1.1 Presheaves

Presheaves are a key concept in sheaf theory, generalizing how data is assigned to open sets of a topological space. They provide a framework for studying local-to-global properties of mathematical objects by examining their behavior on open subsets.

As contravariant functors, presheaves assign objects and morphisms to open sets and inclusions. This structure allows for the construction of presheaf categories, which inherit properties from their target categories and support operations like limits, colimits, and tensor products.

Definition of presheaves

• Presheaves are a fundamental concept in sheaf theory that generalize the notion of assigning data to open sets of a topological space
• Presheaves provide a way to study local-to-global properties of mathematical objects by considering their behavior on open subsets of a space

Presheaves as contravariant functors

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• A presheaf $\mathcal{F}$ on a topological space $X$ is a contravariant functor from the category of open sets of $X$ (denoted as $\mathcal{O}(X)$) to a target category $\mathcal{C}$
• For each open set $U \subseteq X$, the presheaf assigns an object $\mathcal{F}(U)$ in the category $\mathcal{C}$
• For each inclusion of open sets $V \subseteq U$, the presheaf assigns a morphism $\mathcal{F}(U) \to \mathcal{F}(V)$ in $\mathcal{C}$, called the restriction map
• Presheaves satisfy the contravariant functoriality axioms:
  • For each open set $U$, the restriction map $\mathcal{F}(U) \to \mathcal{F}(U)$ is the identity morphism
  • For inclusions $W \subseteq V \subseteq U$, the composition of restriction maps $\mathcal{F}(U) \to \mathcal{F}(V) \to \mathcal{F}(W)$ equals the restriction map $\mathcal{F}(U) \to \mathcal{F}(W)$

Presheaf categories

• For a fixed topological space $X$ and a target category $\mathcal{C}$, the collection of all presheaves on $X$ with values in $\mathcal{C}$ forms a category, denoted as $\mathbf{PSh}(X, \mathcal{C})$
• Objects in the presheaf category are presheaves on $X$ with values in $\mathcal{C}$
• Morphisms in the presheaf category are natural transformations between presheaves
• The presheaf category $\mathbf{PSh}(X, \mathcal{C})$ inherits properties from the target category $\mathcal{C}$, such as limits, colimits, and exactness properties

Morphisms of presheaves

• A morphism of presheaves $\varphi: \mathcal{F} \to \mathcal{G}$ is a natural transformation between the presheaves $\mathcal{F}$ and $\mathcal{G}$
• For each open set $U \subseteq X$, a morphism of presheaves assigns a morphism $\varphi_U: \mathcal{F}(U) \to \mathcal{G}(U)$ in the target category $\mathcal{C}$
• The morphisms $\varphi_U$ are compatible with the restriction maps, i.e., for each inclusion $V \subseteq U$, the diagram commutes:

$$\begin{CD} \mathcal{F}(U) @>{\varphi_U}>> \mathcal{G}(U) \\ @VVV @VVV \\ \mathcal{F}(V) @>{\varphi_V}>> \mathcal{G}(V) \end{CD}$$

Presheaf construction

• Presheaves can be constructed in various ways depending on the mathematical context and the desired properties
• The construction of presheaves often involves assigning data to open sets of a topological space in a functorial manner

Constant presheaves

• A constant presheaf $\underline{A}$ on a topological space $X$ with values in a category $\mathcal{C}$ assigns the same object $A \in \mathcal{C}$ to every open set $U \subseteq X$
• For each inclusion $V \subseteq U$, the restriction map $\underline{A}(U) \to \underline{A}(V)$ is the identity morphism on $A$
• Constant presheaves capture the idea of assigning the same data globally to all open sets of a space

Representable presheaves

• For a topological space $X$ and a point $x \in X$, the representable presheaf $h_x$ is defined as:
  • For each open set $U \subseteq X$, $h_x(U) = \{*\}$ if $x \in U$ and $h_x(U) = \emptyset$ otherwise, where $\{*\}$ is a one-point set
  • For each inclusion $V \subseteq U$, the restriction map $h_x(U) \to h_x(V)$ is the unique map if $x \in V$ and the empty map otherwise
• Representable presheaves are used to study the local behavior of a space around a point and play a crucial role in the Yoneda lemma

