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 \input{preamble} % OK, start here. % \begin{document} \title{Sheaves on Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent Basic properties of sheaves on topological spaces will be explained in this document. A reference is \cite{Godement}. \medskip\noindent This will be superseded by the discussion of sheaves over sites later in the documents. But perhaps it makes sense to briefly define some of the notions here. \section{Basic notions} \label{section-sheaves-basic} \noindent The following is a list of basic notions in topology. \begin{enumerate} \item Let $X$ be a topological space. The phrase: Let $U = \bigcup_{i \in I} U_i$ be an open covering'' means the following: $I$ is a set and for each $i \in I$ we are given an open subset $U_i \subset X$ such that $U$ is the union of the $U_i$. It is allowed to have $I = \emptyset$ in which case there are no $U_i$ and $U = \emptyset$. It is also allowed, in case $I \not = \emptyset$ to have any or all of the $U_i$ be empty. \item etc, etc. \end{enumerate} \section{Presheaves} \label{section-presheaves} \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} \noindent The elements of the set $\mathcal{F}(U)$ are called the {\it sections} of $\mathcal{F}$ over $U$. For every $V \subset U$ the map $\rho^U_V : \mathcal{F}(U) \to \mathcal{F}(V)$ is called the {\it restriction map}. We will use the notation $s|_V := \rho^U_V(s)$ if $s\in \mathcal{F}(U)$. This notation is consistent with the notion of restriction of functions from topology because if $W \subset V \subset U$ and $s$ is a section of $\mathcal{F}$ over $U$ then $s|_W = (s|_V)|_W$ by the property of the restriction maps expressed in the definition above. \medskip\noindent Another notation that is often used is to indicate sections over an open $U$ by the symbol $\Gamma(U, -)$ or by $H^0(U, -)$. In other words, the following equalities are tautological $$\Gamma(U, \mathcal{F}) = \mathcal{F}(U) = H^0(U, \mathcal{F}).$$ In this chapter we will not use this notation, but in others we will. \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} \section{Abelian presheaves} \label{section-abelian-presheaves} \noindent In this section we briefly point out some features of the category of presheaves that allow one to define presheaves of abelian groups. \begin{example} \label{example-singleton-presheaf} Let $X$ be a topological space $X$. Consider a rule $\mathcal{F}$ that associates to every open subset a singleton set. Since every set has a unique map into a singleton set, there exist unique restriction maps $\rho^U_V$. The resulting structure is a presheaf of sets. It is a final object in the category of presheaves of sets, by the property of singleton sets mentioned above. Hence it is also unique up to unique isomorphism. We will sometimes write $*$ for this presheaf. \end{example} \begin{lemma} \label{lemma-product-presheaves} Let $X$ be a topological space. The category of presheaves of sets on $X$ has products (see Categories, Definition \ref{categories-definition-product}). Moreover, the set of sections of the product $\mathcal{F} \times \mathcal{G}$ over an open $U$ is the product of the sets of sections of $\mathcal{F}$ and $\mathcal{G}$ over $U$. \end{lemma} \begin{proof} Namely, suppose $\mathcal{F}$ and $\mathcal{G}$ are presheaves of sets on the topological space $X$. Consider the rule $U \mapsto \mathcal{F}(U) \times \mathcal{G}(U)$, denoted $\mathcal{F} \times \mathcal{G}$. If $V \subset U \subset X$ are open then define the restriction mapping $$(\mathcal{F} \times \mathcal{G})(U) \longrightarrow (\mathcal{F} \times \mathcal{G})(V)$$ by mapping $(s, t) \mapsto (s|_V, t|_V)$. Then it is immediately clear that $\mathcal{F} \times \mathcal{G}$ is a presheaf. Also, there are projection maps $p : \mathcal{F} \times \mathcal{G} \to \mathcal{F}$ and $q : \mathcal{F} \times \mathcal{G} \to \mathcal{G}$. We leave it to the reader to show that for any third presheaf $\mathcal{H}$ we have $\Mor(\mathcal{H}, \mathcal{F} \times \mathcal{G}) = \Mor(\mathcal{H}, \mathcal{F}) \times \Mor(\mathcal{H}, \mathcal{G})$. \end{proof} \noindent Recall that if $(A, + : A \times A \to A, - : A \to A, 0\in A)$ is an abelian group, then the zero and the negation maps are uniquely determined by the addition law. In other words, it makes sense to say let $(A, +)$ be an abelian group''. \begin{lemma} \label{lemma-abelian-presheaves} Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets. Consider the following types of structure on $\mathcal{F}$: \begin{enumerate} \item For every open $U$ the structure of an abelian group on $\mathcal{F}(U)$ such that all restriction maps are abelian group homomorphisms. \item A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$, a map of presheaves $- : \mathcal{F} \to \mathcal{F}$ and a map $0 : * \to \mathcal{F}$ (see Example \ref{example-singleton-presheaf}) satisfying all the axioms of $+, -, 0$ in a usual abelian group. \item A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$, a map of presheaves $- : \mathcal{F} \to \mathcal{F}$ and a map $0 : * \to \mathcal{F}$ such that for each open $U \subset X$ the quadruple $(\mathcal{F}(U), +, -, 0)$ is an abelian group, \item A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ such that for every open $U \subset X$ the map $+ : \mathcal{F}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an abelian group. \end{enumerate} There are natural bijections between the collections of types of data (1) - (4) above. \end{lemma} \begin{proof} Omitted. \end{proof} \noindent The lemma says that to give an abelian group object $\mathcal{F}$ in the category of presheaves is the same as giving a presheaf of sets $\mathcal{F}$ such that all the sets $\mathcal{F}(U)$ are endowed with the structure of an abelian group and such that all the restriction mappings are group homomorphisms. For most algebra structures we will take this approach to (pre)sheaves of such objects, i.e., we will define a (pre)sheaf of such objects to be a (pre)sheaf $\mathcal{F}$ of sets all of whose sets of sections $\mathcal{F}(U)$ are endowed with this structure compatibly with the restriction mappings. \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} \begin{example} \label{example-direct-sum-points} Let $X$ be a topological space. For each $x \in X$ suppose given an abelian group $M_x$. For $U \subset X$ open we set $$\mathcal{F}(U) = \bigoplus\nolimits_{x \in U} M_x.$$ We denote a typical element in this abelian group by $\sum_{i = 1}^n m_{x_i}$, where $x_i \in U$ and $m_{x_i} \in M_{x_i}$. (Of course we may always choose our representation such that $x_1, \ldots, x_n$ are pairwise distinct.) We define for $V \subset U \subset X$ open a restriction mapping $\mathcal{F}(U) \to \mathcal{F}(V)$ by mapping an element $s = \sum_{i = 1}^n m_{x_i}$ to the element $s|_V = \sum_{x_i \in V} m_{x_i}$. We leave it to the reader to verify that this is a presheaf of abelian groups. \end{example} \section{Presheaves of algebraic structures} \label{section-presheaves-structures} \noindent Let us clarify the definition of presheaves of algebraic structures. Suppose that $\mathcal{C}$ is a category and that $F : \mathcal{C} \to \textit{Sets}$ is a faithful functor. Typically $F$ is a forgetful'' functor. For an object $M \in \Ob(\mathcal{C})$ we often call $F(M)$ the {\it underlying set} of the object $M$. If $M \to M'$ is a morphism in $\mathcal{C}$ we call $F(M) \to F(M')$ the {\it underlying map of sets}. In fact, we will often not distinguish between an object and its underlying set, and similarly for morphisms. So we will say a map of sets $F(M) \to F(M')$ is a {\it morphism of algebraic structures}, if it is equal to $F(f)$ for some morphism $f : M \to M'$ in $\mathcal{C}$. \medskip\noindent In analogy with Definition \ref{definition-abelian-presheaves} above a presheaf of objects of $\mathcal{C}$'' could be defined by the following data: \begin{enumerate} \item a presheaf of sets $\mathcal{F}$, and \item for every open $U \subset X$ a choice of an object $A(U) \in \Ob(\mathcal{C})$ \end{enumerate} subject to the following conditions (using the phraseology above) \begin{enumerate} \item for every open $U \subset X$ the set $\mathcal{F}(U)$ is the underlying set of $A(U)$, and \item for every $V \subset U \subset X$ open the map of sets $\rho_V^U: \mathcal{F}(U) \to \mathcal{F}(V)$ is a morphism of algebraic structures. \end{enumerate} In other words, for every $V \subset U$ open in $X$ the restriction mappings $\rho^U_V$ is the image $F(\alpha^U_V)$ for some unique morphism $\alpha^U_V : A(U) \to A(V)$ in the category $\mathcal{C}$. The uniqueness is forced by the condition that $F$ is faithful; it also implies that $\alpha^U_W = \alpha^V_W \circ \alpha^U_V$ whenever $W \subset V \subset U$ are open in $X$. The system $(A(-), \alpha^U_V)$ is what we will define as a presheaf with values in $\mathcal{C}$ on $X$, compare Sites, Definition \ref{sites-definition-presheaf}. We recover our presheaf of sets $(\mathcal{F}, \rho_V^U)$ via the rules $\mathcal{F}(U) = F(A(U))$ and $\rho_V^U = F(\alpha_V^U)$. \begin{definition} \label{definition-presheaf-values-in-category} Let $X$ be a topological space. Let $\mathcal{C}$ be a category. \begin{enumerate} \item A {\it presheaf $\mathcal{F}$ on $X$ with values in $\mathcal{C}$} is given by a rule which assigns to every open $U \subset X$ an object $\mathcal{F}(U)$ of $\mathcal{C}$ and to each inclusion $V \subset U$ a morphism $\rho_V^U : \mathcal{F}(U) \to \mathcal{F}(V)$ in $\mathcal{C}$ such that whenever $W \subset V \subset U$ we have $\rho_W^U = \rho_W^V \circ \rho_V^U$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves with value in $\mathcal{C}$} is given by a morphism $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ in $\mathcal{C}$ compatible with restriction morphisms. \end{enumerate} \end{definition} \begin{definition} \label{definition-underlying-presheaf-sets} Let $X$ be a topological space. Let $\mathcal{C}$ be a category. Let $F : \mathcal{C} \to \textit{Sets}$ be a faithful functor. Let $\mathcal{F}$ be a presheaf on $X$ with values in $\mathcal{C}$. The presheaf of sets $U \mapsto F(\mathcal{F}(U))$ is called the {\it underlying presheaf of sets of $\mathcal{F}$}. \end{definition} \noindent It is customary to use the same letter $\mathcal{F}$ to denote the underlying presheaf of sets, and this makes sense according to our discussion preceding Definition \ref{definition-presheaf-values-in-category}. In particular, the phrase let $s \in \mathcal{F}(U)$'' or let $s$ be a section of $\mathcal{F}$ over $U$'' signifies that $s \in F(\mathcal{F}(U))$. \medskip\noindent This notation and these definitions apply in particular to: {\it Presheaves of (not necessarily abelian) groups, rings, modules over a fixed ring, vector spaces over a fixed field, } etc and {\it morphisms between these}. \section{Presheaves of modules} \label{section-presheaves-modules} \noindent Suppose that $\mathcal{O}$ is a presheaf of rings on $X$. We would like to define the notion of a presheaf of $\mathcal{O}$-modules over $X$. In analogy with Definition \ref{definition-abelian-presheaves} we are tempted to define this as a sheaf of sets $\mathcal{F}$ such that for every open $U \subset X$ the set $\mathcal{F}(U)$ is endowed with the structure of an $\mathcal{O}(U)$-module compatible with restriction mappings (of $\mathcal{F}$ and $\mathcal{O}$). However, it is customary (and equivalent) to define it as in the following definition. \begin{definition} \label{definition-presheaf-modules} Let $X$ be a topological space, and let $\mathcal{O}$ be a presheaf of rings on $X$. \begin{enumerate} \item A {\it presheaf of $\mathcal{O}$-modules} is given by an abelian presheaf $\mathcal{F}$ together with a map of presheaves of sets $$\mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F}$$ such that for every open $U \subset X$ the map $\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an $\mathcal{O}(U)$-module structure on the abelian group $\mathcal{F}(U)$. \item A {\it morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of $\mathcal{O}$-modules} is a morphism of abelian presheaves $\varphi : \mathcal{F} \to \mathcal{G}$ such that the diagram $$\xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi} & \mathcal{F} \ar[d]^{\varphi} \\ \mathcal{O} \times \mathcal{G} \ar[r] & \mathcal{G} }$$ commutes. \item The set of $\mathcal{O}$-module morphisms as above is denoted $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$. \item The category of presheaves of $\mathcal{O}$-modules is denoted $\textit{PMod}(\mathcal{O})$. \end{enumerate} \end{definition} \noindent Suppose that $\mathcal{O}_1 \to \mathcal{O}_2$ is a morphism of presheaves of rings on $X$. In this case, if $\mathcal{F}$ is a presheaf of $\mathcal{O}_2$-modules then we can think of $\mathcal{F}$ as a presheaf of $\mathcal{O}_1$-modules by using the composition $$\mathcal{O}_1 \times \mathcal{F} \to \mathcal{O}_2 \times \mathcal{F} \to \mathcal{F}.