Noun
presheaf (plural presheaves)
(category theory, algebraic geometry) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings. The functorial images of the open sets are sets of things called sections which are said to be "over" those open sets. The (contravariant) functorial images of those inclusion mappings are functions which are called restrictions.
Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Source: Internet
A sheaf on the indiscrete site is the same thing as a presheaf. Source: Internet
For a separated presheaf, the first arrow need only be injective. Source: Internet