TITLE: Is the free group on an empty set defined?
QUESTION [11 upvotes]: I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.

REPLY [3 votes]: Left adjoints preserve colimits, in particular the empty colimits, i.e. initial objects. In particular, the free group functor takes the initial set, i.e. $\emptyset$, to the initial group, i.e. the trivial group. Another example: The polynomial ring without variables is the base ring.
A must read: sexy vacuity