1.3 Morphisms of presheaves and sheaves

Morphisms of presheaves and sheaves are crucial tools for understanding relationships between these mathematical structures. They allow us to compare, transform, and analyze presheaves and sheaves, providing a framework for studying their properties and interactions.

These morphisms form the backbone of categorical approaches to sheaf theory. By defining morphisms, we can explore isomorphisms, compositions, and universal properties, laying the groundwork for deeper insights into the nature of sheaves and their applications in topology and algebra.

Morphisms of presheaves

• Presheaf morphisms play a crucial role in understanding the relationships and transformations between presheaves
• Presheaf morphisms allow for the comparison and manipulation of presheaves, enabling the study of their structural properties and interactions

Definition of presheaf morphism

Top images from around the web for Definition of presheaf morphism

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition of presheaf morphism

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

• 

  Frontiers | On the Design of Social Robots Using Sheaf Theory and Smart Contracts View original

  Is this image relevant?

1 of 3

• A presheaf morphism $f: \mathcal{F} \to \mathcal{G}$ between presheaves $\mathcal{F}$ and $\mathcal{G}$ on a topological space $X$ consists of a collection of morphisms $f_U: \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U \subseteq X$
• The morphisms $f_U$ must be compatible with the restriction maps of the presheaves
  • For any open sets $U \subseteq V$, the following diagram commutes: $f_U \circ \rho^{\mathcal{F}}_{V,U} = \rho^{\mathcal{G}}_{V,U} \circ f_V$
• Presheaf morphisms preserve the presheaf structure and ensure consistency across different open sets

Composition of presheaf morphisms

• Presheaf morphisms can be composed, allowing for the chaining of transformations between presheaves
• Given presheaf morphisms $f: \mathcal{F} \to \mathcal{G}$ and $g: \mathcal{G} \to \mathcal{H}$, their composition $g \circ f: \mathcal{F} \to \mathcal{H}$ is defined by $(g \circ f)_U = g_U \circ f_U$ for each open set $U$
• The composition of presheaf morphisms is associative and respects the identity morphisms, forming a category

Category of presheaves

• The collection of all presheaves on a topological space $X$, along with presheaf morphisms, forms a category denoted as $PSh(X)$
• The objects of $PSh(X)$ are presheaves, and the morphisms are presheaf morphisms
• $PSh(X)$ has rich categorical properties, such as the existence of limits, colimits, and exponential objects

Presheaf isomorphisms

• A presheaf morphism $f: \mathcal{F} \to \mathcal{G}$ is an isomorphism if there exists a presheaf morphism $g: \mathcal{G} \to \mathcal{F}$ such that $g \circ f = id_{\mathcal{F}}$ and $f \circ g = id_{\mathcal{G}}$
• Presheaf isomorphisms capture the notion of equivalence between presheaves
• If two presheaves are isomorphic, they have the same structure and can be considered "the same" up to isomorphism

Morphisms of sheaves

• Sheaf morphisms are a special case of presheaf morphisms that respect the sheaf condition
• Sheaf morphisms allow for the study of relationships and transformations between sheaves, preserving their local-to-global properties

Definition of sheaf morphism

• A sheaf morphism $f: \mathcal{F} \to \mathcal{G}$ between sheaves $\mathcal{F}$ and $\mathcal{G}$ on a topological space $X$ is a presheaf morphism that satisfies the following condition:
  • For any open set $U \subseteq X$ and any open cover $\{U_i\}$ of $U$, if $s_i \in \mathcal{F}(U_i)$ are local sections such that $\rho^{\mathcal{F}}_{U_i,U_i \cap U_j}(s_i) = \rho^{\mathcal{F}}_{U_j,U_i \cap U_j}(s_j)$ for all $i,j$, then $f_U(\rho^{\mathcal{F}}_{U,U_i}(s_i)) = \rho^{\mathcal{G}}_{U,U_i}(f_{U_i}(s_i))$ for all $i$
• Sheaf morphisms preserve the gluing property of sheaves, ensuring that local compatibility is maintained

