I've been saying for a few years that design patterns are algebras seen dimly. People have asked me for references for that, and I haven't had anything to point them to. At this point I would modify it as algebras plus invariants.

I would say category theory is the wrong title for this, though. Category theory is occupied with the algebras that emerge when you put algebras into the same space and look at mappings among them. It emerged in algebraic topology where you would construct a family of algebras that describe aspects of a geometric structure and then establish connections among them to talk about the effects of operations on those geometric structures. Categories gave you one algebra, functors gave you families of algebras, and natural transformations gave you operations on the underlying spaces.

Many of the algebras described here and others that I have found useful emerged earlier in abstract algebra or universal algebra (monoids, lattices) or in parallel in combinatorics (magmas).

The category theoretic view is often an inconvenient setting. Lattices, for example, are best dealt with simply as a set with some operations that obey specific axioms. You get theorems from the axioms, and if you need one of those theorems to apply to your system, you make sure you implement it in such a way that it obeys those axioms.

Or if you have an algebra near to what you need, you may be able to modify it in the abstract setting to keep properties that you want. I ended up doing this with lattices to describe tagging regions of genomes.