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The Kleisli category for the Giry Monad, $G$, has Sets as objects and the morphisms are maps from sets to probability densities over sets. For each pair $(A, G(B))$, there is a special case where $G(B)$ is equal to 1 for a single set element for all $b \in B$, and is 0 for all other elements. This means, the homset $(A, G(B))$ contains a representation of a function from A to B. This means that the Kleisli category of the Giry monad has a subcategory that is all concrete categories. There should be a faithful functor from $CAT$ to Kleis(G), the Kleisli category of the Giry Monad. Is this correct?

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    $\begingroup$ I lose you at "has a subcategory that is all concrete categories". What does this mean? It seems to me that the observations you made before this just show that there is a faithful functor from Set to Kleis(G). $\endgroup$ May 1 '19 at 15:23
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First of all, note that this isn't the usual definition of the Giry monad, which is usually defined on the category of measurable spaces and measurable maps. But of course, a subcategory of that consists of sets with the full $\sigma$-algebra, so it doesn't change your argument : as you noted, there is a faithful functor from $\mathbf{Set}$ to your Kleisli category.

In particular, still as you noted, because faithful functors are stable under composition, any concrete category has a faithful functor to that Kleisli category.

In particular, $\mathbf{Cat}$, the category of small categories, which is concrete (for instance, send $C$ to $\mathbf{Ob}(C)\times \mathbf{Ar}(C)$ and functors to the obvious thing) has a faithful functor to that Kleisli category. But I don't see what sort of information "there exists a faithful functor to the Kleisli category" tells you

It's obviously not the case for $\mathbf{CAT}$, however, which is not locally small.

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