$$ We sometimes denote this by $\mathcal{F}_{\mathcal{O}_1}$ to indicate the restriction of rings. We call this the {\it restriction of $\mathcal{F}$}. We obtain the restriction functor $$\textit{PMod}(\mathcal{O}_2) \longrightarrow \textit{PMod}(\mathcal{O}_1)$$ \medskip\noindent On the other hand, given a presheaf of $\mathcal{O}_1$-modules $\mathcal{G}$ we can construct a presheaf of $\mathcal{O}_2$-modules $\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}$ by the rule $$\left(\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}\right)(U) = \mathcal{O}_2(U) \otimes_{\mathcal{O}_1(U)} \mathcal{G}(U)$$ The index $p$ stands for presheaf'' and not point''. This presheaf is called the tensor product presheaf. We obtain the {\it change of rings} functor $$\textit{PMod}(\mathcal{O}_1) \longrightarrow \textit{PMod}(\mathcal{O}_2)$$ \begin{lemma} \label{lemma-adjointness-tensor-restrict-presheaves} With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection $$\Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} )$$ In other words, the restriction and change of rings functors are adjoint to each other. \end{lemma} \begin{proof} This follows from the fact that for a ring map $A \to B$ the restriction functor and the change of ring functor are adjoint to each other. \end{proof} \section{Sheaves} \label{section-sheaves} \noindent In this section we explain the sheaf condition. \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} \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 section $s_i$'' 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} \begin{example} \label{example-basic-continuous-maps} Let $X$, $Y$ be topological spaces. Consider the rule $\mathcal{F}$ wich 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 $U_i$. Hence clearly $f$ is continuous! \end{example} \noindent We can use the result of the example to define constant sheaves. Namely, suppose that $A$ is a set. Endow $A$ with the discrete topology. Let $U \subset X$ be an open subset. Then we have $$\{ f : U \to A \mid f\text{ continuous}\} = \{ f : U \to A \mid f\text{ locally constant}\}.$$ Thus the rule which assigns to an open all locally constant maps into $A$ is a sheaf. \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} \begin{example} \label{example-sheaf-product-pointwise} Let $X$ be a topological space. Let $(A_x)_{x \in X}$ be a family of sets $A_x$ indexed by points $x \in X$. We are going to construct a sheaf of sets $\Pi$ from this data. For $U \subset X$ open set $$\Pi(U) = \prod\nolimits_{x \in U} A_x.$$ For $V \subset U \subset X$ open define a restriction mapping by the following rule: An element $s = (a_x)_{x\in U} \in \Pi(U)$ restricts to $s|_V = (a_x)_{x \in V}$. It is obvious that this defines a presheaf of sets. We claim this is a sheaf. Namely, let $U = \bigcup U_i$ be an open covering. Suppose that $s_i \in \Pi(U_i)$ are such that $s_i$ and $s_j$ agree over $U_i \cap U_j$. Write $s_i = (a_{i, x})_{x\in U_i}$. The compatibility condition implies that $a_{i, x} = a_{j, x}$ in the set $A_x$ whenever $x \in U_i \cap U_j$. Hence there exists a unique element $s = (a_x)_{x\in U}$ in $\Pi(U) = \prod_{x\in U} A_x$ with the property that $a_x = a_{i, x}$ whenever $x \in U_i$ for some $i$. Of course this element $s$ has the property that $s|_{U_i} = s_i$ for all $i$. \end{example} \begin{example} \label{example-direct-sum-points-not-sheaf} Let $X$ be a topological space. Suppose for each $x\in X$ we are given an abelian group $M_x$. Consider the presheaf $\mathcal{F} : U \mapsto \bigoplus_{x \in U} M_x$ defined in Example \ref{example-direct-sum-points}. This is not a sheaf in general. For example, if $X$ is an infinite set with the discrete topology, then the sheaf condition would imply that $\mathcal{F}(X) = \prod_{x\in X} \mathcal{F}(\{x\})$ but by definition we have $\mathcal{F}(X) = \bigoplus_{x \in X} M_x = \bigoplus_{x \in X} \mathcal{F}(\{x\})$. And an infinite direct sum is in general different from an infinite direct product. \medskip\noindent However, if $X$ is a topological space such that every open of $X$ is quasi-compact, then $\mathcal{F}$ {\it is} a sheaf. This is left as an exercise to the reader. \end{example} \section{Abelian sheaves} \label{section-abelian-sheaves} \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} \noindent Let $X$ be a topological space. In the case of an abelian presheaf $\mathcal{F}$ the sheaf condition with regards to an open covering $U = \bigcup U_i$ is often expressed by saying that the complex of abelian groups $$0 \to \mathcal{F}(U) \to \prod\nolimits_i \mathcal{F}(U_i) \to \prod\nolimits_{(i_0, i_1)} \mathcal{F}(U_{i_0} \cap U_{i_1})$$ is exact. The first map is the usual one, whereas the second maps the element $(s_i)_{i \in I}$ to the element $$( s_{i_0}|_{U_{i_0} \cap U_{i_1}} - s_{i_1}|_{U_{i_0} \cap U_{i_1}} )_{(i_0, i_1)} \in \prod\nolimits_{(i_0, i_1)} \mathcal{F}(U_{i_0} \cap U_{i_1})$$ \section{Sheaves of algebraic structures} \label{section-sheaves-structures} \noindent Let us clarify the definition of sheaves of certain types of structures. First, let us reformulate the sheaf condition. Namely, suppose that $\mathcal{F}$ is a presheaf of sets on the topological space $X$. The sheaf condition can be reformulated as follows. Let $U = \bigcup_{i\in I} U_i$ be an open covering. Consider the diagram $$\xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) }$$ Here the left map is defined by the rule $s \mapsto \prod_{i \in I} s|_{U_i}$. The two maps on the right are the maps $$\prod\nolimits_i s_i \mapsto \prod\nolimits_{(i_0, i_1)} s_{i_0}|_{U_{i_0} \cap U_{i_1}} \text{ resp. } \prod\nolimits_i s_i \mapsto \prod\nolimits_{(i_0, i_1)} s_{i_1}|_{U_{i_0} \cap U_{i_1}}.$$ The sheaf condition exactly says that the left arrow is the equalizer of the right two. This generalizes immediately to the case of presheaves with values in a category as long as the category has products. \begin{definition} \label{definition-sheaf-values-in-category} Let $X$ be a topological space. Let $\mathcal{C}$ be a category with products. A presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $X$ is a {\it sheaf} if for every open covering the diagram $$\xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) }$$ is an equalizer diagram in the category $\mathcal{C}$. \end{definition} \noindent Suppose that $\mathcal{C}$ is a category and that $F : \mathcal{C} \to \textit{Sets}$ is a faithful functor. A good example to keep in mind is the case where $\mathcal{C}$ is the category of abelian groups and $F$ is the forgetful functor. Consider a presheaf $\mathcal{F}$ with values in $\mathcal{C}$ on $X$. We would like to reformulate the condition above in terms of the underlying presheaf of sets (Definition \ref{definition-underlying-presheaf-sets}). Note that the underlying presheaf of sets is a sheaf of sets if and only if all the diagrams $$\xymatrix{ F(\mathcal{F}(U)) \ar[r] & \prod\nolimits_{i\in I} F(\mathcal{F}(U_i)) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i_0, i_1) \in I \times I} F(\mathcal{F}(U_{i_0} \cap U_{i_1})) }$$ of sets -- after applying the forgetful functor $F$ -- are equalizer diagrams! Thus we would like $\mathcal{C}$ to have products and equalizers and we would like $F$ to commute with them. This is equivalent to the condition that $\mathcal{C}$ has limits and that $F$ commutes with them, see Categories, Lemma \ref{categories-lemma-limits-products-equalizers}. But this is not yet good enough (see Example \ref{example-sheaves-topological-spaces}); we also need $F$ to {\it reflect isomorphisms}. This property means that given a morphism $f : A \to A'$ in $\mathcal{C}$, then $f$ is an isomorphism if (and only if) $F(f)$ is a bijection. \begin{lemma} \label{lemma-sheaves-structure} Suppose the category $\mathcal{C}$ and the functor $F : \mathcal{C} \to \textit{Sets}$ have the following properties: \begin{enumerate} \item $F$ is faithful, \item $\mathcal{C}$ has limits and $F$ commutes with them, and \item the functor $F$ reflects isomorphisms. \end{enumerate} Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$. Then $\mathcal{F}$ is a sheaf if and only if the underlying presheaf of sets is a sheaf. \end{lemma} \begin{proof} Assume that $\mathcal{F}$ is a sheaf. Then $\mathcal{F}(U)$ is the equalizer of the diagram above and by assumption we see $F(\mathcal{F}(U))$ is the equalizer of the corresponding diagram of sets. Hence $F(\mathcal{F})$ is a sheaf of sets. \medskip\noindent Assume that $F(\mathcal{F})$ is a sheaf. Let $E \in \Ob(\mathcal{C})$ be the equalizer of the two parallel arrows in Definition \ref{definition-sheaf-values-in-category}. We get a canonical morphism $\mathcal{F}(U) \to E$, simply because $\mathcal{F}$ is a presheaf. By assumption, the induced map $F(\mathcal{F}(U)) \to F(E)$ is an isomorphism, because $F(E)$ is the equalizer of the corresponding diagram of sets. Hence we see $\mathcal{F}(U) \to E$ is an isomorphism by condition (3) of the lemma. \end{proof} \noindent The lemma in particular applies to {\it sheaves of groups, rings, algebras over a fixed ring, modules over a fixed ring, vector spaces over a fixed field, } etc. In other words, these are presheaves of groups, rings, modules over a fixed ring, vector spaces over a fixed field, etc such that the underlying presheaf of sets is a sheaf. \begin{example} \label{example-C0-sheaf-rings} Let $X$ be a topological space. For each open $U \subset X$ consider the $\mathbf{R}$-algebra $\mathcal{C}^{0}(U) = \{ f : U \to \mathbf{R} \mid f\text{ is continuous}\}$. There are obvious restriction mappings that turn this into a presheaf of $\mathbf{R}$-algebras over $X$. By Example \ref{example-basic-continuous-maps} it is a sheaf of sets. Hence by the Lemma \ref{lemma-sheaves-structure} it is a sheaf of $\mathbf{R}$-algebras over $X$. \end{example} \begin{example} \label{example-sheaves-topological-spaces} Consider the category of topological spaces $\textit{Top}$. There is a natural faithful functor $\textit{Top} \to \textit{Sets}$ which commutes with products and equalizers. But it does not reflect isomorphisms. And, in fact it turns out that the analogue of Lemma \ref{lemma-sheaves-structure} is wrong. Namely, suppose $X = \mathbf{N}$ with the discrete topology. Let $A_i$, for $i \in \mathbf{N}$ be a discrete topological space. For any subset $U \subset \mathbf{N}$ define $\mathcal{F}(U) = \prod_{i\in U} A_i$ with the discrete topology. Then this is a presheaf of topological spaces whose underlying presheaf of sets is a sheaf, see Example \ref{example-sheaf-product-pointwise}. However, if each $A_i$ has at least two elements, then this is not a sheaf of topological spaces according to Definition \ref{definition-sheaf-values-in-category}. The reader may check that putting the {\it product topology} on each $\mathcal{F}(U) = \prod_{i\in U} A_i$ does lead to a sheaf of topological spaces over $X$. \end{example} \section{Sheaves of modules} \label{section-sheaves-modules} \begin{definition} \label{definition-sheaf-modules} Let $X$ be a topological space. Let $\mathcal{O}$ be a sheaf of rings on $X$. \begin{enumerate} \item A {\it sheaf of $\mathcal{O}$-modules} is a presheaf of $\mathcal{O}$-modules $\mathcal{F}$, see Definition \ref{definition-presheaf-modules}, such that the underlying presheaf of abelian groups $\mathcal{F}$ is a sheaf. \item A {\it morphism of sheaves of $\mathcal{O}$-modules} is a morphism of presheaves of $\mathcal{O}$-modules. \item Given sheaves of $\mathcal{O}$-modules $\mathcal{F}$ and $\mathcal{G}$ we denote $\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})$ the set of morphism of sheaves of $\mathcal{O}$-modules. \item The category of sheaves of $\mathcal{O}$-modules is denoted $\textit{Mod}(\mathcal{O})$. \end{enumerate} \end{definition} \noindent This definition kind of makes sense even if $\mathcal{O}$ is just a presheaf of rings, although we do not know any examples where this is useful, and we will avoid using the terminology sheaves of $\mathcal{O}$-modules'' in case $\mathcal{O}$ is not a sheaf of rings. \section{Stalks} \label{section-stalks} \noindent Let $X$ be a topological space. Let $x \in X$ be a point. Let $\mathcal{F}$ be a presheaf of sets on $X$. The {\it stalk of $\mathcal{F}$ at $x$} is the set $$\mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U)$$ where the colimit is over the set of open neighbourhoods $U$ of $x$ in $X$. The set of open neighbourhoods is partially ordered by (reverse) inclusion: We say $U \geq U' \Leftrightarrow U \subset U'$. The transition maps in the system are given by the restriction maps of $\mathcal{F}$. See Categories, Section \ref{categories-section-posets-limits} for notation and terminology regarding (co)limits over systems. Note that the colimit is a directed colimit. Thus it is easy to describe $\mathcal{F}_x$. Namely, $$\mathcal{F}_x = \{ (U, s) \mid x\in U, s\in \mathcal{F}(U) \}/\sim$$ with equivalence relation given by $(U, s) \sim (U', s')$ if and only if there exists an open $U'' \subset U \cap U'$ with $x \in U''$ and $s|_{U''} = s'|_{U''}$. By abuse of notation we will often denote $(U, s)$, $s_x$, or even $s$ the corresponding element in $\mathcal{F}_x$. Also we will say $s = s'$ in $\mathcal{F}_x$ for two local sections of $\mathcal{F}$ defined in an open neighbourhood of $x$ to denote that they have the same image in $\mathcal{F}_x$. \medskip\noindent An obvious consequence of this definition is that for any open $U \subset X$ there is a canonical map $$\mathcal{F}(U) \longrightarrow \prod\nolimits_{x \in U} \mathcal{F}_x$$ defined by $s \mapsto \prod_{x \in U} (U, s)$. Think about it! \begin{lemma} \label{lemma-sheaf-subset-stalks} Let $\mathcal{F}$ be a sheaf of sets on the topological space $X$. For every open $U \subset X$ the map $$\mathcal{F}(U) \longrightarrow \prod\nolimits_{x \in U} \mathcal{F}_x$$ is injective. \end{lemma} \begin{proof} Suppose that $s, s' \in \mathcal{F}(U)$ map to the same element in every stalk $\mathcal{F}_x$ for all $x \in U$. This means that for every $x \in U$, there exists an open $V^x \subset U$, $x \in V^x$ such that $s|_{V^x} = s'|_{V^x}$. But then $U = \bigcup_{x \in U} V^x$ is an open covering. Thus by the uniqueness in the sheaf condition we see that $s = s'$. \end{proof} \begin{definition} \label{definition-separated} Let $X$ be a topological space. A presheaf of sets $\mathcal{F}$ on $X$ is {\it separated} if for every open $U \subset X$ the map $\mathcal{F}(U) \to \prod_{x \in U} \mathcal{F}_x$ is injective. \end{definition} \noindent Another observation is that the construction of the stalk $\mathcal{F}_x$ is functorial in the presheaf $\mathcal{F}$. In other words, it gives a functor $$\textit{PSh}(X) \longrightarrow \textit{Sets}, \ \mathcal{F} \longmapsto \mathcal{F}_x.