I'm learning category theory and every definition of a groupoid has required that a groupoid be a small category (or some equivalent requirement) but I don't really know why.
So what is the reason that a groupoid is required to be a small category and what would happen if that requirement were dropped?
A reference would be great if at all possible.
Thanks.
The reason is that almost all groupoids which are encountered in practice are small (or at least essentially small). You can find a collection here. They come from various branches of mathematics.
Of course, for every category we may consider its core, which is a groupoid, which may be large. But when in practice does one consider, for example, the category of all sets with bijections as morphisms?
It is also worthwhile mentioning that Brandt, who invented groupoids, assumed them to be small. Groupoids were conceived as multi-object versions of groups, or as groups with partially defined operations. Although there are, of course, large groups in nature (for instance, the surreal numbers form a large field), most groups in practice are small.