  We prove that the global dimension of the complex algebra of the monoid of
all partial functions on an n-set is $n-1$ for all $n\geq 1$. This is also the
global dimension of the complex algebra of the category of all epimorphisms
between subsets of an $n$-set. In our proof we use standard homological methods
as well as combinatorial techniques associated to the representation theory of
the symmetric group. As part of the proof, we obtain a partial description of
the Cartan matrix of these algebras.