$$ This functor is called the {\it stalk functor}. Namely, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves, then we define $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ by the rule $(U, s) \mapsto (U, \varphi(s))$. To see that this works we have to check that if $(U, s) = (U', s')$ in $\mathcal{F}_x$ then also $(U, \varphi(s)) = (U', \varphi(s'))$ in $\mathcal{G}_x$. This is clear since $\varphi$ is compatible with the restriction mappings. \begin{example} \label{example-stalk-constant-presheaf} Let $X$ be a topological space. Let $A$ be a set. Denote temporarily $A_p$ the constant presheaf with value $A$ ($p$ for presheaf -- not for point). There is a canonical map of presheaves $A_p \to \underline{A}$ into the constant sheaf with value $A$. For every point we have canonical bijections $A = (A_p)_x = \underline{A}_x$, where the second map is induced by functoriality from the map $A_p \to \underline{A}$. \end{example} \begin{example} \label{example-germs-functions} Suppose $X = \mathbf{R}^n$ with the Euclidean topology. Consider the presheaf of $\mathcal{C}^\infty$ functions on $X$, denoted $\mathcal{C}^\infty_{\mathbf{R}^n}$. In other words, $\mathcal{C}^\infty_{\mathbf{R}^n}(U)$ is the set of $\mathcal{C}^\infty$-functions $f : U \to \mathbf{R}$. As in Example \ref{example-basic-continuous-maps} it is easy to show that this is a sheaf. In fact it is a sheaf of $\mathbf{R}$-vector spaces. \medskip\noindent Next, let $x \in X = \mathbf{R}^n$ be a point. How do we think of an element in the stalk $\mathcal{C}^\infty_{\mathbf{R}^n, x}$? Such an element is given by a $\mathcal{C}^\infty$-function $f$ whose domain contains $x$. And a pair of such functions $f$, $g$ determine the same element of the stalk if they agree in a neighbourhood of $x$. In other words, an element if $\mathcal{C}^\infty_{\mathbf{R}^n, x}$ is the same thing as what is sometimes called a {\it germ of a $\mathcal{C}^\infty$-function at $x$}. \end{example} \begin{example} \label{example-sheaf-product-pointwise-stalk} Let $X$ be a topological space. Let $A_x$ be a set for each $x \in X$. Consider the sheaf $\mathcal{F} : U \mapsto \prod_{x\in U} A_x$ of Example \ref{example-sheaf-product-pointwise}. We would just like to point out here that the stalk $\mathcal{F}_x$ of $\mathcal{F}$ at $x$ is in general {\it not} equal to the set $A_x$. Of course there is a map $\mathcal{F}_x \to A_x$, but that is in general the best you can say. For example, suppose $x = \lim x_n$ with $x_n \not = x_m$ for all $n \not = m$ and suppose that $A_y = \{0, 1\}$ for all $y \in X$. Then $\mathcal{F}_x$ maps onto the (infinite) set of tails of sequences of $0$s and $1$s. Namely, every open neighbourhood of $x$ contains almost all of the $x_n$. On the other hand, if every neighbourhood of $x$ contains a point $y$ such that $A_y = \emptyset$, then $\mathcal{F}_x = \emptyset$. \end{example} \section{Stalks of abelian presheaves} \label{section-stalks-abelian-presheaves} \noindent We first deal with the case of abelian groups as a model for the general case. \begin{lemma} \label{lemma-stalk-abelian-presheaf} Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of abelian groups on $X$. There exists a unique structure of an abelian group on $\mathcal{F}_x$ such that for every $U \subset X$ open, $x\in U$ the map $\mathcal{F}(U) \to \mathcal{F}_x$ is a group homomorphism. Moreover, $$\mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U)$$ holds in the category of abelian groups. \end{lemma} \begin{proof} We define addition of a pair of elements $(U, s)$ and $(V, t)$ as the pair $(U \cap V, s|_{U\cap V} + t|_{U \cap V})$. The rest is easy to check. \end{proof} \noindent What is crucial in the proof above is that the partially ordered set of open neighbourhoods is a directed set (Categories, Definition \ref{categories-definition-directed-set}). Namely, the coproduct of two abelian groups $A, B$ is the direct sum $A \oplus B$, whereas the coproduct in the category of sets is the disjoint union $A \amalg B$, showing that colimits in the category of abelian groups do not agree with colimits in the category of sets in general. \section{Stalks of presheaves of algebraic structures} \label{section-stalks-presheaves-structures} \noindent The proof of Lemma \ref{lemma-stalk-abelian-presheaf} will work for any type of algebraic structure such that directed colimits commute with the forgetful functor. \begin{lemma} \label{lemma-stalk-presheaf-values-in-category} Let $\mathcal{C}$ be a category. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $F$ is faithful, and \item directed colimits exist in $\mathcal{C}$ and $F$ commutes with them. \end{enumerate} Let $X$ be a topological space. Let $x \in X$. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$. Then $$\mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U)$$ exists in $\mathcal{C}$. Its underlying set is equal to the stalk of the underlying presheaf of sets of $\mathcal{F}$. Furthermore, the construction $\mathcal{F} \mapsto \mathcal{F}_x$ is a functor from the category of presheaves with values in $\mathcal{C}$ to $\mathcal{C}$. \end{lemma} \begin{proof} Omitted. \end{proof} \noindent By the very definition, all the morphisms $\mathcal{F}(U) \to \mathcal{F}_x$ are morphisms in the category $\mathcal{C}$ which (after applying the forgetful functor $F$) turn into the corresponding maps for the underlying sheaf of sets. As usual we will not distinguish between the morphism in $\mathcal{C}$ and the underlying map of sets, which is permitted since $F$ is faithful. \medskip\noindent This lemma applies in particular to: {\it Presheaves of (not necessarily abelian) groups, rings, modules over a fixed ring, vector spaces over a fixed field}. \section{Stalks of presheaves of modules} \label{section-stalk-presheaves-modules} \begin{lemma} \label{lemma-stalk-module} Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $x \in X$. The canonical map $\mathcal{O}_x \times \mathcal{F}_x \to \mathcal{F}_x$ coming from the multiplication map $\mathcal{O} \times \mathcal{F} \to \mathcal{F}$ defines a $\mathcal{O}_x$-module structure on the abelian group $\mathcal{F}_x$. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-stalk-tensor-presheaf-modules} Let $X$ be a topological space. Let $\mathcal{O} \to \mathcal{O}'$ be a morphism of presheaves of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $x \in X$. We have $$\mathcal{F}_x \otimes_{\mathcal{O}_x} \mathcal{O}'_x = (\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{O}')_x$$ as $\mathcal{O}'_x$-modules. \end{lemma} \begin{proof} Omitted. \end{proof} \section{Algebraic structures} \label{section-algebraic-structures} \noindent In this section we mildly formalize the notions we have encountered in the sections above. \begin{definition} \label{definition-algebraic-structure} A {\it type of algebraic structure} is given by a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \textit{Sets}$ with the following properties \begin{enumerate} \item $F$ is faithful, \item $\mathcal{C}$ has limits and $F$ commutes with limits, \item $\mathcal{C}$ has filtered colimits and $F$ commutes with them, and \item $F$ reflects isomorphisms. \end{enumerate} \end{definition} \noindent We make this definition to point out the properties we will use in a number of arguments below. But we will not actually study this notion in any great detail, since we are prohibited from studying big'' categories by convention, except for those listed in Categories, Remark \ref{categories-remark-big-categories}. Among those the following have the required properties. \begin{lemma} \label{lemma-list-algebraic-structures} The following categories, endowed with the obvious forgetful functor, define types of algebraic structures: \begin{enumerate} \item The category of pointed sets. \item The category of abelian groups. \item The category of groups. \item The category of monoids. \item The category of rings. \item The category of $R$-modules for a fixed ring $R$. \item The category of Lie algebras over a fixed field. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \noindent From now on we will think of a (pre)sheaf of algebraic structures and their stalks, in terms of the underlying (pre)sheaf of sets. This is allowable by Lemmas \ref{lemma-sheaves-structure} and \ref{lemma-stalk-presheaf-values-in-category}. \medskip\noindent In the rest of this section we point out some results on algebraic structures that will be useful in the future. \begin{lemma} \label{lemma-properties-algebraic-structures} Let $(\mathcal{C}, F)$ be a type of algebraic structure. \begin{enumerate} \item $\mathcal{C}$ has a final object $0$ and $F(0) = \{ * \}$. \item $\mathcal{C}$ has products and $F(\prod A_i) = \prod F(A_i)$. \item $\mathcal{C}$ has fibre products and $F(A \times_B C) = F(A)\times_{F(B)}F(C)$. \item $\mathcal{C}$ has equalizers, and if $E \to A$ is the equalizer of $a, b : A \to B$, then $F(E) \to F(A)$ is the equalizer of $F(a), F(b) : F(A) \to F(B)$. \item $A \to B$ is a monomorphism if and only if $F(A) \to F(B)$ is injective. \item if $F(a) : F(A) \to F(B)$ is surjective, then $a$ is an epimorphism. \item given $A_1 \to A_2 \to A_3 \to \ldots$, then $\colim A_i$ exists and $F(\colim A_i) = \colim F(A_i)$, and more generally for any filtered colimit. \end{enumerate} \end{lemma} \begin{proof} Omitted. The only interesting statement is (5) which follows because $A \to B$ is a monomorphism if and only if $A \to A \times_B A$ is an isomorphism, and then applying the fact that $F$ reflects isomorphisms. \end{proof} \begin{lemma} \label{lemma-image-contained-in} Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $A, B, C \in \Ob(\mathcal{C})$. Let $f : A \to B$ and $g : C \to B$ be morphisms of $\mathcal{C}$. If $F(g)$ is injective, and $\Im(F(f)) \subset \Im(F(g))$, then $f$ factors as $f = g \circ t$ for some morphism $t : A \to C$. \end{lemma} \begin{proof} Consider $A \times_B C$. The assumptions imply that $F(A \times_B C) = F(A) \times_{F(B)} F(C) = F(A)$. Hence $A = A \times_B C$ because $F$ reflects isomorphisms. The result follows. \end{proof} \begin{example} \label{example-application-lemma-image-contained-in} The lemma will be applied often to the following situation. Suppose that we have a diagram $$\xymatrix{ A \ar[r] & B \ar[d] \\ C \ar[r] & D }$$ in $\mathcal{C}$. Suppose $C \to D$ is injective on underlying sets, and suppose that the composition $A \to B \to D$ has image on underlying sets in the image of $C \to D$. Then we get a commutative diagram $$\xymatrix{ A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D }$$ in $\mathcal{C}$. \end{example} \begin{example} \label{example-sheaf-product-pointwise-algebraic-structure} Let $F : \mathcal{C} \to \textit{Sets}$ be a type of algebraic structures. Let $X$ be a topological space. Suppose that for every $x \in X$ we are given an object $A_x \in \text{ob}(\mathcal{C})$. Consider the presheaf $\Pi$ with values in $\mathcal{C}$ on $X$ defined by the rule $\Pi(U) = \prod_{x \in U} A_x$ (with obvious restriction mappings). Note that the associated presheaf of sets $U \mapsto F(\Pi(U)) = \prod_{x \in U} F(A_x)$ is a sheaf by Example \ref{example-sheaf-product-pointwise}. Hence $\Pi$ is a sheaf of algebraic structures of type $(\mathcal{C} , F)$. This gives many examples of sheaves of abelian groups, groups, rings, etc. \end{example} \section{Exactness and points} \label{section-exactness-points} \noindent In any category we have the notion of epimorphism, monomorphism, isomorphism, etc. \begin{lemma} \label{lemma-points-exactness} Let $X$ be a topological space. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of sheaves of sets on $X$. \begin{enumerate} \item The map $\varphi$ is a monomorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ is injective. \item The map $\varphi$ is an epimorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ is surjective. \item The map $\varphi$ is an isomorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ is bijective. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \noindent It follows that in the category of sheaves of sets the notions epimorphism and monomorphism can be described as follows. \begin{definition} \label{definition-injective-surjective} Let $X$ be a topological space. \begin{enumerate} \item A presheaf $\mathcal{F}$ is called a {\it subpresheaf} of a presheaf $\mathcal{G}$ if $\mathcal{F}(U) \subset \mathcal{G}(U)$ for all open $U \subset X$ such that the restriction maps of $\mathcal{G}$ induce the restriction maps of $\mathcal{F}$. If $\mathcal{F}$ and $\mathcal{G}$ are sheaves, then $\mathcal{F}$ is called a {\it subsheaf} of $\mathcal{G}$. We sometimes indicate this by the notation $\mathcal{F} \subset \mathcal{G}$. \item A morphism of presheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it injective} if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective for all $U$ open in $X$. \item A morphism of presheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it surjective} if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is surjective for all $U$ open in $X$. \item A morphism of sheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it injective} if and only if $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective for all $U$ open in $X$. \item A morphism of sheaves of sets $\varphi : \mathcal{F} \to \mathcal{G}$ on $X$ is called {\it surjective} if and only if for every open $U$ of $X$ and every section $s$ of $\mathcal{G}(U)$ there exists an open covering $U = \bigcup U_i$ such that $s|_{U_i}$ is in the image of $\mathcal{F}(U_i) \to \mathcal{G}(U)$ for all $i$. \end{enumerate} \end{definition} \begin{lemma} \label{lemma-characterize-epi-mono} Let $X$ be a topological space. \begin{enumerate} \item Epimorphisms (resp.