8.1 Presheaves and their properties

Presheaves on topological spaces bridge local and global properties, assigning sets to open sets and defining restriction maps. They're key to understanding how information flows between different scales in a space.

Presheaves form a rich category with morphisms, properties like gluing and locality, and diverse examples. This structure lays the groundwork for sheaves and their role in geometry and topology.

Presheaves on Topological Spaces

Definition of presheaves and morphisms

• Presheaf definition assigns contravariant functor $F: \mathcal{O}(X)^{op} \rightarrow \mathbf{Set}$ maps open sets of topological space $X$ to category of sets
• Components of a presheaf map open sets $U \subseteq X$ to sets $F(U)$ and inclusions $V \subseteq U$ to restriction maps $\rho_{U,V}: F(U) \rightarrow F(V)$
• Morphisms between presheaves use natural transformations consisting of maps $\eta_U: F(U) \rightarrow G(U)$ compatible with restriction maps $\eta_V \circ \rho_{U,V}^F = \rho_{U,V}^G \circ \eta_U$

Properties of presheaves

• Restriction maps encode local-to-global relationship with functoriality $\rho_{U,U} = id_{F(U)}$ and $\rho_{V,W} \circ \rho_{U,V} = \rho_{U,W}$ for $W \subseteq V \subseteq U$
• Gluing axiom allows unique section $s \in F(U)$ from agreeing sections $s_i \in F(U_i)$ on open cover $\{U_i\}$ of $U$
• Locality property determines sections by restrictions to smaller open sets

Examples of presheaves

• Presheaf of continuous functions maps $F(U) = C(U, \mathbb{R})$ with function restriction maps (real-valued functions)
• Constant presheaf assigns fixed set $A$ to all open sets $U$ with identity restriction maps
• Presheaf of bounded functions uses set of bounded real-valued functions on $U$ with function restrictions
• Presheaf of local constants employs set of locally constant functions on $U$ with function restrictions

Category of presheaves

• Category of presheaves uses presheaves as objects and natural transformations as morphisms
• Properties include completeness, cocompleteness, abelian category structure, and topos structure
• Yoneda embedding functor $y: \mathcal{O}(X) \rightarrow \mathbf{PSh}(X)$ maps open sets to representable presheaves
• Subobject classifier $\Omega$ presheaf of local truth values maps $\Omega(U)$ to set of open subsets of $U$
• Exponential objects $G^F$ exist for presheaves $F$ and $G$ with $(G^F)(U)$ as set of natural transformations $F|_U \rightarrow G|_U$