Abstract
In this talk, I'll discuss graphical languages that give syntax for regular and abelian categories. Every regular or abelian category has an associated regular or abelian calculus, a kind of graphical language with manipulation rules. This construction is 2-functorial and has a left adjoint given by a "syntactic category" construction; indeed the 2-category of regular categories is reflective in that of regular calculi, and similarly for abelian categories and abelian calculi. I will discuss these notions, and give a flavor of the graphical languages involved.