Categories of fractions and infinitary structures

Steve Lack – 14 August 1996

This talk was a gentle introduction to categories of fractions and coinverters, and conditions under which structure on a category passes to the category of fractions. Most of the talk was taken up in a summary of left and right caculi of fractions (Gabriel-Zisman), pullback congruences (BĂ©nabou) and the theorem of Kelly-Lack-Walters that for T a strongly-finitary 2-monad on Cat, the forgetful 2-functor from T-Alg to Cat creates reflexive coinverters. It was then noted that this can be extended to 2-monads involving operations of possibly infinite but still discrete arity, at the expense of making the assumption that the class $\Sigma$ of arrows to be inverted admit a calculus of left or right fractions.