Presheaves of sections

• For a continuous map $f: Y \to X$ between topological spaces, the presheaf of sections $\Gamma(f)$ assigns to each open set $U \subseteq X$ the set of continuous sections of $f$ over $U$:
  • $\Gamma(f)(U) = \{s: U \to Y \mid f \circ s = \mathrm{id}_U\}$
  • For each inclusion $V \subseteq U$, the restriction map $\Gamma(f)(U) \to \Gamma(f)(V)$ is given by restricting a section $s: U \to Y$ to $V$
• Presheaves of sections are used to study the local properties of continuous maps and vector bundles

Limits and colimits in presheaf categories

• The presheaf category $\mathbf{PSh}(X, \mathcal{C})$ inherits the limits and colimits from the target category $\mathcal{C}$
• Limits (products, equalizers, pullbacks) and colimits (coproducts, coequalizers, pushouts) in $\mathbf{PSh}(X, \mathcal{C})$ are computed pointwise, i.e., for each open set $U \subseteq X$
• For example, the product of presheaves $\mathcal{F} \times \mathcal{G}$ is defined by $(\mathcal{F} \times \mathcal{G})(U) = \mathcal{F}(U) \times \mathcal{G}(U)$ for each open set $U$
• The existence of limits and colimits in presheaf categories allows for the construction of new presheaves from existing ones

Sheafification of presheaves

• Not all presheaves satisfy the gluing axioms required for sheaves, which capture the idea of local data uniquely determining global data
• Sheafification is a process that assigns to each presheaf a sheaf that best approximates it while preserving its essential properties

Presheaf of sections vs sheaf of sections

• For a continuous map $f: Y \to X$, the presheaf of sections $\Gamma(f)$ may not always be a sheaf
• The sheaf condition for $\Gamma(f)$ requires that for any open cover $\{U_i\}$ of an open set $U \subseteq X$ and any collection of sections $s_i \in \Gamma(f)(U_i)$ that agree on overlaps, there exists a unique global section $s \in \Gamma(f)(U)$ restricting to $s_i$ on each $U_i$
• The sheaf of sections is the sheafification of the presheaf of sections, denoted as $\Gamma^+(f)$, which satisfies the sheaf axioms

Sheafification functor

• The sheafification functor $(\cdot)^+: \mathbf{PSh}(X, \mathcal{C}) \to \mathbf{Sh}(X, \mathcal{C})$ assigns to each presheaf $\mathcal{F}$ a sheaf $\mathcal{F}^+$
• The sheafification functor is left adjoint to the forgetful functor $\mathbf{Sh}(X, \mathcal{C}) \to \mathbf{PSh}(X, \mathcal{C})$, which means that there is a natural bijection:
  • $\mathrm{Hom}_{\mathbf{Sh}(X, \mathcal{C})}(\mathcal{F}^+, \mathcal{G}) \cong \mathrm{Hom}_{\mathbf{PSh}(X, \mathcal{C})}(\mathcal{F}, \mathcal{G})$ for any presheaf $\mathcal{F}$ and sheaf $\mathcal{G}$
• The sheafification functor preserves colimits, as it is a left adjoint

Universal property of sheafification

• The sheafification $\mathcal{F}^+$ of a presheaf $\mathcal{F}$ satisfies a universal property: for any sheaf $\mathcal{G}$ and any morphism of presheaves $\varphi: \mathcal{F} \to \mathcal{G}$, there exists a unique morphism of sheaves $\varphi^+: \mathcal{F}^+ \to \mathcal{G}$ such that the following diagram commutes:

$$\begin{CD} \mathcal{F} @>{\eta_{\mathcal{F}}}>> \mathcal{F}^+ \\ @V{\varphi}VV @VV{\varphi^+}V \\ \mathcal{G} @= \mathcal{G} \end{CD}$$

• The universal property characterizes the sheafification as the best approximation of a presheaf by a sheaf

Stalks of presheaves

• The stalk of a presheaf $\mathcal{F}$ at a point $x \in X$ is defined as the colimit:
  • $\mathcal{F}_x = \varinjlim_{x \in U} \mathcal{F}(U)$ where the colimit is taken over all open sets $U$ containing $x$
• The stalk of a presheaf captures the local behavior of the presheaf around a point
• A morphism of presheaves $\varphi: \mathcal{F} \to \mathcal{G}$ induces a morphism of stalks $\varphi_x: \mathcal{F}_x \to \mathcal{G}_x$ for each $x \in X$
• A presheaf $\mathcal{F}$ is a sheaf if and only if for any open set $U \subseteq X$ and any family of elements $s_x \in \mathcal{F}_x$ for $x \in U$ that are compatible on overlaps, there exists a unique section $s \in \mathcal{F}(U)$ whose germs are $s_x$

