Pawel Sobocinski: "Algebraic theories with string diagrams"

[@JosiahWarren]

That was great presentation

[@funktorial]

I don't buy the claim that considering algebraic theories as Lawvere theories escapes the reliance on particular presentation. In the commutative monoid example, the theory is described using *some presentation*, namely the free category generated by m,e quotiented by the congruence given the eqs. This is just as presentation-dependent as before. And of course it is, giving generators and relations to define some algebraic object is a presentation and there can be many presentations that give rise to the same object. I have a hard time seeing how any defense of this way of doing things wouldn't also be a defense for the classical universal algebraist's way of doing things (with term algebras and so on). Of course, there's still utility in considering things using Lawvere theories, it's a nice way of bundling and neatly describing some familiar constructions and points the way to further generalizations and progress. But it's not magic. We'll still have to describe infinite objects in finitistic ways.