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I am reading Patterson's paper Knowledge representation in bicategories of relations. It looks like it has many of the properties of the Markov categories which Fritz has been detailing in A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Is the bicategory of sets and relations actually a Markov category?

[Edit] Someone has pointed out that I may be asking about the bicategory version of Markov categories. So, the question is, is the bicategory Rel a Markov bicategory.

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  • $\begingroup$ If I read the Fritz link, I read that a Markov category is a category endowed with some additional structure (symmetric monoidal + ...). So given a category, the question "is this a Markov category" doesn't make sense (as "is this topological space a category?" is senseless). The closest I could imagine is whether it admits such a structure (which might be or not be unique). Is it the desired meaning of the question. $\endgroup$
    – YCor
    Commented Nov 22, 2021 at 0:13
  • $\begingroup$ @YCor that is the standard interpretation of what the question is asking, it's not a type mismatch, past phrasing the answer as "yes, using the [blah] monoidal structure" (if this is the case). $\endgroup$
    – David Roberts ♦
    Commented Nov 22, 2021 at 0:54
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    $\begingroup$ Do you mean a bicategorical version of Markov category? Or a truncation of Rel to a 1-category? And if the first, do you mean the definition of $\mathbf{Rel}$ as a 2-category at ncatlab.org/nlab/show/Rel#definition ? $\endgroup$
    – David Roberts ♦
    Commented Nov 22, 2021 at 0:58
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    $\begingroup$ OK, thanks for the clarification. Thankfully, $\mathbf{Rel}$ is a (1,2)-category (it is poset-enriched, not category-enriched, so questions of coherence for a 2-categorical version of a Markov category will not be so scary. $\endgroup$
    – David Roberts ♦
    Commented Nov 22, 2021 at 5:41
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    $\begingroup$ I have no answer to this question because I don't know what a Markov 2-category would be, but I'm interested in working on this problem. $\endgroup$
    – xuq01
    Commented Nov 23, 2021 at 15:42

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