\ monomorphisms) in the category of presheaves are exactly the surjective (resp.\ injective) maps of presheaves. \item Epimorphisms (resp.\ monomorphisms) in the category of sheaves are exactly the surjective (resp.\ injective) maps of sheaves, and are exactly those maps with are surjective (resp.\ injective) on all the stalks. \item The sheafification of a surjective (resp.\ injective) morphism of presheaves of sets is surjective (resp.\ injective). \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-check-homomorphism-stalks} let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $\mathcal{F}$, $\mathcal{G}$ are sheaves on $X$ with values in $\mathcal{C}$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of the underlying sheaves of sets. If for all points $x \in X$ the map $\mathcal{F}_x \to \mathcal{G}_x$ is a morphism of algebraic structures, then $\varphi$ is a morphism of sheaves of algebraic structures. \end{lemma} \begin{proof} Let $U$ be an open subset of $X$. Consider the diagram of (underlying) sets $$\xymatrix{ \mathcal{F}(U) \ar[r] \ar[d] & \prod_{x \in U} \mathcal{F}_x \ar[d] \\ \mathcal{G}(U) \ar[r] & \prod_{x \in U} \mathcal{G}_x }$$ By assumption, and previous results, all but the left vertical arrow are morphisms of algebraic structures. In addition the bottom horizontal arrow is injective, see Lemma \ref{lemma-sheaf-subset-stalks}. Hence we conclude by Lemma \ref{lemma-image-contained-in}, see also Example \ref{example-application-lemma-image-contained-in} \end{proof} \noindent Short exact sequences of abelian sheaves, etc will be discussed in the chapter on sheaves of modules. See Modules, Section \ref{modules-section-kernels}. \section{Sheafification} \label{section-sheafification} \noindent In this section we explain how to get the sheafification of a presheaf on a topological space. We will use stalks to describe the sheafification in this case. This is different from the general procedure described in Sites, Section \ref{sites-section-sheafification}, and perhaps somewhat easier to understand. \medskip\noindent The basic construction is the following. Let $\mathcal{F}$ be a presheaf of sets on a topological space $X$. For every open $U \subset X$ we define $$\mathcal{F}^{\#}(U) = \{ (s_u) \in \prod\nolimits_{u \in U} \mathcal{F}_u \text{ such that }(*) \}$$ where $(*)$ is the property: \begin{itemize} \item[$(*)$] For every $u \in U$, there exists an open neighbourhood $u \in V \subset U$, and a section $\sigma \in \mathcal{F}(V)$ such that for all $v \in V$ we have $s_v = (V, \sigma)$ in $\mathcal{F}_v$. \end{itemize} Note that $(*)$ is a condition for each $u \in U$, and that given $u \in U$ the truth of this condition depends only on the values $s_v$ for $v$ in any open neighbourhood of $u$. Thus it is clear that, if $V \subset U \subset X$ are open, the projection maps $$\prod\nolimits_{u \in U} \mathcal{F}_u \longrightarrow \prod\nolimits_{v \in V} \mathcal{F}_v$$ maps elements of $\mathcal{F}^{\#}(U)$ into $\mathcal{F}^{\#}(V)$. In other words, we get the structure of a presheaf of sets on $\mathcal{F}^{\#}$. \medskip\noindent Furthermore, the map $\mathcal{F}(U) \to \prod_{u \in U} \mathcal{F}_u$ described in Section \ref{section-stalks} clearly has image in $\mathcal{F}^{\#}(U)$. In addition, if $V \subset U \subset X$ are open then we have the following commutative diagram $$\xymatrix{ \mathcal{F}(U) \ar[r] \ar[d] & \mathcal{F}^{\#}(U) \ar[r] \ar[d] & \prod_{u\in U} \mathcal{F}_u \ar[d] \\ \mathcal{F}(V) \ar[r] & \mathcal{F}^{\#}(V) \ar[r] & \prod_{v\in V} \mathcal{F}_v }$$ where the vertical maps are induced from the restriction mappings. Thus we see that there is a canonical morphism of presheaves $\mathcal{F} \to \mathcal{F}^{\#}$. \medskip\noindent In Example \ref{example-sheaf-product-pointwise} we saw that the rule $\Pi(\mathcal{F}) : U \mapsto \prod_{u\in U} \mathcal{F}_u$ is a sheaf, with obvious restriction mappings. And by construction $\mathcal{F}^{\#}$ is a subpresheaf of this. In other words, we have morphisms of presheaves $$\mathcal{F} \to \mathcal{F}^\# \to \Pi(\mathcal{F}).$$ In addition the rule that associates to $\mathcal{F}$ the sequence above is clearly functorial in the presheaf $\mathcal{F}$. This notation will be used in the proofs of the lemmas below. \begin{lemma} \label{lemma-sheafification-sheaf} The presheaf $\mathcal{F}^{\#}$ is a sheaf. \end{lemma} \begin{proof} It is probably better for the reader to find their own explanation of this than to read the proof here. In fact the lemma is true for the same reason as why the presheaf of continuous function is a sheaf, see Example \ref{example-basic-continuous-maps} (and this analogy can be made precise using the espace \'etal\'e''). \medskip\noindent Anyway, let $U = \bigcup U_i$ be an open covering. Suppose that $s_i = (s_{i, u})_{u \in U_i} \in \mathcal{F}^{\#}(U_i)$ such that $s_i$ and $s_j$ agree over $U_i \cap U_j$. Because $\Pi(\mathcal{F})$ is a sheaf, we find an element $s = (s_u)_{u\in U}$ in $\prod_{u\in U} \mathcal{F}_u$ restricting to $s_i$ on $U_i$. We have to check property $(*)$. Pick $u \in U$. Then $u \in U_i$ for some $i$. Hence by $(*)$ for $s_i$, there exists a $V$ open, $u \in V \subset U_i$ and a $\sigma \in \mathcal{F}(V)$ such that $s_{i, v} = (V, \sigma)$ in $\mathcal{F}_v$ for all $v \in V$. Since $s_{i, v} = s_v$ we get $(*)$ for $s$. \end{proof} \begin{lemma} \label{lemma-stalk-sheafification} Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets on $X$. Let $x \in X$. Then $\mathcal{F}_x = \mathcal{F}^\#_x$. \end{lemma} \begin{proof} The map $\mathcal{F}_x \to \mathcal{F}^\#_x$ is injective, since already the map $\mathcal{F}_x \to \Pi(\mathcal{F})_x$ is injective. Namely, there is a canonical map $\Pi(\mathcal{F})_x \to \mathcal{F}_x$ which is a left inverse to the map $\mathcal{F}_x \to \Pi(\mathcal{F})_x$, see Example \ref{example-sheaf-product-pointwise-stalk}. To show that it is surjective, suppose that $\overline{s} \in \mathcal{F}^\#_x$. We can find an open neighbourhood $U$ of $x$ such that $\overline{s}$ is the equivalence class of $(U, s)$ with $s \in \mathcal{F}^\#(U)$. By definition, this means there exists an open neighbourhood $V \subset U$ of $x$ and a section $\sigma \in \mathcal{F}(V)$ such that $s|_V$ is the image of $\sigma$ in $\Pi(\mathcal{F})(V)$. Clearly the class of $(V, \sigma)$ defines an element of $\mathcal{F}_x$ mapping to $\overline{s}$. \end{proof} \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} \begin{proof} Clearly, there is a commutative diagram $$\xymatrix{ \mathcal{F} \ar[r] \ar[d] & \mathcal{F}^\# \ar[r] \ar[d] & \Pi(\mathcal{F}) \ar[d] \\ \mathcal{G} \ar[r] & \mathcal{G}^\# \ar[r] & \Pi(\mathcal{G}) \\ }$$ So it suffices to prove that $\mathcal{G} = \mathcal{G}^\#$. To see this it suffices to prove, for every point $x \in X$ the map $\mathcal{G}_x \to \mathcal{G}^\#_x$ is bijective, by Lemma \ref{lemma-points-exactness}. And this is Lemma \ref{lemma-stalk-sheafification} above. \end{proof} \noindent This lemma really says that there is an adjoint pair of functors: $i : \Sh(X) \to \textit{PSh}(X)$ (inclusion) and $\# : \textit{PSh}(X) \to \Sh(X)$ (sheafification). The formula is that $$\Mor_{\textit{PSh}(X)}(\mathcal{F}, i(\mathcal{G})) = \Mor_{\Sh(X)}(\mathcal{F}^\#, \mathcal{G})$$ which says that sheafification is a left adjoint of the inclusion functor. See Categories, Section \ref{categories-section-adjoint}. \begin{example} \label{example-sheafify-constant} See Example \ref{example-stalk-constant-presheaf} for notation. The map $A_p \to \underline{A}$ induces a map $A_p^\# \to \underline{A}$. It is easy to see that this is an isomorphism. In words: The sheafification of the constant presheaf with value $A$ is the constant sheaf with value $A$. \end{example} \begin{lemma} \label{lemma-separated-presheaf-into-sheaf} Let $X$ be a topological space. A presheaf $\mathcal{F}$ is separated (see Definition \ref{definition-separated}) if and only if the canonical map $\mathcal{F} \to \mathcal{F}^\#$ is injective. \end{lemma} \begin{proof} This is clear from the construction of $\mathcal{F}^\#$ in this section. \end{proof} \section{Sheafification of abelian presheaves} \label{section-sheafify-abelian-presheaves} \noindent The following strange looking lemma is likely unnecessary, but very convenient to deal with sheafification of presheaves of algebraic structures. \begin{lemma} \label{lemma-diagram-fibre-product} Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets on $X$. Let $U \subset X$ be open. There is a canonical fibre product diagram $$\xymatrix{ \mathcal{F}^\#(U) \ar[d] \ar[r] & \Pi(\mathcal{F})(U) \ar[d] \\ \prod_{x \in U} \mathcal{F}_x \ar[r] & \prod_{x \in U} \Pi(\mathcal{F})_x }$$ where the maps are the following: \begin{enumerate} \item The left vertical map has components $\mathcal{F}^\#(U) \to \mathcal{F}^\#_x = \mathcal{F}_x$ where the equality is Lemma \ref{lemma-stalk-sheafification}. \item The top horizontal map comes from the map of presheaves $\mathcal{F} \to \Pi(\mathcal{F})$ described in Section \ref{section-sheafification}. \item The right vertical map has obvious component maps $\Pi(\mathcal{F})(U) \to \Pi(\mathcal{F})_x$. \item The bottom horizontal map has components $\mathcal{F}_x \to \Pi(\mathcal{F})_x$ which come from the map of presheaves $\mathcal{F} \to \Pi(\mathcal{F})$ described in Section \ref{section-sheafification}. \end{enumerate} \end{lemma} \begin{proof} It is clear that the diagram commutes. We have to show it is a fibre product diagram. The bottom horizontal arrow is injective since all the maps $\mathcal{F}_x \to \Pi(\mathcal{F})_x$ are injective (see beginning proof of Lemma \ref{lemma-stalk-sheafification}). A section $s \in \Pi(\mathcal{F})(U)$ is in $\mathcal{F}^\#$ if and only if $(*)$ holds. But $(*)$ says that around every point the section $s$ comes from a section of $\mathcal{F}$. By definition of the stalk functors, this is equivalent to saying that the value of $s$ in every stalk $\Pi(\mathcal{F})_x$ comes from an element of the stalk $\mathcal{F}_x$. Hence the lemma. \end{proof} \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} \begin{proof} Recall the sheaf of sets $\Pi(\mathcal{F})$ defined in Section \ref{section-sheafification}. All the stalks $\mathcal{F}_x$ are abelian groups, see Lemma \ref{lemma-stalk-abelian-presheaf}. Hence $\Pi(\mathcal{F})$ is a sheaf of abelian groups by Example \ref{example-sheaf-product-pointwise-algebraic-structure}. Also, it is clear that the map $\mathcal{F} \to \Pi(\mathcal{F})$ is a morphism of abelian presheaves. If we show that condition $(*)$ of Section \ref{section-sheafification} defines a subgroup of $\Pi(\mathcal{F})(U)$ for all open subsets $U \subset X$, then $\mathcal{F}^\#$ canonically inherits the structure of abelian sheaf. This is quite easy to do by hand, and we leave it to the reader to find a good simple argument. The argument we use here, which generalizes to presheaves of algebraic structures is the following: Lemma \ref{lemma-diagram-fibre-product} show that $\mathcal{F}^\#(U)$ is the fibre product of a diagram of abelian groups. Thus $\mathcal{F}^\#$ is an abelian subgroup as desired. \medskip\noindent Note that at this point $\mathcal{F}^\#_x$ is an abelian group by Lemma \ref{lemma-stalk-abelian-presheaf} and that $\mathcal{F}_x \to \mathcal{F}^\#_x$ is a bijection (Lemma \ref{lemma-stalk-sheafification}) and a homomorphism of abelian groups. Hence $\mathcal{F}_x \to \mathcal{F}^\#_x$ is an isomorphism of abelian groups. This will be used below without further mention. \medskip\noindent To prove the adjointness property we use the adjointness property of sheafification of presheaves of sets. For example if $\psi : \mathcal{F} \to i(\mathcal{G})$ is morphism of presheaves then we obtain a morphism of sheaves $\psi' : \mathcal{F}^\# \to \mathcal{G}$. What we have to do is to check that this is a morphism of abelian sheaves. We may do this for example by noting that it is true on stalks, by Lemma \ref{lemma-stalk-sheafification}, and then using Lemma \ref{lemma-check-homomorphism-stalks} above. \end{proof} \section{Sheafification of presheaves of algebraic structures} \label{section-sheafification-presheaves-structures} \begin{lemma} \label{lemma-sheafify-presheaf-structures} Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$ on $X$. Then there exists a sheaf $\mathcal{F}^\#$ with values in $\mathcal{C}$ and a morphism $\mathcal{F} \to \mathcal{F}^\#$ of presheaves with values in $\mathcal{C}$ with the following properties: \begin{enumerate} \item The map $\mathcal{F} \to \mathcal{F}^\#$ identifies the underlying sheaf of sets of $\mathcal{F}^\#$ with the sheafification of the underlying presheaf of sets of $\mathcal{F}$. \item For any morphism $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf with values in $\mathcal{C}$ there exists a unique factorization $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$. \end{enumerate} \end{lemma} \begin{proof} The proof is the same as the proof of Lemma \ref{lemma-sheafify-abelian-presheaf}, with repeated application of Lemma \ref{lemma-image-contained-in} (see