Composition of sheaf morphisms

• Sheaf morphisms can be composed, inheriting the composition of presheaf morphisms
• The composition of sheaf morphisms is well-defined and yields another sheaf morphism
• The category of sheaves on a topological space $X$, denoted as $Sh(X)$, has sheaves as objects and sheaf morphisms as morphisms

Category of sheaves

• The category of sheaves $Sh(X)$ is a full subcategory of the category of presheaves $PSh(X)$
• $Sh(X)$ inherits many categorical properties from $PSh(X)$, such as the existence of limits, colimits, and exponential objects
• The inclusion functor $i: Sh(X) \to PSh(X)$ is fully faithful, reflecting the fact that sheaves are a special case of presheaves

Sheaf isomorphisms

• A sheaf morphism $f: \mathcal{F} \to \mathcal{G}$ is an isomorphism if it has an inverse sheaf morphism $g: \mathcal{G} \to \mathcal{F}$ such that $g \circ f = id_{\mathcal{F}}$ and $f \circ g = id_{\mathcal{G}}$
• Sheaf isomorphisms capture the notion of equivalence between sheaves, preserving both the presheaf structure and the sheaf condition
• Isomorphic sheaves have the same local and global properties, and can be considered "the same" up to isomorphism

Sheafification functor

• Sheafification is a process that converts a presheaf into a sheaf, preserving its essential properties
• The sheafification functor provides a systematic way to construct sheaves from presheaves, establishing a connection between the categories $PSh(X)$ and $Sh(X)$

Definition of sheafification

• Given a presheaf $\mathcal{F}$ on a topological space $X$, its sheafification is a sheaf $\mathcal{F}^+$ together with a presheaf morphism $\theta: \mathcal{F} \to \mathcal{F}^+$ satisfying the following universal property:
  • For any sheaf $\mathcal{G}$ and presheaf morphism $f: \mathcal{F} \to \mathcal{G}$, there exists a unique sheaf morphism $f^+: \mathcal{F}^+ \to \mathcal{G}$ such that $f = f^+ \circ \theta$
• The sheafification $\mathcal{F}^+$ is constructed by "patching together" the local sections of $\mathcal{F}$ in a way that enforces the sheaf condition

Sheafification as left adjoint

• The sheafification functor $(-)^+: PSh(X) \to Sh(X)$ is left adjoint to the inclusion functor $i: Sh(X) \to PSh(X)$
• The adjunction $(-)^+ \dashv i$ captures the idea that sheafification is the "best approximation" of a presheaf by a sheaf
• The unit of the adjunction is the presheaf morphism $\theta: \mathcal{F} \to i(\mathcal{F}^+)$, which is universal among presheaf morphisms from $\mathcal{F}$ to sheaves

Universal property of sheafification

• The universal property of sheafification states that for any presheaf $\mathcal{F}$ and sheaf $\mathcal{G}$, there is a bijection between the set of presheaf morphisms $Hom_{PSh(X)}(\mathcal{F},i(\mathcal{G}))$ and the set of sheaf morphisms $Hom_{Sh(X)}(\mathcal{F}^+,\mathcal{G})$
• This bijection is natural in both $\mathcal{F}$ and $\mathcal{G}$, establishing an adjunction between the categories $PSh(X)$ and $Sh(X)$
• The universal property characterizes sheafification as the "optimal" way to convert a presheaf into a sheaf

Morphisms of sheaves vs presheaves

• Sheaf morphisms are a special case of presheaf morphisms, satisfying an additional compatibility condition with respect to the sheaf structure
• Understanding the relationship between sheaf morphisms and presheaf morphisms is crucial for studying the connections between sheaves and presheaves

Comparison of definitions

• A sheaf morphism is a presheaf morphism that preserves the gluing property of sheaves
• Every sheaf morphism is a presheaf morphism, but not every presheaf morphism between sheaves is a sheaf morphism
• The sheaf condition imposes an additional constraint on the morphisms, ensuring that they respect the local-to-global nature of sheaves

