While the standard definition of a finite element is the one given by Ciarlet in his 1978 book, we have been developing an alternative approach, called Finite Element Systems, based on category theory. Then the mesh is interpreted as a category, and the discrete function spaces attached to cells, provide a contravariant functor to the category of finite dimensional vector spaces. In other words a finite element is a particular kind of presheaf. Restrictions from cells in the mesh to subcells play a crucial role. The unisolvence of degrees of freedom in Ciarlets definition corresponds to a softness condition on the sheaf. Sheaf theory provides a useful setting in which to compare continuous and discrete theories. Indeed the continuous theory can also be described with sheaves of Banach spaces. The language works well for sequences such as differential forms and elasticity, and unifying de Rham theorems on cohomology have been proved in this setting.