also Example \ref{example-application-lemma-image-contained-in}). The main idea however, is to define $\mathcal{F}^\#(U)$ as the fibre product in $\mathcal{C}$ of the diagram $$\xymatrix{ & \Pi(\mathcal{F})(U) \ar[d] \\ \prod_{x \in U} \mathcal{F}_x \ar[r] & \prod_{x \in U} \Pi(\mathcal{F})_x }$$ compare Lemma \ref{lemma-diagram-fibre-product}. \end{proof} \section{Sheafification of presheaves of modules} \label{section-sheafification-presheaves-modules} \begin{lemma} \label{lemma-sheafification-presheaf-modules} Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf $\mathcal{O}$-modules. Let $\mathcal{O}^\#$ be the sheafification of $\mathcal{O}$. Let $\mathcal{F}^\#$ be the sheafification of $\mathcal{F}$ as a presheaf of abelian groups. There exists a map of sheaves of sets $$\mathcal{O}^\# \times \mathcal{F}^\# \longrightarrow \mathcal{F}^\#$$ which makes the diagram $$\xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{O}^\# \times \mathcal{F}^\# \ar[r] & \mathcal{F}^\# }$$ commute and which makes $\mathcal{F}^\#$ into a sheaf of $\mathcal{O}^\#$-modules. In addition, if $\mathcal{G}$ is a sheaf of $\mathcal{O}^\#$-modules, then any morphism of presheaves of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ (into the restriction of $\mathcal{G}$ to a $\mathcal{O}$-module) factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ where $\mathcal{F}^\# \to \mathcal{G}$ is a morphism of $\mathcal{O}^\#$-modules. \end{lemma} \begin{proof} Omitted. \end{proof} \noindent This actually means that the functor $i : \textit{Mod}(\mathcal{O}^\#) \to \textit{PMod}(\mathcal{O})$ (combining restriction and including sheaves into presheaves) and the sheafification functor of the lemma ${}^\# : \textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}^\#)$ are adjoint. In a formula $$\Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \Mor_{\textit{Mod}(\mathcal{O}^\#)}(\mathcal{F}^\#, \mathcal{G})$$ \medskip\noindent Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a morphism of sheaves of rings on $X$. In Section \ref{section-presheaves-modules} we defined a restriction functor and a change of rings functor on presheaves of modules associated to this situation. \medskip\noindent If $\mathcal{F}$ is a sheaf of $\mathcal{O}_2$-modules then the restriction $\mathcal{F}_{\mathcal{O}_1}$ of $\mathcal{F}$ is clearly a sheaf of $\mathcal{O}_1$-modules. We obtain the restriction functor $$\textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Mod}(\mathcal{O}_1)$$ \medskip\noindent On the other hand, given a sheaf of $\mathcal{O}_1$-modules $\mathcal{G}$ the presheaf of $\mathcal{O}_2$-modules $\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}$ is in general not a sheaf. Hence we define the {\it tensor product sheaf} $\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}$ by the formula $$\mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G} = (\mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G})^\#$$ as the sheafification of our construction for presheaves. We obtain the {\it change of rings} functor $$\textit{Mod}(\mathcal{O}_1) \longrightarrow \textit{Mod}(\mathcal{O}_2)$$ \begin{lemma} \label{lemma-adjointness-tensor-restrict} With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection $$\Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}, \mathcal{F} )$$ In other words, the restriction and change of rings functors are adjoint to each other. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-adjointness-tensor-restrict-presheaves} and the fact that $\Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} )$ because $\mathcal{F}$ is a sheaf. \end{proof} \begin{lemma} \label{lemma-stalk-tensor-sheaf-modules} Let $X$ be a topological space. Let $\mathcal{O} \to \mathcal{O}'$ be a morphism of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf $\mathcal{O}$-modules. Let $x \in X$. We have $$\mathcal{F}_x \otimes_{\mathcal{O}_x} \mathcal{O}'_x = (\mathcal{F} \otimes_\mathcal{O} \mathcal{O}')_x$$ as $\mathcal{O}'_x$-modules. \end{lemma} \begin{proof} Follows directly from Lemma \ref{lemma-stalk-tensor-presheaf-modules} and the fact that taking stalks commutes with sheafification. \end{proof} \section{Continuous maps and sheaves} \label{section-presheaves-functorial} \noindent Let $f : X \to Y$ be a continuous map of topological spaces. We will define the pushforward and pullback functors for presheaves and sheaves. \medskip\noindent Let $\mathcal{F}$ be a presheaf of sets on $X$. We define the {\it pushforward} of $\mathcal{F}$ by the rule $$f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}(V))$$ for any open $V \subset Y$. Given $V_1 \subset V_2 \subset Y$ open the restriction map is given by the commutativity of the diagram $$\xymatrix{ f_*\mathcal{F}(V_2) \ar[d] \ar@{=}[r] & \mathcal{F}(f^{-1}(V_2)) \ar[d]^{\text{restriction for }\mathcal{F}} \\ f_*\mathcal{F}(V_1) \ar@{=}[r] & \mathcal{F}(f^{-1}(V_1)) }$$ It is clear that this defines a presheaf of sets. The construction is clearly functorial in the presheaf $\mathcal{F}$ and hence we obtain a functor $$f_* : \textit{PSh}(X) \longrightarrow \textit{PSh}(Y).$$ \begin{lemma} \label{lemma-pushforward-sheaf} Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$. Then $f_*\mathcal{F}$ is a sheaf on $Y$. \end{lemma} \begin{proof} This immediately follows from the fact that if $V = \bigcup V_j$ is an open covering in $Y$, then $f^{-1}(V) = \bigcup f^{-1}(V_j)$ is an open covering in $X$. \end{proof} \noindent As a consequence we obtain a functor $$f_* : \Sh(X) \longrightarrow \Sh(Y).$$ This is compatible with composition in the following strong sense. \begin{lemma} \label{lemma-pushforward-composition} Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps of topological spaces. The functors $(g \circ f)_*$ and $g_* \circ f_*$ are equal (on both presheaves and sheaves of sets). \end{lemma} \begin{proof} This is because $(g \circ f)_*\mathcal{F}(W) = \mathcal{F}((g \circ f)^{-1}W)$ and $(g_* \circ f_*)\mathcal{F}(W) = \mathcal{F}(f^{-1} g^{-1} W)$ and $(g \circ f)^{-1}W = f^{-1} g^{-1} W$. \end{proof} \noindent Let $\mathcal{G}$ be a presheaf of sets on $Y$. The {\it pullback presheaf} $f_p\mathcal{G}$ of a given presheaf $\mathcal{G}$ is defined as the left adjoint of the pushforward $f_*$ on presheaves. In other words it should be a presheaf $f_p \mathcal{G}$ on $X$ such that $$\Mor_{\textit{PSh}(X)}(f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PSh}(Y)}(\mathcal{G}, f_*\mathcal{F}).$$ By the Yoneda lemma this determines the pullback uniquely. It turns out that it actually exists. \begin{lemma} \label{lemma-pullback-presheaves} Let $f : X \to Y$ be a continuous map. There exists a functor $f_p : \textit{PSh}(Y) \to \textit{PSh}(X)$ which is left adjoint to $f_*$. For a presheaf $\mathcal{G}$ it is determined by the rule $$f_p\mathcal{G}(U) = \colim_{f(U) \subset V} \mathcal{G}(V)$$ where the colimit is over the collection of open neighbourhoods $V$ of $f(U)$ in $Y$. The colimits are over directed partially ordered sets. (The restriction mappings of $f_p\mathcal{G}$ are explained in the proof.) \end{lemma} \begin{proof} The colimit is over the partially ordered set consisting of open subsets $V \subset Y$ which contain $f(U)$ with ordering by reverse inclusion. This is a directed partially ordered set, since if $V, V'$ are in it then so is $V \cap V'$. Furthermore, if $U_1 \subset U_2$, then every open neighbourhood of $f(U_2)$ is an open neighbourhood of $f(U_1)$. Hence the system defining $f_p\mathcal{G}(U_2)$ is a subsystem of the one defining $f_p\mathcal{G}(U_1)$ and we obtain a restriction map (for example by applying the generalities in Categories, Lemma \ref{categories-lemma-functorial-colimit}). \medskip\noindent Note that the construction of the colimit is clearly functorial in $\mathcal{G}$, and similarly for the restriction mappings. Hence we have defined $f_p$ as a functor. \medskip\noindent A small useful remark is that there exists a canonical map $\mathcal{G}(U) \to f_p\mathcal{G}(f^{-1}(U))$, because the system of open neighbourhoods of $f(f^{-1}(U))$ contains the element $U$. This is compatible with restriction mappings. In other words, there is a canonical map $i_\mathcal{G} : \mathcal{G} \to f_* f_p \mathcal{G}$. \medskip\noindent Let $\mathcal{F}$ be a presheaf of sets on $X$. Suppose that $\psi : f_p\mathcal{G} \to \mathcal{F}$ is a map of presheaves of sets. The corresponding map $\mathcal{G} \to f_*\mathcal{F}$ is the map $f_*\psi \circ i_\mathcal{G} : \mathcal{G} \to f_* f_p \mathcal{G} \to f_* \mathcal{F}$. \medskip\noindent Another small useful remark is that there exists a canonical map $c_\mathcal{F} : f_p f_* \mathcal{F} \to \mathcal{F}$. Namely, let $U \subset X$ open. For every open neighbourhood $V \supset f(U)$ in $Y$ there exists a map $f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}(V))\to \mathcal{F}(U)$, namely the restriction map on $\mathcal{F}$. And this is compatible with the restriction mappings between values of $\mathcal{F}$ on $f^{-1}$ of varying opens containing $f(U)$. Thus we obtain a canonical map $f_p f_* \mathcal{F}(U) \to \mathcal{F}(U)$. Another trivial verification shows that these maps are compatible with restriction maps and define a map $c_\mathcal{F}$ of presheaves of sets. \medskip\noindent Suppose that $\varphi : \mathcal{G} \to f_*\mathcal{F}$ is a map of presheaves of sets. Consider $f_p\varphi : f_p \mathcal{G} \to f_p f_* \mathcal{F}$. Postcomposing with $c_\mathcal{F}$ gives the desired map $c_\mathcal{F} \circ f_p\varphi : f_p\mathcal{G} \to \mathcal{F}$. We omit the verification that this construction is inverse to the construction in the other direction given above. \end{proof} \begin{lemma} \label{lemma-stalk-pullback-presheaf} Let $f : X \to Y$ be a continuous map. Let $x \in X$. Let $\mathcal{G}$ be a presheaf of sets on $Y$. There is a canonical bijection of stalks $(f_p\mathcal{G})_x = \mathcal{G}_{f(x)}$. \end{lemma} \begin{proof} This you can see as follows \begin{eqnarray*} (f_p\mathcal{G})_x & = & \colim_{x \in U} f_p\mathcal{G}(U) \\ & = & \colim_{x \in U} \colim_{f(U) \subset V} \mathcal{G}(V) \\ & = & \colim_{f(x) \in V} \mathcal{G}(V) \\ & = & \mathcal{G}_{f(x)} \end{eqnarray*} Here we have used Categories, Lemma \ref{categories-lemma-colimits-commute}, and the fact that any $V$ open in $Y$ containing $f(x)$ occurs in the third description above. Details omitted. \end{proof} \noindent Let $\mathcal{G}$ be a sheaf of sets on $Y$. The {\it pullback sheaf} $f^{-1}\mathcal{G}$ is defined by the formula $$f^{-1}\mathcal{G} = (f_p\mathcal{G})^\# .$$ Sheafification is a left adjoint to the inclusion of sheaves in presheaves, and $f_p$ is a left adjoint to $f_*$ on presheaves. As a formal consequence we obtain that $f^{-1}$ is a left adjoint of pushforward on sheaves. In other words, $$\Mor_{\Sh(X)}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\Sh(Y)}(\mathcal{G}, f_*\mathcal{F}).$$ The formal argument is given in the setting of abelian sheaves in the next section. \begin{lemma} \label{lemma-stalk-pullback} Let $x \in X$. Let $\mathcal{G}$ be a sheaf of sets on $Y$. There is a canonical bijection of stalks $(f^{-1}\mathcal{G})_x = \mathcal{G}_{f(x)}$. \end{lemma} \begin{proof} This is a combination of Lemmas \ref{lemma-stalk-sheafification} and \ref{lemma-stalk-pullback-presheaf}. \end{proof} \begin{lemma} \label{lemma-pullback-composition} Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps of topological spaces. The functors $(g \circ f)^{-1}$ and $f^{-1} \circ g^{-1}$ are canonically isomorphic. Similarly $(g \circ f)_p \cong f_p \circ g_p$ on presheaves. \end{lemma} \begin{proof} To see this use that adjoint functors are unique up to unique isomorphism, and Lemma \ref{lemma-pushforward-composition}. \end{proof} \begin{definition} \label{definition-f-map} Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$ and let $\mathcal{G}$ be a sheaf of sets on $Y$. An {\it $f$-map $\xi : \mathcal{G} \to \mathcal{F}$} is a collection of maps $\xi_V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}(V))$ indexed by open subsets $V \subset Y$ such that $$\xymatrix{ \mathcal{G}(V) \ar[r]_{\xi_V} \ar[d]_{\text{restriction of }\mathcal{G}} & \mathcal{F}(f^{-1}V) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[r]^{\xi_{V'}} & \mathcal{F}(f^{-1}V') }$$ commutes for all $V' \subset V \subset Y$ open. \end{definition} \begin{lemma} \label{lemma-f-map} Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$ and let $\mathcal{G}$ be a sheaf of sets on $Y$. There are canonical bijections between the following three sets: \begin{enumerate} \item The set of maps $\mathcal{G} \to f_*\mathcal{F}$. \item The set of maps $f^{-1}\mathcal{G} \to \mathcal{F}$. \item The set of $f$-maps $\xi : \mathcal{G} \to \mathcal{F}$. \end{enumerate} \end{lemma} \begin{proof} We leave the easy verification to the reader. \end{proof} \noindent It is sometimes convenient to think about $f$-maps instead of maps between sheaves either on $X$ or on $Y$. We define composition of $f$-maps as follows. \begin{definition} \label{definition-composition-f-maps} Suppose that $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. Suppose that $\mathcal{F}$ is a sheaf on $X$, $\mathcal{G}$ is a sheaf on $Y$, and $\mathcal{H}$ is a sheaf on $Z$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map. Let $\psi : \mathcal{H} \to \mathcal{G}$ be an $g$-map. The {\it composition of $\varphi$ and $\psi$} is the $(g \circ f)$-map $\varphi \circ \psi$ defined by the commutativity of the diagrams $$\xymatrix{ \mathcal{H}(W) \ar[rr]_{(\varphi \circ \psi)_W} \ar[rd]_{\psi_W} & & \mathcal{F}(f^{-1}g^{-1}W) \\ & \mathcal{G}(g^{-1}W) \ar[ru]_{\varphi_{g^{-1}W}} }$$ \end{definition} \noindent We leave it to the reader to verify that this works. Another way to think about this is to think of $\varphi \circ \psi$ as the composition $$\mathcal{H} \xrightarrow{\psi} g_*\mathcal{G} \xrightarrow{g_*\varphi} g_* f_* \mathcal{F} = (g \circ f)_* \mathcal{F}$$ Now, doesn't it seem that thinking about $f$-maps is somehow easier? \medskip\noindent Finally, given a continuous map $f : X \to Y$, and an $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$ there is a natural map on stalks $$\varphi_x : \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_x$$ for all $x \in X$. The image of a representative $(V, s)$ of an element in $\mathcal{G}_{f(x)}$ is mapped to the element in $\mathcal{F}_x$ with representative $(f^{-1}V, \varphi_V(s))$. We leave it to the reader to see that this is well defined. Another way to state it is that it is the unique map such that all diagrams $$\xymatrix{ \mathcal{F}(f^{-1}V) \ar[r] & \mathcal{F}_x \\ \mathcal{G}(V) \ar[r] \ar[u]^{\varphi_V} & \mathcal{G}_{f(x)} \ar[u]^{\varphi_x} }$$ (for $x \in V \subset Y$ open) commute. \begin{lemma} \label{lemma-compose-f-maps-stalks} Suppose that $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. Suppose that $\mathcal{F}$ is a sheaf on $X$, $\mathcal{G}$ is a sheaf on $Y$, and $\mathcal{H}$ is a sheaf on $Z$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map. Let $\psi : \mathcal{H} \to \mathcal{G}$ be an $g$-map. Let $x \in X$ be a point. The map on stalks $(\varphi \circ \psi)_x : \mathcal{H}_{g(f(x))} \to \mathcal{F}_x$ is the composition $$\mathcal{H}_{g(f(x))} \xrightarrow{\psi_{f(x)}} \mathcal{G}_{f(x)} \xrightarrow{\varphi_x} \mathcal{F}_x$$ \end{lemma} \begin{proof} Immediate from Definition \ref{definition-composition-f-maps} and the definition of the map on stalks above. \end{proof} \section{Continuous maps and abelian sheaves} \label{section-abelian-presheaves-functorial} \noindent Let $f : X \to Y$ be a continuous map. We claim there are functors \begin{eqnarray*} f_* : \textit{PAb}(X) & \longrightarrow & \textit{PAb}(Y) \\ f_* : \textit{Ab}(X) & \longrightarrow & \textit{Ab}(Y) \\ f_p : \textit{PAb}(Y) & \longrightarrow & \textit{PAb}(X) \\ f^{-1} : \textit{Ab}(Y) & \longrightarrow & \textit{Ab}(X) \end{eqnarray*} with similar properties to their counterparts in Section \ref{section-presheaves-functorial}. To see this we argue in the following way. \medskip\noindent Each of the functors will be constructed in the same way as the corresponding functor in Section \ref{section-presheaves-functorial}. This works because all the colimits in that section are directed colimits (but we will work through it below). \medskip\noindent First off, given an abelian presheaf $\mathcal{F}$ on $X$ and an abelian presheaf $\mathcal{G}$ on $Y$ we define \begin{eqnarray*} f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}(V)) \\ f_p\mathcal{G}(U) & = & \colim_{f(U) \subset V} \mathcal{G}(V) \end{eqnarray*} as abelian groups. The restriction mappings are the same as the restriction mappings for presheaves of sets (and they are all homomorphisms of abelian groups). \medskip\noindent The assignments $\mathcal{F} \mapsto f_*\mathcal{F}$ and $\mathcal{G} \to f_p\mathcal{G}$ are functors on the categories of presheaves of abelian groups. This is clear, as (for example) a map of abelian presheaves $\mathcal{G}_1 \to \mathcal{G}_2$ gives rise to a map of directed systems $\{\mathcal{G}_1(V)\}_{f(U) \subset V} \to \{\mathcal{G}_2(V)\}_{f(U) \subset V}$ all of whose maps are homomorphisms and hence gives rise to a homomorphism of abelian groups $f_p\mathcal{G}_1(U) \to f_p\mathcal{G}_2(U)$. \medskip\noindent The functors $f_*$ and $f_p$ are adjoint on the category of presheaves of abelian groups, i.e., we have $$\Mor_{\textit{PAb}(X)}(f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PAb}(Y)}(\mathcal{G}, f_*\mathcal{F}).$$ To prove this, note that the map $i_\mathcal{G} : \mathcal{G} \to f_* f_p\mathcal{G}$ from the proof of Lemma \ref{lemma-pullback-presheaves} is a map of abelian presheaves. Hence if $\psi : f_p\mathcal{G} \to \mathcal{F}$ is a map of abelian presheaves, then the corresponding map $\mathcal{G} \to f_*\mathcal{F}$ is the map $f_*\psi \circ i_\mathcal{G} : \mathcal{G} \to f_* f_p \mathcal{G} \to f_* \mathcal{F}$ is also a map of abelian presheaves. For the other direction we point out that the map $c_\mathcal{F} : f_p f_* \mathcal{F} \to \mathcal{F}$ from the proof of Lemma \ref{lemma-pullback-presheaves} is a map of abelian presheaves as well (since it is made out of restriction mappings of $\mathcal{F}$ which are all homomorphisms). Hence given a map of abelian presheaves $\varphi : \mathcal{G} \to f_*\mathcal{F}$ the map $c_\mathcal{F} \circ f_p\varphi : f_p\mathcal{G} \to \mathcal{F}$ is a map of abelian presheaves as well. Since these constructions $\psi \mapsto f_*\psi$ and $\varphi \mapsto c_\mathcal{F} \circ f_p\varphi$ are inverse to each other as constructions on maps of presheaves of sets we see they are also inverse to each other on maps of abelian presheaves. \medskip\noindent If $\mathcal{F}$ is an abelian sheaf on $Y$, then $f_*\mathcal{F}$ is an abelian sheaf on $X$. This is true because of the definition of an abelian sheaf and because this is true for sheaves of sets, see Lemma \ref{lemma-pushforward-sheaf}. This defines the functor $f_*$ on the category of abelian sheaves. \medskip\noindent We define $f^{-1}\mathcal{G} = (f_p\mathcal{G})^\#$ as before. Adjointness of $f_*$ and $f^{-1}$ follows formally as in the case of presheaves of sets. Here is the argument: \begin{eqnarray*} \Mor_{\textit{Ab}(X)}(f^{-1}\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{PAb}(X)}(f_p\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{PAb}(Y)}(\mathcal{G}, f_*\mathcal{F}) \\ & = & \Mor_{\textit{Ab}(Y)}(\mathcal{G}, f_*\mathcal{F}) \end{eqnarray*} \begin{lemma} \label{lemma-pullback-abelian-stalk} Let $f : X \to Y$ be a continuous map. \begin{enumerate} \item Let $\mathcal{G}$ be an abelian presheaf on $Y$. Let $x \in X$. The bijection $\mathcal{G}_{f(x)} \to (f_p\mathcal{G})_x$ of Lemma \ref{lemma-stalk-pullback-presheaf} is an isomorphism of abelian groups. \item Let $\mathcal{G}$ be an abelian sheaf on $Y$. Let $x \in X$. The bijection $\mathcal{G}_{f(x)} \to (f^{-1}\mathcal{G})_x$ of Lemma \ref{lemma-stalk-pullback} is an isomorphism of abelian groups. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \noindent Given a continuous map $f : X \to Y$ and sheaves of abelian groups $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$, the notion of an {\it $f$-map $\mathcal{G} \to \mathcal{F}$ of sheaves of abelian groups} makes sense. We can just define it exactly as in Definition \ref{definition-f-map} (replacing maps of sets with homomorphisms of abelian groups) or we can simply say that it is the same as a map of abelian sheaves $\mathcal{G} \to f_*\mathcal{F}$. We will use this notion freely in the following. The group of $f$-maps between $\mathcal{G}$ and $\mathcal{F}$ will be in canonical bijection with the groups $\Mor_{\textit{Ab}(X)}(f^{-1}\mathcal{G}, \mathcal{F})$ and $\Mor_{\textit{Ab}(Y)}(\mathcal{G}, f_*\mathcal{F})$. \medskip\noindent Composition of $f$-maps is defined in exactly the same manner as in the case of $f$-maps of sheaves of sets. In addition, given an $f$-map $\mathcal{G} \to \mathcal{F}$ as above, the induced maps on stalks $$\varphi_x : \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_x$$ are abelian group homomorphisms. \section{Continuous maps and sheaves of algebraic structures} \label{section-presheaves-structures-functorial} \noindent Let $(\mathcal{C}, F)$ be a type of algebraic structure. For a topological space $X$ let us introduce the notation: \begin{enumerate} \item $\textit{PSh}(X, \mathcal{C})$ will be the category of presheaves with values in $\mathcal{C}$. \item $\Sh(X, \mathcal{C})$ will be the category of sheaves with values in $\mathcal{C}$. \end{enumerate} Let $f : X \to Y$ be a continuous map of topological spaces. The same arguments as in the previous section show there are functors \begin{eqnarray*} f_* : \textit{PSh}(X, \mathcal{C}) & \longrightarrow & \textit{PSh}(Y, \mathcal{C}) \\ f_* : \Sh(X, \mathcal{C}) & \longrightarrow & \Sh(Y, \mathcal{C}) \\ f_p : \textit{PSh}(Y, \mathcal{C}) & \longrightarrow & \textit{PSh}(X, \mathcal{C}) \\ f^{-1} : \Sh(Y, \mathcal{C}) & \longrightarrow & \Sh(X, \mathcal{C}) \end{eqnarray*} constructed in the same manner and with the same properties as the functors constructed for abelian (pre)sheaves. In particular there are commutative diagrams $$\xymatrix{ \textit{PSh}(X, \mathcal{C}) \ar[r]^{f_*} \ar[d]^F & \textit{PSh}(Y, \mathcal{C}) \ar[d]^F & \Sh(X, \mathcal{C}) \ar[r]^{f_*} \ar[d]^F & \Sh(Y, \mathcal{C}) \ar[d]^F \\ \textit{PSh}(X) \ar[r]^{f_*} & \textit{PSh}(Y) & \Sh(X) \ar[r]^{f_*} & \Sh(Y) \\ \textit{PSh}(Y, \mathcal{C}) \ar[r]^{f_p} \ar[d]^F & \textit{PSh}(X, \mathcal{C}) \ar[d]^F & \Sh(Y, \mathcal{C}) \ar[r]^{f^{-1}} \ar[d]^F & \Sh(X, \mathcal{C}) \ar[d]^F \\ \textit{PSh}(Y) \ar[r]^{f_p} & \textit{PSh}(X) & \Sh(Y) \ar[r]^{f^{-1}} & \Sh(X) }$$ \medskip\noindent The main formulas to keep in mind are the following \begin{eqnarray*} f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}(V)) \\ f_p\mathcal{G}(U) & = & \colim_{f(U) \subset V} \mathcal{G}(V) \\ f^{-1}\mathcal{G} & = & (f_p\mathcal{G})^\# \\ (f_p\mathcal{G})_x & = & \mathcal{G}_{f(x)} \\ (f^{-1}\mathcal{G})_x & = & \mathcal{G}_{f(x)} \end{eqnarray*} Each of these formulas has the property that they hold in the category $\mathcal{C}$ and that upon taking underlying sets we get the corresponding formula for presheaves of sets. In addition we have the adjointness properties \begin{eqnarray*} \Mor_{\textit{PSh}(X, \mathcal{C})}(f_p\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{PSh}(Y, \mathcal{C})}(\mathcal{G}, f_*\mathcal{F}) \\ \Mor_{\Sh(X, \mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F}) & = & \Mor_{\Sh(Y, \mathcal{C})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} To prove these, the main step is to construct the maps $$i_\mathcal{G} : \mathcal{G} \longrightarrow f_*f_p\mathcal{G}$$ and $$c_\mathcal{F} : f_p f_* \mathcal{F} \longrightarrow \mathcal{F}$$ which occur in the proof of Lemma \ref{lemma-pullback-presheaves} as morphisms of presheaves with values in $\mathcal{C}$. This may be safely left to the reader since the constructions are exactly the same as in the case of presheaves of sets. \medskip\noindent Given a continuous map $f : X \to Y$ and sheaves of algebraic structures $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$, the notion of an {\it $f$-map $\mathcal{G} \to \mathcal{F}$ of sheaves of algebraic structures} makes sense. We can just define it exactly as in Definition \ref{definition-f-map} (replacing maps of sets with morphisms in $\mathcal{C}$) or we can simply say that it is the same as a map of sheaves of algebraic structures $\mathcal{G} \to f_*\mathcal{F}$. We will use this notion freely in the following. The set of $f$-maps between $\mathcal{G}$ and $\mathcal{F}$ will be in canonical bijection with the sets $\Mor_{\Sh(X, \mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F})$ and $\Mor_{\Sh(Y, \mathcal{C})}(\mathcal{G}, f_*\mathcal{F})$. \medskip\noindent Composition of $f$-maps is defined in exactly the same manner as in the case of $f$-maps of sheaves of sets. In addition, given an $f$-map $\mathcal{G} \to \mathcal{F}$ as above, the induced maps on stalks $$\varphi_x : \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_x$$ are homomorphisms of algebraic structures. \begin{lemma} \label{lemma-f-map-sets-algebraic-structures} Let $f : X \to Y$ be a continuous map of topological spaces. Suppose given sheaves of algebraic structures $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map of underlying sheaves of sets. If for every $V \subset Y$ open the map of sets $\varphi_V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$ is the effect of a morphism in $\mathcal{C}$ on underlying sets, then $\varphi$ comes from a unique $f$-morphism between sheaves of algebraic structures. \end{lemma} \begin{proof} Omitted. \end{proof} \section{Continuous maps and sheaves of modules} \label{section-presheaves-modules-functorial} \noindent The case of sheaves of modules is more complicated. The reason is that the natural setting for defining the pullback and pushforward functors, is the setting of ringed spaces, which we will define below. First we state a few obvious lemmas. \begin{lemma} \label{lemma-pushforward-presheaf-module} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$f_*\mathcal{O} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F}$$ which turns $f_*\mathcal{F}$ into a presheaf of $f_*\mathcal{O}$-modules. This construction is functorial in $\mathcal{F}$. \end{lemma} \begin{proof} Let $V \subset Y$ is open. We define the map of the lemma to be the map $$f_*\mathcal{O}(V) \times f_*\mathcal{F}(V) = \mathcal{O}(f^{-1}V) \times \mathcal{F}(f^{-1}V) \to \mathcal{F}(f^{-1}V) = f_*\mathcal{F}(V).$$ Here the arrow in the middle is the multiplication map on $X$. