One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a sense, the notion of independence is what distinguishes probablity theory from the theory of finite measures.

Is there a categorial way to make sense of and enlighten the notion of independent products in category theory?

It is possible to formulate independence in Lawvere's category of probabilistic mappings (Borel spaces as objects and Markov kernels as morphisms) in terms of constant morphisms, but I think this is not very enlightening, conditional independence is built into the morphisms. Maybe, this is what one has to do when putting probability center stage?

I do know the rudiments of categry theory, but I would prefer an answer that does not require too much immersion in category thory, provided that is possible.

One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a sense, the notion of independence is what distinguishes probablity theory from the theory of finite measures.

Is there a categorial way to make sense of and enlighten the notion of independent products in category theory?

It is possible to formulate independence in Lawvere's category of probabilistic mappings (Borel spaces as objects and Markov kernels as morphisms) in terms of constant morphisms, but I think this is not very enlightening, conditional independence is built into the morphisms. Maybe, this is what one has to do when putting probability center stage?

I do know the rudiments of categry theory, but I would prefer an answer that does not require too much immersion in category thory, provided that is possible.