presheaves

Presheaves are a basic structural tool in topology and geometry. Given a topological space X and a category C (commonly Sets, Ab, Rings, etc.), a presheaf F on X with values in C assigns to every open subset U ⊆ X an object F(U) and to every inclusion V ⊆ U a restriction morphism res^U_V: F(U) → F(V). These restrictions satisfy res^U_U = id and res^W_V ∘ res^U_W = res^U_V whenever V ⊆ W ⊆ U. The construction is contravariant: larger opens map to the smaller ones via restriction.

Typical examples include: the presheaf of continuous real-valued functions U ↦ C^0(U, R), with restriction given by restricting functions to smaller domains; the presheaf of sections of a fiber bundle; and the presheaf of locally constant or smooth functions, all with restriction maps given by restriction of functions.

A presheaf F is called a sheaf if it satisfies a gluing condition: for any open cover {U_i} of U, a family s_i ∈ F(U_i) that agree on overlaps (i.e., their restrictions to U_i ∩ U_j coincide) comes from a unique global section s ∈ F(U) whose restrictions equal s_i. The notion of gluing makes sheaves locally determined objects.

Associated to a presheaf is its stalk at a point x ∈ X, defined as the colimit F_x = colim_{x ∈ U} F(U) over all neighborhoods U of x; stalks encode germs of sections and capture local behavior.

Morphisms of presheaves are natural transformations between the corresponding contravariant functors. The category of presheaves PSh(X, C) often has a full subcategory Sh(X, C) of sheaves. There is a left adjoint, called sheafification, that assigns to each presheaf the "best" associated sheaf.

Presheaves provide a flexible framework for organizing locally defined data and are foundational in algebraic topology, algebraic geometry, and related fields.