations to linguistics to understand the way a given language emerges from the way of life of its speakers and how it influences it back. I'm indeed convinced that no proper translation tool can exist without taking into account accurately the actual existence/life of the speakers of the considered languages. I'm sorry if MO is not the right place to ask such a question, but I have no idea where I could ask it otherwise as far as I feel the need for categories to tackle my problem (maybe a forum about computer science or artificial intelligence would be more appropriate, if so, feel free to tell me about such a website).

Thanks in advance for any reference and/or insight.

Edit: here comes an illustration of what I mean:

Linguistic graph Description

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    $\begingroup$ Could you give a hint on how you hope to use a concept such a random category theory in understanding the evolution of human language? $\endgroup$ – Liviu Nicolaescu Apr 25 '15 at 15:04
  • $\begingroup$ For example one could view what I denote by ENV and EX as categories such that there exists a random functor F from ENV to EX, viewed as a direct influence, mapping an object A of ENV to an object F(A) of EX with a given probability. The arrow from WAL to ENV, hence a feedback, would be another random functor, while the functor from EX to WAL would not be random. The very existence of feedbacks requires a dynamical point of view, setting categories as vertices of a random graph, and random functors as edges thereof. $\endgroup$ – Sylvain JULIEN Apr 25 '15 at 15:52
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    $\begingroup$ You might like some recent work by Alex Simpson. Motivated by an answer to a question here on MO he developed a categorical framework for treating the probabilistic notion of conditional independence for random variables. Online are slides of his talk "Category-theoretic structure for conditional independence" at the Cambridge category theory seminar. $\endgroup$ – მამუკა ჯიბლაძე Apr 25 '15 at 17:18
  • $\begingroup$ Thank you very much. By the way, could you tell me how to pronounce your name? I already noticed some valuable comments from you and I feel intrigued. Is your mother tongue Kannada? $\endgroup$ – Sylvain JULIEN Apr 25 '15 at 18:00
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    $\begingroup$ I do not understand what these categories are because the links you included point to suspicious looking sites. $\endgroup$ – Liviu Nicolaescu Apr 25 '15 at 22:27
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Let me try to talk you out of your current viewpoint. I am interested in language translation also and am somewhat familiar with the existing techniques. The most popularly the n-gram algorithm - look at the n words behind and n words in front of a word to "determine its meaning". These fail catastrophically in trying to translate from a language such as Thai where the order is critical. In learning Thai, or any of the Asian languages, you quickly come to learn just how much of the conversation is implied. The word "I" is typically dropped in Thai... So how does this relate to what you are after? I believe you are in fact interested in Probability Theory from the Categorical Viewpoint, as it allows you to use conditionals. When you talk to person A you will respond to a question differently than you might respond to the same question from person B because there are more/less implicit assumptions you make when talking to someone. These conditionals require (obviously) probability theory to even begin modeling the problem. There are many aspect to the problem you are considering - to have a "complete" picture of what's occuring you also need a "Theory of Mind" (i.e., I need to model your thought process in replying to you appropriately - talking to a 10 year old child is vastly different than replying to a PhD with 20 years experience). For my money, I would start with the categorical viewpoint of probability. Which is precisely what I am doing.

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Instead of a self-contained answer, here is a "random" idea: I think that what you're asking is not the dual of the question you mention, but the internalisation. The reason is that "meta category theory", that is, the theory of category theory, is naturally treated with the tools of category theory itself.

Let's study random functors between two categories $\mathcal{C}$ and $\mathcal{D}$. What you're actually doing is equipping the set of functors $\operatorname{Cat}(\mathcal{C}, \mathcal{D})$ with a probability measure. But there already is a giant machinery to treat functor categories and the like with any amount of category theory you like, the only difference here is that we have enriched it with some suitable category of probability measures.

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