Riemann noted that the hypergeometric function can be characterized by its global properties, without reference to power series, differential equations, or any other sort of explicit expression. His characterization is conveniently restated in terms of sheaves:

Suppose that we have a sheaf of holomorphic functions over $\mathbb{C} \setminus \{0,1\}$ which satisfy the following properties:

• It is closed under analytic continuation.
• It is closed under taking linear combinations.
• The space of function elements over any open set is two dimensional.
• If one analytically continues a function element along a line towards one of the points $0,1$ or towards infinity, it is bounded by a power.