David Roberts
bicategory of fractions

Given a bicategory and a class of 1-arrows, one may be interested in adjoining pseudoinverses to the bicategory while only adding the minimal structure so this is again a bicategory. More concretely, given a bicatgory $B$ with a class of 1-arrows $W$, find the (weakly - these are bicategories after all) universal solution to the problem of finding a homomorphism

$$B\to C$$B \to C

that sends elements of $W$ to equivalences. In a paper in 1996, Pronk gave conditions for when this is (nicely) do-able, and constructed the ‘bicategory of fractions’ $B[W^{-1}]$.

Although Pronk’s construction is quite explicit (modulo some choices, all of which give biequivalent results), it is not necessarily the only solution. Hence she gives a characterisation theorem for when a bicategory $D$ equipped with a $W$-inverting homomorphism $B\to D$ is biequivalent to $B[W^{-1}]$.

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Details here.