Sheaves on Spaces

en · official · sha256:e728b50fec00b5a51ffd85b7518d87d42e03d9b363884124cfca05a3aa077563

\begin{definition} \label{definition-presheaf} Let $X$ be a topological space. \begin{enumerate} \item A {\it presheaf $\mathcal{F}$ of sets on $X$} is a rule which assigns to each open $U \subset X$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ a map $\rho^U_V : \mathcal{F}(U) \to \mathcal{F}(V)$ such that $\rho^U_U = \text{id}_{\mathcal{F}(U)}$ and whenever $W \subset V \subset U$ we have $\rho^U_W = \rho^V_W \circ \rho ^U_V$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of sets on $X$} is a rule which assigns to each open $U \subset X$ a map of sets $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps, i.e., whenever $V \subset U \subset X$ are open the diagram $$ \xymatrix{ \mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho^U_V} & \mathcal{G}(U) \ar[d]^{\rho^U_V} \\ \mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V) } $$ commutes. \item The category of presheaves of sets on $X$ will be denoted $\textit{PSh}(X)$. \end{enumerate} \end{definition}

Referencing norms: H0IsGlobalSections, RestrictionByValues, defined_on, global_sections, global_sections_functor, open_of, presheaf_of_sets_on, restricts_to, section_over

en · official · sha256:1678d42b0eb9230b7149cc456cbf74baa8353baa6d5c6b119500d0fefd2bf9f6

\begin{definition} \label{definition-constant-presheaf} Let $X$ be a topological space. Let $A$ be a set. The {\it constant presheaf with value $A$} is the presheaf that assigns the set $A$ to every open $U \subset X$, and such that all restriction mappings are $\text{id}_A$. \end{definition}

Referencing norms: constant_presheaf

en · official · sha256:b8388b54fb29de485ebd7c91f38c607f6005d41b4d427265a75b163118fd70d6

\begin{definition} \label{definition-abelian-presheaves} Let $X$ be a topological space. \begin{enumerate} \item A {\it presheaf of abelian groups on $X$} or an {\it abelian presheaf over $X$} is a presheaf of sets $\mathcal{F}$ such that for each open $U \subset X$ the set $\mathcal{F}(U)$ is endowed with the structure of an abelian group, and such that all restriction maps $\rho^U_V$ are homomorphisms of abelian groups, see Lemma \ref{lemma-abelian-presheaves} above. \item A {\it morphism of abelian presheaves over $X$} $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets which induces a homomorphism of abelian groups $\mathcal{F}(U) \to \mathcal{G}(U)$ for every open $U \subset X$. \item The category of presheaves of abelian groups on $X$ is denoted $\textit{PAb}(X)$. \end{enumerate} \end{definition}

Referencing norms: abelian_group_structure, abelian_presheaf_on, abelian_presheaf_on/necessary, abelian_presheaf_on/sufficient, power_of_two

en · official · sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76

\begin{definition} \label{definition-sheaf} Let $X$ be a topological space. \begin{enumerate} \item A {\it sheaf $\mathcal{F}$ of sets on $X$} is a presheaf of sets which satisfies the following additional property: Given any open covering $U = \bigcup_{i \in I} U_i$ and any collection of sections $s_i \in \mathcal{F}(U_i)$, $i \in I$ such that $\forall i, j\in I$ $$ s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j} $$ there exists a unique section $s \in \mathcal{F}(U)$ such that $s_i = s|_{U_i}$ for all $i \in I$. \item A {\it morphism of sheaves of sets} is simply a morphism of presheaves of sets. \item The category of sheaves of sets on $X$ is denoted $\Sh(X)$. \end{enumerate} \end{definition}

Referencing norms: DeclaredSheafNotRefutedByData, FamilyCompatible, GluingNotUnique, NoGluing, NotASheafByFailingCovering, NotSheafByFailedGluing, NotSheafByFailedUniqueness, NotSheafByRefutingCovering, RestrictionsAgreeOnOverlap, RestrictsToMember, SectionGluesFamily, SheafByGluing, SheafConditionFailsByExistence, SheafConditionFailsByUniqueness, SheafConditionHoldsOnCovering, UniqueGluing, compatible_sections_glue, covers, families_presented, family_for, family_member, gluing_is_unique, intersection, member

en · official · sha256:792139165e62cd6a503272e7c266371bf232620f144512476457b940aee7f5c9

\begin{remark} \label{remark-confusion} There is always a bit of confusion as to whether it is necessary to say something about the set of sections of a sheaf over the empty set $\emptyset \subset X$. It is necessary, and we already did if you read the definition right. Namely, note that the empty set is covered by the empty open covering, and hence the ``collection of sections $s_i$'' from the definition above actually form an element of the empty product which is the final object of the category the sheaf has values in. In other words, if you read the definition right you automatically deduce that $\mathcal{F}(\emptyset) = \textit{a final object}$, which in the case of a sheaf of sets is a singleton. If you do not like this argument, then you can just require that $\mathcal{F}(\emptyset) = \{*\}$. \medskip\noindent In particular, this condition will then ensure that if $U, V \subset X$ are open and {\it disjoint} then $$ \mathcal{F}(U \cup V) = \mathcal{F}(U) \times \mathcal{F}(V). $$ (Because the fibre product over a final object is a product.) \end{remark}

