I half-agree with you. The agreement part is that the algebraic structures that naturally arise in my programming experience are indeed ultra-basic: monoids, l

I half-agree with you. The agreement part is that the algebraic structures that naturally arise in my programming experience are indeed ultra-basic: monoids, lattices, semirings, partial orders, and so on. (When I need elementary linear algebra things are getting fancy!) The disagreement arises from the fact that the right home for these basic concepts is often surprisingly exotic. For example, it's often the case that what you need is not a monoid in the category of sets, but in the category of sets and relations or partial orders and monotone functions. Another example is when you want to represent syntax trees with scoped binding structure. Now a syntax tree is the dumbest possible algebraic structure (it's a free algebra with all generators and no relations), but when you have binding you need a free algebra in the category of presheaves over finite sets and injections.