I have made the following observation (hopefully a correct one) when reading the paper Orbifolds as stacks:
They start with the strict $2$-category category of Lie groupoids, functors, natural transformations between functors. After realising (I think so) the rigid ness, and that it is not of interesting nature, they construct a weak $2$-category of Lie groupoids, bibundles and isomorphism of bibundles. Then, they consider to embed this weak $2$-category into a strict $2$-category of Stacks.
So, we start with a strict $2$-category, realise that it is difficult to work with, embedd this in a weak $2$-category and then strictify it to get a strict $2$-category. Surprisingly, this strict $2$-category is “interesting“ and “good enough to work on”.
Are there other “interesting” strict $2$-categories obtained from strictification of a weak $2$-category?
Are there other “interesting” strict $2$-categories obtained from strictification of a weak $2$-category obtained from “generalising” a strict $2$-category?
