Algebraic Subtyping$^1$ presents types as an initial algebra in the category of distributive lattices. This informs the design of the type inference algorithm and various correctness properties.
In the Nominal Sets literature$^2$, an initial algebra in the category of nominal sets is used to create datatypes that allow recursion modulo alpha equivalence.
Are there any other examples of initial algebra semantics in categories other than Set that help accomplish computational goals?
[$^1$]: Dolan, S. (2017). Algebraic Subtyping. BCS, The Chartered Institute for IT.
[$^2$]: Pitts, A. (2016). Nominal techniques. ACM SIGLOG News, 3(1), 57-72.