Induced morphisms on stalks

• Given a presheaf morphism $f: \mathcal{F} \to \mathcal{G}$ between sheaves, it induces a morphism on the stalks $f_x: \mathcal{F}_x \to \mathcal{G}_x$ for each point $x \in X$
• The induced morphism on stalks is defined by $f_x([s]) = [f_U(s)]$, where $[s]$ is the germ of a section $s \in \mathcal{F}(U)$ at $x$
• If $f$ is a sheaf morphism, then the induced morphisms on stalks are well-defined and compatible with the restriction maps

Sheafification of presheaf morphisms

• Given a presheaf morphism $f: \mathcal{F} \to \mathcal{G}$ between presheaves, it induces a sheaf morphism $f^+: \mathcal{F}^+ \to \mathcal{G}^+$ between their sheafifications
• The sheafification functor $(-)^+$ is functorial, meaning that it preserves the composition and identity of presheaf morphisms
• The sheafification of a presheaf morphism is the unique sheaf morphism that makes the following diagram commute: $\mathcal{F} \stackrel{f}{\to} \mathcal{G}$ $\theta_{\mathcal{F}} \downarrow \quad \downarrow \theta_{\mathcal{G}}$ $\mathcal{F}^+ \stackrel{f^+}{\to} \mathcal{G}^+$

Applications of sheaf morphisms

• Sheaf morphisms play a fundamental role in various constructions and applications of sheaf theory
• They allow for the manipulation and comparison of sheaves, enabling the study of their structural properties and relationships

Restriction and extension of sheaves

• Given a sheaf $\mathcal{F}$ on a topological space $X$ and an open subset $U \subseteq X$, the restriction of $\mathcal{F}$ to $U$, denoted as $\mathcal{F}|_U$, is a sheaf on $U$
• The restriction functor $(-)_U: Sh(X) \to Sh(U)$ is defined by $\mathcal{F} \mapsto \mathcal{F}|_U$ and $f \mapsto f|_U$, where $f|_U$ is the restriction of a sheaf morphism $f$ to $U$
• The extension functor $i_*: Sh(U) \to Sh(X)$ is defined by $i_*(\mathcal{G})(V) = \mathcal{G}(V \cap U)$ for any open set $V \subseteq X$ and sheaf $\mathcal{G}$ on $U$
• The restriction and extension functors form an adjunction $(-)_U \dashv i_*$, capturing the relationship between sheaves on $X$ and sheaves on $U$

Gluing of sheaves

• Sheaf morphisms enable the gluing of sheaves, allowing for the construction of global sheaves from local data
• Given a topological space $X$ and an open cover $\{U_i\}$ of $X$, suppose we have sheaves $\mathcal{F}_i$ on each $U_i$ and sheaf isomorphisms $\varphi_{ij}: \mathcal{F}_i|_{U_i \cap U_j} \to \mathcal{F}_j|_{U_i \cap U_j}$ satisfying the cocycle condition $\varphi_{jk} \circ \varphi_{ij} = \varphi_{ik}$ on triple intersections
• The gluing data $(\mathcal{F}_i, \varphi_{ij})$ determines a unique sheaf $\mathcal{F}$ on $X$, up to isomorphism, such that $\mathcal{F}|_{U_i} \cong \mathcal{F}_i$ and the isomorphisms are compatible with the $\varphi_{ij}$
• The gluing construction is a powerful tool for constructing sheaves with prescribed local behavior

Exact sequences of sheaves

• Sheaf morphisms allow for the construction of exact sequences of sheaves, which capture important structural information
• A sequence of sheaf morphisms $0 \to \mathcal{F} \stackrel{f}{\to} \mathcal{G} \stackrel{g}{\to} \mathcal{H} \to 0$ is called a short exact sequence if it is exact at each object, meaning:
  • $f$ is injective (kernel is zero)
  • $g$ is surjective (cokernel is zero)
  • $im(f) = ker(g)$
• Exact sequences of sheaves arise naturally in various contexts, such as the study of sheaf cohomology and the classification of vector bundles
• The snake lemma is a powerful tool for studying the relationship between exact sequences of sheaves and their associated long exact sequences in cohomology