  A minifold is a smooth projective $n$-dimensional variety such that its
bounded derived category of coherent sheaves $\D^b(X)$ admits a semi-orthogonal
decomposition into an exceptional collection of $n+1$ exceptional objects. In
this paper we classify minifolds of dimension $n \leq 4$. We conjecture that
the derived category of fake projective spaces have a similar semi-orthogonal
decomposition into a collection of $n+1$ exceptional objects and a category
with vanishing Hochschild homology. We prove this for fake projective planes
with non-abelian automorphism group. We construct new examples of phantom
categories with both Hochschild homology and Grothendieck group vanishing.
