The sheaf condition has several different equivalent formulations. The official definition chosen here is in terms of grothendieck topologies so that the results on sites could be applied here easily, and this condition does not require additional constraints on the value category. The equivalent formulations of the sheaf condition on presheaf C X are as follows :

1. Top.presheaf.is_sheaf: (the official definition) It is a sheaf with respect to the grothendieck topology on opens X, which is to say: For each open cover { Uᵢ } of U, and a family of compatible functions A ⟶ F(Uᵢ) for an A : X, there exists an unique gluing A ⟶ F(U) compatible with the restriction.

2. Top.presheaf.is_sheaf_equalizer_products: (requires C to have all products) For each open cover { Uᵢ } of U, F(U) ⟶ ∏ F(Uᵢ) is the equalizer of the two morphisms ∏ F(Uᵢ) ⟶ ∏ F(Uᵢ ∩ Uⱼ). See Top.presheaf.is_sheaf_iff_is_sheaf_equalizer_products.

3. Top.presheaf.is_sheaf_opens_le_cover: For each open cover { Uᵢ } of U, F(U) is the limit of the diagram consisting of arrows F(V₁) ⟶ F(V₂) for every pair of open sets V₁ ⊇ V₂ that are contained in some Uᵢ. See Top.presheaf.is_sheaf_iff_is_sheaf_opens_le_cover.

4. Top.presheaf.is_sheaf_pairwise_intersections: For each open cover { Uᵢ } of U, F(U) is the limit of the diagram consisting of arrows from F(Uᵢ) and F(Uⱼ) to F(Uᵢ ∩ Uⱼ) for each pair (i, j). See Top.presheaf.is_sheaf_iff_is_sheaf_pairwise_intersections.

The following requires C to be concrete and complete, and forget C to reflect isomorphisms and preserve limits. This applies to most "algebraic" categories, e.g. groups, abelian groups and rings.

5. Top.presheaf.is_sheaf_unique_gluing: (requires C to be concrete and complete; forget C to reflect isomorphisms and preserve limits) For each open cover { Uᵢ } of U, and a compatible family of elements x : F(Uᵢ), there exists a unique gluing x : F(U) that restricts to the given elements. See Top.presheaf.is_sheaf_iff_is_sheaf_unique_gluing.

6. The underlying sheaf of types is a sheaf. See Top.presheaf.is_sheaf_iff_is_sheaf_comp and category_theory.presheaf.is_sheaf_iff_is_sheaf_forget.

A sheaf C X is a presheaf of objects from C over a (bundled) topological space X, satisfying the sheaf condition.

The underlying presheaf of a sheaf

The forgetful functor from sheaves to presheaves.