Saturation for classes of morphisms (Q2869329)

In this short note the authors prove that an externally saturated class \(\Sigma\) of morphisms in a category \(C\) is an internally saturated class (in the sense of P. J. Freyd and G. M. Kelly) iff it is externally saturated and admits a calculus of left fractions. They show, using a suitable shape functor, that every internal saturated class is also externally saturated. In a previous paper [Glas. Mat., III. Ser. 42, No. 2, 309--318 (2007; Zbl 1152.18001)], the second author proved that every internally saturated class has a calculus of left fractions, and in this paper the authors prove that the converse holds true provided that the category \(C\) has finite colimits and a terminal object.