The notion of state and models for interaction, categorically

In a previous work we have investigated the world of TreeL-categories, where L is a meetsemilattice monoid, with particular attention to lifting factorization properties for TreeL-functors. Our method consisted in lifting constructions as far as possible through the different levels of our structures, namely: the lp 2-category L, the monoidal category TreeL of symmetric L- categories and TreeL-Cat. We did apply the same method to the factorization reflection property in order to shift Conduch´e’s theorem in this enriched setting. In this context we did prove the equivalence between the existence of a right adjoint to the inverse image functor and the factorization reflection property. The further condition imposing that we have to consider only Conduch´e enrichments is trivially satisfied in the Set case, when enriched categories are ordinary categories. In fact Set is a category of trees on the trivial meet-semilattice monoid I, freely generated by the empty set [17]. There, no factorization in the sense of Definition 2 is possible. TreeL-categories are particularly useful in modeling non-deterministic concurrent processes; in particular BehL is a powerful tool to represent operational semantics, both in the case of interleaving and true concurrent approach.