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of $f_*\mathcal{O}$-module on $f_*\mathcal{F}$. \end{proof} \begin{lemma} \label{lemma-pullback-presheaf-module} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $Y$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$f_p\mathcal{O} \times f_p\mathcal{G} \longrightarrow f_p\mathcal{G}$$ which turns $f_p\mathcal{G}$ into a presheaf of $f_p\mathcal{O}$-modules. This construction is functorial in $\mathcal{G}$. \end{lemma} \begin{proof} Let $U \subset X$ is open. We define the map of the lemma to be the map \begin{eqnarray*} f_p\mathcal{O}(U) \times f_p\mathcal{G}(U) & = & \colim_{f(U) \subset V} \mathcal{O}(V) \times \colim_{f(U) \subset V} \mathcal{G}(V) \\ & = & \colim_{f(U) \subset V} (\mathcal{O}(V)\times \mathcal{G}(V)) \\ & \to & \colim_{f(U) \subset V} \mathcal{G}(V) \\ & = & f_p\mathcal{G}(U). \end{eqnarray*} Here the arrow in the middle is the multiplication map on $Y$. The second equality holds because directed colimits commute with finite limits, see Categories, Lemma \ref{categories-lemma-directed-commutes}. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of $f_p\mathcal{O}$-module on $f_p\mathcal{G}$. \end{proof} \noindent Let $f : X \to Y$ be a continuous map. Let $\mathcal{O}_X$ be a presheaf of rings on $X$ and let $\mathcal{O}_Y$ be a presheaf of rings on $Y$. So at the moment we have defined functors \begin{eqnarray*} f_* : \textit{PMod}(\mathcal{O}_X) & \longrightarrow & \textit{PMod}(f_*\mathcal{O}_X) \\ f_p : \textit{PMod}(\mathcal{O}_Y) & \longrightarrow & \textit{PMod}(f_p\mathcal{O}_Y) \end{eqnarray*} These satisfy some compatibilities as follows. \begin{lemma} \label{lemma-adjoint-push-pull-presheaves-modules} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $Y$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a presheaf of $f_p\mathcal{O}$-modules. Then $$\Mor_{\textit{PMod}(f_p\mathcal{O})}(f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).$$ Here we use Lemmas \ref{lemma-pullback-presheaf-module} and \ref{lemma-pushforward-presheaf-module}, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module via the map $i_\mathcal{O} : \mathcal{O} \to f_*f_p\mathcal{O}$ (defined first in the proof of Lemma \ref{lemma-pullback-presheaves}). \end{lemma} \begin{proof} Note that we have $$\Mor_{\textit{PAb}(X)}(f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PAb}(Y)}(\mathcal{G}, f_*\mathcal{F}).$$ according to Section \ref{section-abelian-presheaves-functorial}. So what we have to prove is that under this correspondence, the subsets of module maps correspond. In addition, the correspondence is determined by the rule $$(\psi : f_p\mathcal{G} \to \mathcal{F}) \longmapsto (f_*\psi \circ i_\mathcal{G} : \mathcal{G} \to f_* \mathcal{F})$$ and in the other direction by the rule $$(\varphi : \mathcal{G} \to f_* \mathcal{F}) \longmapsto (c_\mathcal{F} \circ f_p\varphi : f_p\mathcal{G} \to \mathcal{F})$$ where $i_\mathcal{G}$ and $c_\mathcal{F}$ are as in Section \ref{section-abelian-presheaves-functorial}. Hence, using the functoriality of $f_*$ and $f_p$ we see that it suffices to check that the maps $i_\mathcal{G} : \mathcal{G} \to f_* f_p \mathcal{G}$ and $c_\mathcal{F} : f_p f_* \mathcal{F} \to \mathcal{F}$ are compatible with module structures, which we leave to the reader. \end{proof} \begin{lemma} \label{lemma-adjoint-pull-push-presheaves-modules} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a presheaf of $f_*\mathcal{O}$-modules. Then $$\Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{O} \otimes_{p, f_pf_*\mathcal{O}} f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).$$ Here we use Lemmas \ref{lemma-pullback-presheaf-module} and \ref{lemma-pushforward-presheaf-module}, and we use the map $c_\mathcal{O} : f_pf_*\mathcal{O} \to \mathcal{O}$ in the definition of the tensor product. \end{lemma} \begin{proof} This follows from the equalities \begin{eqnarray*} \Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{O} \otimes_{p, f_pf_*\mathcal{O}} f_p\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{PMod}(f_pf_*\mathcal{O})}( f_p\mathcal{G}, \mathcal{F}_{f_pf_*\mathcal{O}}) \\ & = & \Mor_{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*(\mathcal{F}_{f_pf_*\mathcal{O}})) \\ & = & \Mor_{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} The first equality is Lemma \ref{lemma-adjointness-tensor-restrict-presheaves}. The second equality is Lemma \ref{lemma-adjoint-push-pull-presheaves-modules}. The third equality is given by the equality $f_*(\mathcal{F}_{f_pf_*\mathcal{O}}) = f_*\mathcal{F}$ of abelian sheaves which is $f_*\mathcal{O}$-linear. Namely, $\text{id}_{f_*\mathcal{O}}$ corresponds to $c_\mathcal{O}$ under the adjunction described in the proof of Lemma \ref{lemma-pullback-presheaves} and thus $\text{id}_{f_*\mathcal{O}} = f_*c_\mathcal{O} \circ i_{f_*\mathcal{O}}$. \end{proof} \begin{lemma} \label{lemma-pushforward-module} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. The pushforward $f_*\mathcal{F}$, as defined in Lemma \ref{lemma-pushforward-presheaf-module} is a sheaf of $f_*\mathcal{O}$-modules. \end{lemma} \begin{proof} Obvious from the definition and Lemma \ref{lemma-pushforward-sheaf}. \end{proof} \begin{lemma} \label{lemma-pullback-module} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G}$$ which turns $f^{-1}\mathcal{G}$ into a sheaf of $f^{-1}\mathcal{O}$-modules. \end{lemma} \begin{proof} Recall that $f^{-1}$ is defined as the composition of the functor $f_p$ and sheafification. Thus the lemma is a combination of Lemma \ref{lemma-pullback-presheaf-module} and Lemma \ref{lemma-sheafification-presheaf-modules}. \end{proof} \noindent Let $f : X \to Y$ be a continuous map. Let $\mathcal{O}_X$ be a sheaf of rings on $X$ and let $\mathcal{O}_Y$ be a sheaf of rings on $Y$. So now we have defined functors \begin{eqnarray*} f_* : \textit{Mod}(\mathcal{O}_X) & \longrightarrow & \textit{Mod}(f_*\mathcal{O}_X) \\ f^{-1} : \textit{Mod}(\mathcal{O}_Y) & \longrightarrow & \textit{Mod}(f^{-1}\mathcal{O}_Y) \end{eqnarray*} These satisfy some compatibilities as follows. \begin{lemma} \label{lemma-adjoint-push-pull-modules} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$-modules. Then $$\Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).$$ Here we use Lemmas \ref{lemma-pullback-module} and \ref{lemma-pushforward-module}, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module by restriction via $\mathcal{O} \to f_*f^{-1}\mathcal{O}$. \end{lemma} \begin{proof} Argue by the equalities \begin{eqnarray*} \Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(f_p\mathcal{O})}(f_p\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} where the second is Lemmas \ref{lemma-adjoint-push-pull-presheaves-modules} and the first is by Lemma \ref{lemma-sheafification-presheaf-modules}. \end{proof} \begin{lemma} \label{lemma-adjoint-pull-push-modules} Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a sheaf of $f_*\mathcal{O}$-modules. Then $$\Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).$$ Here we use Lemmas \ref{lemma-pullback-module} and \ref{lemma-pushforward-module}, and we use the canonical map $f^{-1}f_*\mathcal{O} \to \mathcal{O}$ in the definition of the tensor product. \end{lemma} \begin{proof} This follows from the equalities \begin{eqnarray*} \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(f^{-1}f_*\mathcal{O})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}f_*\mathcal{O}}) \\ & = & \Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} which are a combination of Lemma \ref{lemma-adjointness-tensor-restrict} and \ref{lemma-adjoint-push-pull-modules}. \end{proof} \noindent Let $f : X \to Y$ be a continuous map. Let $\mathcal{O}_X$ be a (pre)sheaf of rings on $X$ and let $\mathcal{O}_Y$ be a (pre)sheaf of rings on $Y$. So at the moment we have defined functors \begin{eqnarray*} f_* : \textit{PMod}(\mathcal{O}_X) & \longrightarrow & \textit{PMod}(f_*\mathcal{O}_X) \\ f_* : \textit{Mod}(\mathcal{O}_X) & \longrightarrow & \textit{Mod}(f_*\mathcal{O}_X) \\ f_p : \textit{PMod}(\mathcal{O}_Y) & \longrightarrow & \textit{PMod}(f_p\mathcal{O}_Y) \\ f^{-1} : \textit{Mod}(\mathcal{O}_Y) & \longrightarrow & \textit{Mod}(f^{-1}\mathcal{O}_Y) \end{eqnarray*} Clearly, usually the pair of functors $(f_*, f^{-1})$ on sheaves of modules are not adjoint, because their target categories do not match. Namely, as we saw above, it works only if by some miracle the sheaves of rings $\mathcal{O}_X, \mathcal{O}_Y$ satisfy the relations $\mathcal{O}_X = f^{-1}\mathcal{O}_Y$ and $\mathcal{O}_Y = f_*\mathcal{O}_X$. This is almost never true in practice. We interrupt the discussion to define the correct notion of morphism for which a suitable adjoint pair of functors on sheaves of modules exists. \section{Ringed spaces} \label{section-ringed-spaces} \noindent Let $X$ be a topological space and let $\mathcal{O}_X$ be a sheaf of rings on $X$. We are supposed to think of the sheaf of rings $\mathcal{O}_X$ as a sheaf of functions on $X$. And if $f : X \to Y$ is a suitable'' map, then by composition a function on $Y$ turns into a function on $X$. Thus there should be a natural $f$-map from $\mathcal{O}_Y$ to $\mathcal{O}_X$ See Definition \ref{definition-f-map}, and the remarks in previous sections for terminology. For a precise example, see Example \ref{example-continuous-map-ringed} below. Here is the relevant abstract definition. \begin{definition} \label{definition-ringed-space} A {\it ringed space} is a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$ on $X$. A {\it morphism of ringed spaces} $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a pair consisting of a continuous map $f : X \to Y$ and an $f$-map of sheaves of rings $f^\sharp : \mathcal{O}_Y \to \mathcal{O}_X$. \end{definition} \begin{example} \label{example-continuous-map-ringed} Let $f : X \to Y$ be a continuous map of topological spaces. Consider the sheaves of continuous real valued functions $\mathcal{C}^0_X$ on $X$ and $\mathcal{C}^0_Y$ on $Y$, see Example \ref{example-C0-sheaf-rings}. We claim that there is a natural $f$-map $f^\sharp : \mathcal{C}^0_Y \to \mathcal{C}^0_X$ associated to $f$. Namely, we simply define it by the rule \begin{eqnarray*} \mathcal{C}^0_Y(V) & \longrightarrow & \mathcal{C}^0_X(f^{-1}V) \\ h & \longmapsto & h \circ f \end{eqnarray*} Strictly speaking we should write $f^\sharp(h) = h \circ f|_{f^{-1}(V)}$. It is clear that this is a family of maps as in Definition \ref{definition-f-map} and compatible with the $\mathbf{R}$-algebra structures. Hence it is an $f$-map of sheaves of $\mathbf{R}$-algebras, see Lemma \ref{lemma-f-map-sets-algebraic-structures}. \medskip\noindent Of course there are lots of other situations where there is a canonical morphism of ringed spaces associated to a geometrical type of morphism. For example, if $M$, $N$ are $\mathcal{C}^\infty$-manifolds and $f : M \to N$ is a infinitely differentiable map, then $f$ induces a canonical morphism of ringed spaces $(M, \mathcal{C}_M^\infty) \to (N, \mathcal{C}^\infty_N)$. The construction (which is identical to the above) is left to the reader. \end{example} \noindent It may not be completely obvious how to compose morphisms of ringed spaces hence we spell it out here. \begin{definition} \label{definition-composition-maps-ringed-spaces} Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ and $(g, g^\sharp) : (Y, \mathcal{O}_Y) \to (Z, \mathcal{O}_Z)$ be morphisms of ringed spaces. Then we define the {\it composition of morphisms of ringed spaces} by the rule $$(g, g^\sharp) \circ (f, f^\sharp) = (g \circ f, f^\sharp \circ g^\sharp).$$ Here we use composition of $f$-maps defined in Definition \ref{definition-composition-f-maps}. \end{definition} \section{Morphisms of ringed spaces and modules} \label{section-ringed-spaces-functoriality-modules} \noindent We have now introduced enough notation so that we are able to define the pullback and pushforward of modules along a morphism of ringed spaces. \begin{definition} \label{definition-pushforward} Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We define the {\it pushforward} of $\mathcal{F}$ as the sheaf of $\mathcal{O}_Y$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$ of the module structure given in Lemma \ref{lemma-pushforward-module}. \item Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. We define the {\it pullback} $f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_X$-modules defined by the formula $$f^*\mathcal{G} = \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\mathcal{G}$$ where the ring map $f^{-1}\mathcal{O}_Y \to \mathcal{O}_X$ is the map corresponding to $f^\sharp$, and where the module structure is given by Lemma \ref{lemma-pullback-module}. \end{enumerate} \end{definition} \noindent Thus we have defined functors \begin{eqnarray*} f_* : \textit{Mod}(\mathcal{O}_X) & \longrightarrow & \textit{Mod}(\mathcal{O}_Y) \\ f^* : \textit{Mod}(\mathcal{O}_Y) & \longrightarrow & \textit{Mod}(\mathcal{O}_X) \end{eqnarray*} The final result on these functors is that they are indeed adjoint as expected. \begin{lemma} \label{lemma-adjoint-pullback-pushforward-modules} Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. There is a canonical bijection $$\Hom_{\mathcal{O}_X}(f^*\mathcal{G}, \mathcal{F}) = \Hom_{\mathcal{O}_Y}(\mathcal{G}, f_*\mathcal{F}).