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This paper adresses two issues in dealing with bicategories of fractions. Thefirst is to introduce a set of conditions on a class of arrows in a bicategorywhich is weaker than the one given in Pronk, Etendues and stacks asbicategories of fractions, but still allows a bicalculus of fractions. Theseconditions allow us to invert a smaller collection of arrows so that in somecases we may obtain a bicategory of fractions with small hom-categories. Weadapt the construction of the bicategory of fractions to work with the weakerconditions. The second issue is the difficulty in dealing with 2-cells, whichare defined by equivalence classes. We discuss conditions under which there arecanonical representatives for 2-cells, and how pasting of 2-cells can besimplified in the presence of certain pseudo pullbacks. We also discuss howboth of these improvements apply in the category of orbispaces.

Bicategories of fractions revisited: towards small homs and canonical 2-cells