2.1 Sheafification

Sheafification transforms presheaves into sheaves, ensuring local data can be consistently glued together. This process is crucial in sheaf theory, allowing us to work with objects that respect local-to-global consistency principles.

The universal property of sheafification makes it the canonical way to associate a sheaf to a presheaf. This property ensures that any map from a presheaf to a sheaf factors uniquely through its sheafification, making it a fundamental tool in sheaf theory.

Definition of sheafification

• Sheafification is the process of converting a presheaf into a sheaf, which is a fundamental construction in sheaf theory
• It ensures that the local data encoded by the presheaf can be consistently glued together to form a global object
• Sheafification is characterized by a universal property, making it a canonical way to associate a sheaf to a given presheaf

Presheaves and sheaves

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• A presheaf $\mathcal{F}$ on a topological space $X$ assigns to each open set $U \subset X$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ of open sets a restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$
  • These assignments are subject to certain compatibility conditions
• A sheaf is a presheaf satisfying two additional conditions:
  1. (Local identity) If $\{U_i\}$ is an open cover of $U$ and $s, t \in \mathcal{F}(U)$ agree on each $U_i$, then $s = t$
  2. (Gluing) If $\{U_i\}$ is an open cover of $U$ and we have elements $s_i \in \mathcal{F}(U_i)$ that agree on overlaps, then there exists a unique $s \in \mathcal{F}(U)$ restricting to each $s_i$
• Intuitively, sheaves capture the idea of local-to-global consistency, allowing compatible local data to be uniquely glued into global sections

Universal property of sheafification

• Given a presheaf $\mathcal{F}$, its sheafification $\mathcal{F}^+$ is a sheaf equipped with a morphism of presheaves $\theta: \mathcal{F} \to \mathcal{F}^+$ satisfying the following universal property:
  • For any sheaf $\mathcal{G}$ and morphism of presheaves $\varphi: \mathcal{F} \to \mathcal{G}$, there exists a unique morphism of sheaves $\psi: \mathcal{F}^+ \to \mathcal{G}$ such that $\varphi = \psi \circ \theta$
• The universal property ensures that $\mathcal{F}^+$ is the "best approximation" of $\mathcal{F}$ by a sheaf, in the sense that any map from $\mathcal{F}$ to a sheaf factors uniquely through $\mathcal{F}^+$

Construction of sheafification

• There are several equivalent ways to construct the sheafification of a presheaf, each offering a different perspective on the process
• The constructions involve the notion of stalks and germs, which capture the local behavior of the presheaf at each point

Stalks and germs

• The stalk of a presheaf $\mathcal{F}$ at a point $x \in X$ is the direct limit $\mathcal{F}_x := \varinjlim_{x \in U} \mathcal{F}(U)$ over all open neighborhoods $U$ of $x$
  • Elements of the stalk are called germs and represent the local behavior of the presheaf near $x$
• The restriction maps of the presheaf induce maps between the stalks, making the assignment $x \mapsto \mathcal{F}_x$ a functor from the category of points of $X$ to the target category of $\mathcal{F}$
• The sheaf condition can be phrased in terms of stalks: a presheaf $\mathcal{F}$ is a sheaf if and only if for every open set $U$ and every family of germs $(s_x)_{x \in U}$ with $s_x \in \mathcal{F}_x$, there exists a unique section $s \in \mathcal{F}(U)$ whose germs are the given $(s_x)$

Étalé space approach

• The étalé space of a presheaf $\mathcal{F}$ is the disjoint union $E(\mathcal{F}) := \bigsqcup_{x \in X} \mathcal{F}_x$ of its stalks, equipped with a natural topology
  • A subset $V \subset E(\mathcal{F})$ is open if for every $s_x \in V$, there exists an open neighborhood $U$ of $x$ and a section $s \in \mathcal{F}(U)$ such that the germ of $s$ at every point of $U$ belongs to $V$
• The sheafification of $\mathcal{F}$ can be obtained as the sheaf of continuous sections of the canonical projection $\pi: E(\mathcal{F}) \to X$
  • Explicitly, $\mathcal{F}^+(U) := \{ \sigma: U \to E(\mathcal{F}) \mid \pi \circ \sigma = \mathrm{id}_U \text{ and } \sigma \text{ is continuous} \}$

Sheaf space approach

• An alternative construction of the sheafification uses the notion of sheaf spaces
• A sheaf space is a pair $(E, \pi)$ consisting of a topological space $E$ and a local homeomorphism $\pi: E \to X$
• The sheaf of sections of a sheaf space $(E, \pi)$ is the sheaf $\mathcal{F}$ defined by $\mathcal{F}(U) := \{ \sigma: U \to E \mid \pi \circ \sigma = \mathrm{id}_U \text{ and } \sigma \text{ is continuous} \}$
• The sheafification of a presheaf $\mathcal{F}$ can be obtained as the sheaf of sections of a suitable sheaf space constructed from $\mathcal{F}$

Comparison of constructions

• The étalé space and sheaf space approaches are closely related, as the étalé space of a presheaf naturally carries the structure of a sheaf space
• Both constructions can be seen as ways of encoding the local behavior of the presheaf and ensuring that the sheaf condition is satisfied
• The universal property of sheafification guarantees that these constructions yield essentially the same result, up to unique isomorphism

Properties of sheafification functor

• The sheafification construction defines a functor $(-)^+$ from the category of presheaves on $X$ to the category of sheaves on $X$
• This functor has several important properties that reflect its role as a best approximation of presheaves by sheaves