$$ In other words: the functor $f^*$ is the left adjoint to $f_*$. \end{lemma} \begin{proof} This follows from the work we did before: \begin{eqnarray*} \Hom_{\mathcal{O}_X}(f^*\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(\mathcal{O}_X)}( \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{Mod}(f^{-1}\mathcal{O}_Y)}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_Y}) \\ & = & \Hom_{\mathcal{O}_Y}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} Here we use Lemmas \ref{lemma-adjointness-tensor-restrict} and \ref{lemma-adjoint-push-pull-modules}. \end{proof} \begin{lemma} \label{lemma-push-pull-composition-modules} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. The functors $(g \circ f)_*$ and $g_* \circ f_*$ are equal. There is a canonical isomorphism of functors $(g \circ f)^* \cong f^* \circ g^*$. \end{lemma} \begin{proof} The result on pushforwards is a consequence of Lemma \ref{lemma-pushforward-composition} and our definitions. The result on pullbacks follows from this by the same argument as in the proof of Lemma \ref{lemma-pullback-composition}. \end{proof} \noindent Given a morphism of ringed spaces $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$, and a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$, a sheaf of $\mathcal{O}_Y$-modules $\mathcal{G}$ on $Y$, the notion of an {\it $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$ of sheaves of modules} makes sense. We can just define it as an $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$ of abelian sheaves such that for all open $V \subset Y$ the map $$\mathcal{G}(V) \longrightarrow \mathcal{F}(f^{-1}V)$$ is an $\mathcal{O}_Y(V)$-module map. Here we think of $\mathcal{F}(f^{-1}V)$ as an $\mathcal{O}_Y(V)$-module via the map $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V)$. The set of $f$-maps between $\mathcal{G}$ and $\mathcal{F}$ will be in canonical bijection with the sets $\Mor_{\textit{Mod}(\mathcal{O}_X)}(f^*\mathcal{G}, \mathcal{F})$ and $\Mor_{\textit{Mod}(\mathcal{O}_Y)}(\mathcal{G}, f_*\mathcal{F})$. See above. \medskip\noindent Composition of $f$-maps is defined in exactly the same manner as in the case of $f$-maps of sheaves of sets. In addition, given an $f$-map $\mathcal{G} \to \mathcal{F}$ as above, and $x \in X$ the induced map on stalks $$\varphi_x : \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_x$$ is an $\mathcal{O}_{Y, f(x)}$-module map where the $\mathcal{O}_{Y, f(x)}$-module structure on $\mathcal{F}_x$ comes from the $\mathcal{O}_{X, x}$-module structure via the map $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$. Here is a related lemma. \begin{lemma} \label{lemma-stalk-pullback-modules} Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $x \in X$. Then $$(f^*\mathcal{G})_x = \mathcal{G}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x}$$ as $\mathcal{O}_{X, x}$-modules where the tensor product on the right uses $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-stalk-tensor-sheaf-modules} and the identification of the stalks of pullback sheaves at $x$ with the corresponding stalks at $f(x)$. See the formulae in Section \ref{section-presheaves-structures-functorial} for example. \end{proof} \section{Skyscraper sheaves and stalks} \label{section-skyscraper-sheaves} \begin{definition} \label{definition-skyscraper-sheaf} Let $X$ be a topological space. \begin{enumerate} \item Let $x \in X$ be a point. Denote $i_x : \{x\} \to X$ the inclusion map. Let $A$ be a set and think of $A$ as a sheaf on the one point space $\{x\}$. We call $i_{x, *}A$ the {\it skyscraper sheaf at $x$ with value $A$}. \item If in (1) above $A$ is an abelian group then we think of $i_{x, *}A$ as a sheaf of abelian groups on $X$. \item If in (1) above $A$ is an algebraic structure then we think of $i_{x, *}A$ as a sheaf of algebraic structures. \item If $(X, \mathcal{O}_X)$ is a ringed space, then we think of $i_x : \{x\} \to X$ as a morphism of ringed spaces $(\{x\}, \mathcal{O}_{X, x}) \to (X, \mathcal{O}_X)$ and if $A$ is a $\mathcal{O}_{X, x}$-module, then we think of $i_{x, *}A$ as a sheaf of $\mathcal{O}_X$-modules. \item We say a sheaf of sets $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists an point $x$ of $X$ and a set $A$ such that $\mathcal{F} \cong i_{x, *}A$. \item We say a sheaf of abelian groups $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists an point $x$ of $X$ and an abelian group $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of abelian groups. \item We say a sheaf of algebraic structures $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists an point $x$ of $X$ and an algebraic structure $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of algebraic structures. \item If $(X, \mathcal{O}_X)$ is a ringed space and $\mathcal{F}$ is a sheaf of $\mathcal{O}_X$-modules, then we say $\mathcal{F}$ is a {\it skyscraper sheaf} if there exists a point $x \in X$ and a $\mathcal{O}_{X, x}$-module $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of $\mathcal{O}_X$-modules. \end{enumerate} \end{definition} \begin{lemma} \label{lemma-skyscraper-stalks} Let $X$ be a topological space, $x \in X$ a point, and $A$ a set. For any point $x' \in X$ the stalk of the skyscraper sheaf at $x$ with value $A$ at $x'$ is $$(i_{x, *}A)_{x'} = \left\{ \begin{matrix} A & \text{if} & x' \in \overline{\{x\}} \\ \{*\} & \text{if} & x' \not\in \overline{\{x\}} \end{matrix} \right.$$ A similar description holds for the case of abelian groups, algebraic structures and sheaves of modules. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-stalk-skyscraper-adjoint} Let $X$ be a topological space, and let $x \in X$ a point. The functors $\mathcal{F} \mapsto \mathcal{F}_x$ and $A \mapsto i_{x, *}A$ are adjoint. In a formula $$\Mor_{\textit{Sets}}(\mathcal{F}_x, A) = \Mor_{\Sh(X)}(\mathcal{F}, i_{x, *}A).$$ A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have $$\Hom_{\mathcal{O}_{X, x}}(\mathcal{F}_x, A) = \Hom_{\mathcal{O}_X}(\mathcal{F}, i_{x, *}A).$$ \end{lemma} \begin{proof} Omitted. Hint: The stalk functor can be seen as the pullback functor for the morphism $i_x : \{x\} \to X$. Then the adjointness follows from adjointness of $i_x^{-1}$ and $i_{x, *}$ (resp.\ $i_x^*$ and $i_{x, *}$ in the case of sheaves of modules). \end{proof} \section{Limits and colimits of presheaves} \label{section-limits-presheaves} \noindent Let $X$ be a topological space. Let $\mathcal{I} \to \textit{PSh}(X)$, $i \mapsto \mathcal{F}_i$ be a diagram. \begin{enumerate} \item Both $\lim_i \mathcal{F}_i$ and $\colim_i \mathcal{F}_i$ exist. \item For any open $U \subset X$ we have $$(\lim_i \mathcal{F}_i)(U) = \lim_i \mathcal{F}_i(U)$$ and $$(\colim_i \mathcal{F}_i)(U) = \colim_i \mathcal{F}_i(U).$$ \item Let $x \in X$ be a point. In general the stalk of $\lim_i \mathcal{F}_i$ at $x$ is not equal to the limit of the stalks. But if the diagram category is finite then it is the case. In other words, the stalk functor is left exact (see Categories, Definition \ref{categories-definition-exact}). \item Let $x \in X$. We always have $$(\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x}.$$ \end{enumerate} The proofs are all easy. \section{Limits and colimits of sheaves} \label{section-limits-sheaves} \noindent Let $X$ be a topological space. Let $\mathcal{I} \to \Sh(X)$, $i \mapsto \mathcal{F}_i$ be a diagram. \begin{enumerate} \item Both $\lim_i \mathcal{F}_i$ and $\colim_i \mathcal{F}_i$ exist. \item The inclusion functor $i : \Sh(X) \to \textit{PSh}(X)$ commutes with limits. In other words, we may compute the limit in the category of sheaves as the limit in the category of presheaves. In particular, for any open $U \subset X$ we have $$(\lim_i \mathcal{F}_i)(U) = \lim_i \mathcal{F}_i(U).$$ \item The inclusion functor $i : \Sh(X) \to \textit{PSh}(X)$ does not commute with colimits in general (not even with finite colimits -- think surjections). The colimit is computed as the sheafification of the colimit in the category of presheaves: $$\colim_i \mathcal{F}_i = \Big(U \mapsto \colim_i \mathcal{F}_i(U)\Big)^\#.$$ \item Let $x \in X$ be a point. In general the stalk of $\lim_i \mathcal{F}_i$ at $x$ is not equal to the limit of the stalks. But if the diagram category is finite then it is the case. In other words, the stalk functor is left exact. \item Let $x \in X$. We always have $$(\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x}.$$ \item The sheafification functor ${}^\# : \textit{PSh}(X) \to \Sh(X)$ commutes with all colimits, and with finite limits. But it does not commute with all limits. \end{enumerate} The proofs are all easy. Here is an example of what is true for directed colimits of sheaves. \begin{lemma} \label{lemma-directed-colimits-sections} Let $X$ be a topological space. Let $I$ be a directed set. Let $(\mathcal{F}_i, \varphi_{ii'})$ be a system of sheaves of sets over $I$, see Categories, Section \ref{categories-section-posets-limits}. Let $U \subset X$ be an open subset. Consider the canonical map $$\Psi : \colim_i \mathcal{F}_i(U) \longrightarrow \left(\colim_i \mathcal{F}_i\right)(U)$$ \begin{enumerate} \item If all the transition maps are injective then $\Psi$ is injective for any open $U$. \item If $U$ is quasi-compact, then $\Psi$ is injective. \item If $U$ is quasi-compact and all the transition maps are injective then $\Psi$ is an isomorphism. \item If $U$ has a cofinal system of open coverings $\mathcal{U} : U = \bigcup_{j\in J} U_j$ with $J$ finite and $U_j \cap U_{j'}$ quasi-compact for all $j, j' \in J$, then $\Psi$ is bijective. \end{enumerate} \end{lemma} \begin{proof} Assume all the transition maps are injective. In this case the presheaf $\mathcal{F}' : V \mapsto \colim_i \mathcal{F}_i(V)$ is separated (see Definition \ref{definition-separated}). By the discussion above we have $(\mathcal{F}')^\# = \colim_i \mathcal{F}_i$. By Lemma \ref{lemma-separated-presheaf-into-sheaf} we see that $\mathcal{F}' \to (\mathcal{F}')^\#$ is injective. This proves (1). \medskip\noindent Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_i(U)$ and $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on the left hand side which have the same image under $\Psi$. Since $U$ is quasi-compact this means there exists a finite open covering $U = \bigcup_{j = 1, \ldots, m} U_j$ and for each $j$ an index $i_j \in I$, $i_j \geq i$, $i_j \geq i'$ such that $\varphi_{ii_j}(s) = \varphi_{i'i_j}(s')$. Let $i''\in I$ be $\geq$ than all of the $i_j$. We conclude that $\varphi_{ii''}(s)$ and $\varphi_{i'i''}(s)$ agree on the opens $U_j$ for all $j$ and hence that $\varphi_{ii''}(s) = \varphi_{i'i''}(s)$. This proves (2). \medskip\noindent Assume $U$ is quasi-compact and all transition maps injective. Let $s$ be an element of the target of $\Psi$. Since $U$ is quasi-compact there exists a finite open covering $U = \bigcup_{j = 1, \ldots, m} U_j$, for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$ such that $s|_{U_j}$ comes from $s_j$ for all $j$. Pick $i \in I$ which is $\geq$ than all of the $i_j$. By (1) the sections $\varphi_{i_ji}(s_j)$ agree over the overlaps $U_j \cap U_{j'}$. Hence they glue to a section $s' \in \mathcal{F}_i(U)$ which maps to $s$ under $\Psi$. This proves (3). \medskip\noindent Assume the hypothesis of (4). Let $s$ be an element of the target of $\Psi$. By assumption there exists a finite open covering $U = \bigcup_{j = 1, \ldots, m} U_j$, with $U_j \cap U_{j'}$ quasi-compact for all $j, j' \in J$ and for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$ such that $s|_{U_j}$ is the image of $s_j$ for all $j$. Since $U_j \cap U_{j'}$ is quasi-compact we can apply (2) and we see that there exists an $i_{jj'} \in I$, $i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that $\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_j \cap U_{j'}$. Choose an index $i \in I$ wich is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue to a section of $\mathcal{F}_i$ over $U$. This section is mapped to the element $s$ as desired. \end{proof} \begin{example} \label{example-conditions-needed-colimit} Let $X = \{s_1, s_2, \xi_1, \xi_2, \xi_3, \ldots\}$ as a set. Declare a subset $U \subset X$ to be open if $s_1 \in U$ or $s_2 \in U$ implies $U$ contains all of the $\xi_i$. Let $U_n = \{\xi_n, \xi_{n + 1}, \ldots\}$, and let $j_n : U_n \to X$ be the inclusion map. Set $\mathcal{F}_n = j_{n, *}\underline{\mathbf{Z}}$. There are transition maps $\mathcal{F}_n \to \mathcal{F}_{n + 1}$. Let $\mathcal{F} = \colim \mathcal{F}_n$. Note that $\mathcal{F}_{n, \xi_m} = 0$ if $m < n$ because $\{\xi_m\}$ is an open subset of $X$ which misses $U_n$. Hence we see that $\mathcal{F}_{\xi_n} = 0$ for all $n$. On the other hand the stalk $\mathcal{F}_{s_i}$, $i = 1, 2$ is the colimit $$M = \colim_n \prod\nolimits_{m \geq n} \mathbf{Z}$$ which is not zero. We conclude that the sheaf $\mathcal{F}$ is the direct sum of the skyscraper sheaves with value $M$ at the closed points $s_1$ and $s_2$. Hence $\Gamma(X, \mathcal{F}) = M \oplus M$. On the other hand, the reader can verify that $\colim_n \Gamma(X, \mathcal{F}_n) = M$. Hence some condition is necessary in part (4) of Lemma \ref{lemma-directed-colimits-sections} above. \end{example} \noindent There is a version of the previous lemma dealing with sheaves on a diagram of spectral spaces. To state it we introduce some notation. Let $\mathcal{I}$ be a cofiltered index category. Let $i \mapsto X_i$ be a diagram of spectral spaces over $\mathcal{I}$ such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_a : X_j \to X_i$ is spectral. Set $X = \lim X_i$ and denote $p_i : X \to X_i$ the projection. \begin{lemma} \label{lemma-compute-pullback-to-limit} In the situation described above, let $i \in \Ob(\mathcal{I})$ and let $\mathcal{G}$ be a sheaf on $X_i$. For $U_i \subset X_i$ quasi-compact open we have $$p_i^{-1}\mathcal{G}(p_i^{-1}(U_i)) = \colim_{a : j \to i} f_a^{-1}\mathcal{G}(f_a^{-1}(U_i))$$ \end{lemma} \begin{proof} Let us prove the canonical map $\colim_{a : j \to i} f_a^{-1}\mathcal{G}(f_a^{-1}(U_i)) \to p_i^{-1}\mathcal{G}(p_i^{-1}(U_i))$ is injective. Let $s, s'$ be sections of $f_a^{-1}\mathcal{G}$ over $f_a^{-1}(U_i)$ for some $a : j \to i$. For $b : k \to j$ let $Z_k \subset f_{a \circ b}^{-1}(U_i)$ be the closed subset of points $x$ such that the image of $s$ and $s'$ in the stalk $(f_{a \circ b}^{-1}\mathcal{G})_x$ are different. If $Z_k$ is nonempty for all $b : k \to j$, then by Topology, Lemma \ref{topology-lemma-inverse-limit-spectral-spaces-nonempty} we see that $\lim_{b : k \to j} Z_k$ is nonempty too. Then for \$x \in \