# Is there a faithful functor from $CAT$ to the Kleisli category for the Giry Monad?

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?

• 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). May 1, 2019 at 15:23

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, $$\mathbf{Cat}$$, 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.