Let \(\ts{\mcf_i}\) be a direct system of sheaves and morphisms on \(X\). Define the direct limit of the system, denoted \(\varinjlim \mcf_i\), to be the sheaf associated to the presheaf \(U \mapsto \varinjlim \mcf_i(U)\).
Show that this is a direct limit in the category of sheaves on \(X\): given a sheaf \(\mcg\) and a collection of morphisms \(\mcf_i \to \mcg\) compatible with the maps of the direct system, there exists a unique map \(\varinjlim \mcf_i \to \mcg\) such that for each \(i\) the original map \(\mcf_i \to \mcg\) is the composite \(\mcf_i \to \varinjlim \mcf_i \to \mcg\).
Solution
Let \[ \mcp(U)=\varinjlim_i\mcf_i(U) \] with restriction maps induced from those of the \(\mcf_i\). Thus \(\mcp\) is the sectionwise direct-limit presheaf, and by definition \[ \mcf:=\varinjlim_i\mcf_i=\mcp^+ \] is its associated sheaf. Write \[ \eta:\mcp\longrightarrow\mcp^+ \] for the sheafification map.
1. For every \(i\) there is a canonical sheaf morphism\[ \lambda_i:\mcf_i\longrightarrow\mcf. \]These morphisms are compatible with the transition maps of the direct system.
For every open \(U\subseteq X\), the direct-limit maps give \[ \mcf_i(U)\longrightarrow\varinjlim_j\mcf_j(U)=\mcp(U). \] These commute with restriction maps, so they define a presheaf morphism \[ \mcf_i\longrightarrow\mcp. \] Composing with \[ \eta:\mcp\longrightarrow\mcp^+=\mcf \] gives \(\lambda_i\). Compatibility follows sectionwise from compatibility of the canonical maps into a direct limit.
2. Let \(\mcg\) be a sheaf and suppose we are given compatible morphisms\[ \phi_i:\mcf_i\longrightarrow\mcg. \]There is a unique presheaf morphism\[ \phi:\mcp\longrightarrow\mcg \]whose composite with \(\mcf_i\to\mcp\) is \(\phi_i\) for every \(i\).
Fix an open set \(U\subseteq X\). The maps on sections \[ \phi_i(U):\mcf_i(U)\longrightarrow\mcg(U) \] are compatible with the direct system. By the universal property of the direct limit in the category of groups, modules, or sets under consideration, there is a unique map \[ \phi(U):\varinjlim_i\mcf_i(U)\longrightarrow\mcg(U) \] whose restriction to each \(\mcf_i(U)\) is \(\phi_i(U)\).
For an inclusion \(V\subseteq U\), both composites \[ \mcp(U)\rightrightarrows\mcg(V) \] obtained by restricting before or after \(\phi\) agree on the image of every \(\mcf_i(U)\), because each \(\phi_i\) is a morphism of sheaves. The direct-limit universal property therefore makes the two composites equal. Hence the maps \(\phi(U)\) assemble to a presheaf morphism \(\phi:\mcp\to\mcg\).
Uniqueness also holds open set by open set by the same direct-limit universal property.
3. The presheaf morphism \(\phi\) factors uniquely through the sheafification:\[ \boxed{ \begin{array}{ccc} \mcp&\xrightarrow{\eta}&\mcf=\mcp^+\\ &\searrow_{\phi}&\downarrow{\scriptstyle\Phi}\\ &&\mcg, \end{array} } \]for a unique sheaf morphism\[ \Phi:\mcf\longrightarrow\mcg. \]
Sheafification is left adjoint to the inclusion of sheaves into presheaves. Equivalently, for every presheaf \(\mcp\) and every sheaf \(\mcg\), composition with the sheafification map induces a bijection \[ \operatorname{Hom}_{\mathrm{Sh}(X)}(\mcp^+,\mcg) \xrightarrow{\sim} \operatorname{Hom}_{\mathrm{PSh}(X)}(\mcp,\mcg). \] Apply this universal property to the presheaf morphism \(\phi\) from 2.
4. The morphism \(\Phi\) satisfies\[ \Phi\circ\lambda_i=\phi_i \qquad \text{for every }i. \]
By 1, \(\lambda_i\) is the composite \[ \mcf_i\longrightarrow\mcp\xrightarrow{\eta}\mcf. \] By 3, \[ \Phi\circ\eta=\phi. \] Therefore \[ \Phi\circ\lambda_i =\phi\circ(\mcf_i\to\mcp) =\phi_i \] by the defining property of \(\phi\) in 2.
5. The morphism \(\Phi\) is the unique sheaf morphism with the property in
4.
Suppose \[ \Psi:\mcf\longrightarrow\mcg \] is another sheaf morphism satisfying \[ \Psi\circ\lambda_i=\phi_i \] for every \(i\).
Precompose with the sheafification map: \[ \Psi\circ\eta:\mcp\longrightarrow\mcg. \] For every \(i\), its composite with \(\mcf_i\to\mcp\) is \(\phi_i\). By uniqueness in 2, \[ \Psi\circ\eta=\phi=\Phi\circ\eta. \] The sheafification universal property in 3 then implies \[ \Psi=\Phi. \]
6. Hence\[ \boxed{ \mcf=\left(U\longmapsto\varinjlim_i\mcf_i(U)\right)^+ } \]with the maps \(\lambda_i\) is the direct limit of the system \(\{\mcf_i\}\) in the category of sheaves on \(X\).
Steps 4 and 5 establish exactly the existence and uniqueness demanded by the categorical universal property of the colimit.
7. Q.E.D.
Step 6 is the assertion of the exercise.