Adjoint to forgetful functor

• The sheafification functor is the left adjoint to the forgetful functor $U$ from sheaves to presheaves
  • For any presheaf $\mathcal{F}$ and sheaf $\mathcal{G}$, there is a natural bijection $\mathrm{Hom}_{\mathrm{Sh}}(\mathcal{F}^+, \mathcal{G}) \cong \mathrm{Hom}_{\mathrm{PSh}}(\mathcal{F}, U(\mathcal{G}))$
• The unit of the adjunction is the natural transformation $\theta: \mathrm{id}_{\mathrm{PSh}} \to U \circ (-)^+$ given by the universal morphisms $\theta_{\mathcal{F}}: \mathcal{F} \to \mathcal{F}^+$
• The adjunction expresses the universal property of sheafification in categorical terms

Preservation of finite limits

• The sheafification functor preserves finite limits, meaning that it commutes with finite products and equalizers
  • For presheaves $\mathcal{F}, \mathcal{G}$, there are natural isomorphisms $(\mathcal{F} \times \mathcal{G})^+ \cong \mathcal{F}^+ \times \mathcal{G}^+$ and $(\mathrm{eq}(f,g))^+ \cong \mathrm{eq}(f^+,g^+)$
• This property is a consequence of the fact that sheaves are defined by local conditions and finite limits are computed pointwise in the category of presheaves

Interaction with global sections functor

• The global sections functor $\Gamma: \mathrm{Sh}(X) \to \mathrm{Set}$ assigns to each sheaf $\mathcal{F}$ the set $\mathcal{F}(X)$ of global sections
• There is a natural comparison map $\Gamma(\mathcal{F}) \to \Gamma(\mathcal{F}^+)$ induced by the universal morphism $\theta_{\mathcal{F}}: \mathcal{F} \to \mathcal{F}^+$
  • This map is always injective, but not necessarily surjective
• The sheafification functor can be seen as a way of enlarging the presheaf to ensure that all compatible local sections can be glued into global sections

Sheafification in categories

• The notion of sheafification can be generalized to settings beyond topological spaces, using the concept of Grothendieck topologies and sites

Grothendieck topologies and sites

• A Grothendieck topology on a category $\mathcal{C}$ is a collection of families of morphisms $\{U_i \to U\}_{i \in I}$, called covering families, satisfying certain axioms
  • Intuitively, these families play the role of open covers in the topological setting
• A site $(\mathcal{C}, J)$ is a category $\mathcal{C}$ equipped with a Grothendieck topology $J$
• Presheaves and sheaves can be defined on a site, with the sheaf condition reformulated using the covering families

Sheafification in general categories

• Given a site $(\mathcal{C}, J)$, the category of presheaves $\mathrm{PSh}(\mathcal{C})$ is the functor category $[\mathcal{C}^{\mathrm{op}}, \mathrm{Set}]$
• The sheafification of a presheaf $\mathcal{F}$ on $(\mathcal{C}, J)$ is a sheaf $\mathcal{F}^+$ equipped with a morphism $\theta: \mathcal{F} \to \mathcal{F}^+$ satisfying the universal property, as in the topological case
• The construction of sheafification in this general setting can be carried out using a suitable generalization of the étalé space or sheaf space approach

Comparison to topological case

• When $\mathcal{C}$ is the category of open sets of a topological space $X$ and $J$ is the usual topology, the notions of presheaves, sheaves, and sheafification coincide with the ones defined earlier
• The site-theoretic approach provides a unified framework for studying sheaves in various contexts, such as algebraic geometry, where the underlying category is not necessarily topological
• Many of the properties of sheafification, such as its adjointness to the forgetful functor and preservation of finite limits, hold in the general setting of sites

Applications of sheafification

• Sheafification is a fundamental tool in sheaf theory and finds applications in various areas of mathematics

Sheaf cohomology

• Sheaf cohomology is a powerful invariant that associates to each sheaf $\mathcal{F}$ on a topological space (or more generally, a site) a sequence of abelian groups $H^i(X, \mathcal{F})$
• The construction of sheaf cohomology relies on the notion of injective resolutions, which can be obtained by applying the sheafification functor to certain presheaf resolutions
• Sheaf cohomology has applications in algebraic topology, complex analysis, and algebraic geometry, among other fields

Sheaf-theoretic approach to manifolds

• Sheaves provide a convenient language for studying manifolds and their local-to-global properties
• The sheaf of smooth functions on a manifold $M$ is a fundamental object that encodes the differential structure of $M$
  • This sheaf is obtained by sheafifying the presheaf of smooth functions defined on open subsets of $M$
• Other important sheaves on manifolds, such as the sheaf of differential forms or the sheaf of vector fields, can be constructed using similar techniques

Sheaves in algebraic geometry

• In algebraic geometry, sheaves are used to study schemes, which are a generalization of algebraic varieties
• The structure sheaf $\mathcal{O}_X$ of a scheme $X$ is a sheaf of rings that encodes the local algebraic properties of $X$
  • It is obtained by sheafifying the presheaf of regular functions on open subsets of $X$
• Coherent sheaves, which are sheaves of $\mathcal{O}_X$-modules satisfying certain finiteness conditions, play a central role in the study of schemes and their properties
• Sheaf cohomology in the context of schemes provides important invariants, such as the cohomology groups of coherent sheaves, which are used to study questions in algebraic geometry and number theory