Referencing norms: FamilyCompatible, GluingNotUnique, NotASheafByFailingCovering, SheafConditionFailsByUniqueness

en · official · sha256:90e7a4291238cb0fe8e5784fd2bf7ee21c45b936b75a3fa06ed0d11a2eb2a041

\begin{example} \label{example-basic-continuous-maps} Let $X$, $Y$ be topological spaces. Consider the rule $\mathcal{F}$ which associates to the open $U \subset X$ the set $$ \mathcal{F}(U) = \{ f : U \to Y \mid f \text{ is continuous}\} $$ with the obvious restriction mappings. We claim that $\mathcal{F}$ is a sheaf. To see this suppose that $U = \bigcup_{i\in I} U_i$ is an open covering, and $f_i \in \mathcal{F}(U_i)$, $i\in I$ with $f_i |_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$ for all $i, j \in I$. In this case define $f : U \to Y$ by setting $f(u)$ equal to the value of $f_i(u)$ for any $i \in I$ such that $u \in U_i$. This is well defined by assumption. Moreover, $f : U \to Y$ is a map such that its restriction to $U_i$ agrees with the continuous map $f_i$. Hence clearly $f$ is continuous! \end{example}

Referencing norms: ConstantSheafIsSheaf

en · official · sha256:dd05660d523049b43c908e00357b644b49aecdc06ba7eb53adb2c325e626e2c3

\begin{definition} \label{definition-constant-sheaf} Let $X$ be a topological space. Let $A$ be a set. The {\it constant sheaf with value $A$} denoted $\underline{A}$, or $\underline{A}_X$ is the sheaf that assigns to an open $U \subset X$ the set of all locally constant maps $U \to A$ with restriction mappings given by restrictions of functions. \end{definition}

Referencing norms: AgreeAtPoint, ConstantSheafIsSheaf, LocallyConstantAlongSpecialization, RestrictionByValues, SameValueAtTwoPoints, SectionsOfConstantSheaf, all_functions_presented, constant_sheaf, value_at, z2_constant_sheaf

en · official · sha256:3260f7d5e9cbfe266160539b8375cd23ac2429762aaa12eb860ad707dc28a6cd

\begin{definition} \label{definition-abelian-sheaf} Let $X$ be a topological space. \begin{enumerate} \item An {\it abelian sheaf on $X$} or {\it sheaf of abelian groups on $X$} is an abelian presheaf on $X$ such that the underlying presheaf of sets is a sheaf. \item The category of sheaves of abelian groups is denoted $\textit{Ab}(X)$. \end{enumerate} \end{definition}

Referencing norms: AbelianSheafByDefinition, AbelianSheafIsObjectOfAbX, sheaves_category

en · official · sha256:282a15f0475c3f68a862ce8346d7972c4f62ab14daa09a1922c02bb9f01c558e

\begin{lemma} \label{lemma-sheafification-sheaf} The presheaf $\mathcal{F}^{\#}$ is a sheaf. \end{lemma}

Referencing norms: SheafificationIsSheaf, plus_construction

en · official · sha256:2ba5dec16073f1fdafcf8a352bfc30f215d20caf060e768c51658875dc939220

\begin{lemma} \label{lemma-sheafify-universal} Let $\mathcal{F}$ be a presheaf of sets on $X$. Any map $\mathcal{F} \to \mathcal{G}$ into a sheaf of sets factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$. \end{lemma}

Referencing norms: SheafifyUniversal, plus_construction

en · official · sha256:6cfa0751723700062760868c89628b4aa1d8d91b0ca0b69f61af805272569a0e

\begin{lemma} \label{lemma-sheafify-abelian-presheaf} Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. Then there exists a unique structure of abelian sheaf on $\mathcal{F}^\#$ such that $\mathcal{F} \to \mathcal{F}^\#$ is a morphism of abelian presheaves. Moreover, the following adjointness property holds $$ \Mor_{\textit{PAb}(X)}(\mathcal{F}, i(\mathcal{G})) = \Mor_{\textit{Ab}(X)}(\mathcal{F}^\#, \mathcal{G}). $$ \end{lemma}

Referencing norms: SheafifyAbelianPresheaf