Philosophy of mathematics seems to focus primarily on set theory, which is probably a historical accident (just like it is a historical accident that set theory became the prevalent "foundation" in the 20th century). If you want to see things from other perspectives you could read things like:
S. Awodey: Structure in mathematics and logic: a categorical perspective. Philosophia Mathematica (3), vol. 4 (1996), pp. 209--237. (See also the subsequent discussion.)
F. W. Lawvere's "Down with "Foundations"! Up with algebra!" posting on the categories mailing list.
Perhaps other can suggest additional references in this direction.