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.