Operations on presheaves

• Presheaves support various operations that allow for the construction of new presheaves from existing ones
• These operations are functorial and often have geometric or algebraic interpretations

Restriction and extension of presheaves

• For a presheaf $\mathcal{F}$ on a topological space $X$ and an open subset $U \subseteq X$, the restriction of $\mathcal{F}$ to $U$ is a presheaf $\mathcal{F}|_U$ on $U$ defined by:
  • $(\mathcal{F}|_U)(V) = \mathcal{F}(V)$ for each open set $V \subseteq U$
  • The restriction maps of $\mathcal{F}|_U$ are the same as those of $\mathcal{F}$
• For a presheaf $\mathcal{G}$ on an open subset $U \subseteq X$, the extension by zero of $\mathcal{G}$ to $X$ is a presheaf $j_!\mathcal{G}$ on $X$ defined by:
  • $(j_!\mathcal{G})(V) = \mathcal{G}(V)$ if $V \subseteq U$ and $(j_!\mathcal{G})(V) = 0$ otherwise, where $0$ is an initial object in the target category
  • The restriction maps of $j_!\mathcal{G}$ are the same as those of $\mathcal{G}$ when defined and the unique maps to $0$ otherwise

Direct and inverse image functors

• For a continuous map $f: X \to Y$ between topological spaces, the direct image functor $f_*: \mathbf{PSh}(X, \mathcal{C}) \to \mathbf{PSh}(Y, \mathcal{C})$ assigns to each presheaf $\mathcal{F}$ on $X$ a presheaf $f_*\mathcal{F}$ on $Y$ defined by:
  • $(f_*\mathcal{F})(V) = \mathcal{F}(f^{-1}(V))$ for each open set $V \subseteq Y$
  • The restriction maps of $f_*\mathcal{F}$ are induced by the restriction maps of $\mathcal{F}$
• The inverse image functor $f^{-1}: \mathbf{PSh}(Y, \mathcal{C}) \to \mathbf{PSh}(X, \mathcal{C})$ assigns to each presheaf $\mathcal{G}$ on $Y$ a presheaf $f^{-1}\mathcal{G}$ on $X$ defined by:
  • $(f^{-1}\mathcal{G})(U) = \varinjlim_{[f(U)](https://www.fiveableKeyTerm:f(u)) \subseteq V} \mathcal{G}(V)$ for each open set $U \subseteq X$, where the colimit is taken over all open sets $V \subseteq Y$ containing $f(U)$
  • The restriction maps of $f^{-1}\mathcal{G}$ are induced by the restriction maps of $\mathcal{G}$
• The direct and inverse image functors are adjoint: $f^{-1}$ is left adjoint to $f_*$

Tensor product of presheaves

• For presheaves $\mathcal{F}$ and $\mathcal{G}$ on a topological space $X$ with values in a monoidal category $(\mathcal{C}, \otimes)$, the tensor product presheaf $\mathcal{F} \otimes \mathcal{G}$ is defined by:
  • $(\mathcal{F} \otimes \mathcal{G})(U) = \mathcal{F}(U) \otimes \mathcal{G}(U)$ for each open set $U \subseteq X$
  • The restriction maps of $\mathcal{F} \otimes \mathcal{G}$ are induced by the restriction maps of $\mathcal{F}$ and $\mathcal{G}$ and the monoidal structure of $\mathcal{C}$
• The tensor product of presheaves is associative and unital, with the constant presheaf at the monoidal unit serving as the unit for the tensor product

Internal Hom of presheaves

• For presheaves $\mathcal{F}$ and $\mathcal{G}$ on a topological space $X$ with values in a category $\mathcal{C}$, the internal Hom presheaf $\underline{\mathrm{Hom}}(\mathcal{F}, \mathcal{G})$ is defined by:
  • $\underline{\mathrm{Hom}}(\mathcal{F}, \mathcal{G})(U) = \mathrm{Hom}_{\mathbf{PSh}(U, \mathcal{C})}(\mathcal{F}|_U, \mathcal{G}|_U)$ for each open set $U \subseteq X$