Categories with sums and right distributive tensor product (Q1861509)

Models for parallel and concurrent processes lead quite naturally to the study of monoidal categories. In a previous paper the author has studied a particular category \({\mathcal T}ree\) of trees, equipped with a non-symmetric tensor product, interpreted as a concatenation. This category seems to be very useful to represent (local) behavior of agents able to communicate. The category \({\mathcal T}ree\) is also provided with a coproduct (corresponding to choice between behaviors) and the tensor product is only partially distributive with respect to it, in order to preserve non-determinism. Such a category can be properly defined as the category of the (finite) symmetric category on a free monoid, when this free monoid is considered as a 2-category. The monoidal structure is inherited from the concatenation in the monoid. In the present paper the author proves that for every alphabet \(A\), \({\mathcal T}ree(A)\), the category of finite \(A\)-labeled trees, is equivalent to the free category that is generated by \(A\) and enjoys the afore-mentioned properties. The related category \({\mathcal B}eh(A)\), corresponding to global behaviors, is also proved to be equivalent to the free category that is generated by \(A\) and enjoys a smaller set of properties.