Godement's "Topologie Algebrique et Theorie des Faisceaux", Part II
I will quote at length from Godement to begin with. We will start by developing a cross product in sheaf cohomology, similar to the one present in singular cohomology.
Let \(X,Y\) be a pair of spaces. We will work with sheaves of Abelian groups over \(X,Y\) and \(X\times Y\); everything we are saying of course is easily adaptable to \(A\)-modules for a commutative ring \(A\). If \(\mathcal{F}\) is a sheaf of Abelian groups over \(X\) and \(\mathcal{G}\) is a sheaf of Abelian groups (henceforth simply "sheaf") over \(Y\), then it is possible to construct over \(X\times Y\) a "total tensor product sheaf" \(\mathcal{F}\hat{\otimes}\mathcal{G}\). (We use the hat to distinguish it from a tensor product of sheaves over a single space.) The definition of this sheaf is by the standard process: we will define a bundle \(\pi : B\to X\times Y\), which will turn out to be espace étalé of the sheaf \(\mathcal{F}\hat{\otimes}\mathcal{G}\). We construct \(B\) by defining the fiber over \(x,y\) to be \(\mathcal{F}_x \otimes\mathcal{G}_y\), and we confer upon this space the weakest topology such that for any pair of open sets \(U\subset X\), \(V\subset Y\), and for any sections \(f_1,\dots, f_n \in \mathcal{F}(U)\),\(g_1 ,\dots, g_n \in \mathcal{F}(V)\), the section \(\sum_i f_i\otimes g_i\) of \(\pi\) over \(U\times V\) which sends \((x,y)\) to \(\sum_i f_i(x)\otimes g_i(y)\) is continuous. We then define \(\mathcal{F}\hat{\otimes}\mathcal{G}\) as the sheaf of sections of the bundle \(\pi\).
In the case where \(X = Y\), then we can pull back the sheaf \(\mathcal{F}\hat{\otimes} \mathcal{G}\) along the diagonal \(\delta: X\to X\times X\) to give a tensor product of sheaves over \(X\); this sheaf, \( \delta^\ast (\mathcal{F}\hat{\otimes}\mathcal{G})\), we refer to simply as \(\mathcal{F}\otimes\mathcal{G}\) (without the hat.) This is what is more commonly known as the tensor product of sheaves.
It is clear that for each pair of open sets \(U \subset X, V\subset Y\), there is a canonical bilinear map of Abelian groups
\begin{equation}
\mathcal{F}(U)\otimes \mathcal{G}(V)\to
(\mathcal{F}\hat{\otimes}\mathcal{G})(U\times
V)\label{eq:174}
\end{equation}
This bilinear map is natural with respect to pairs \((U,V)\) in the product of the topologies. In particular we will be interested in the case \(U=X,V=Y\), where we get a bilinear map
\begin{equation}
\Gamma(\mathcal{F})\otimes\Gamma(\mathcal{G})\to
\Gamma(\mathcal{F}\hat{\otimes}\mathcal{G})\label{eq:173}
\end{equation}
Indeed, as with the standard tensor product of Abelian groups, this total tensor product of sheaves is characterized by a universal property with respect to bilinear maps: whenever \(\mathcal{P}\) is any sheaf on \(X\times Y\), and if we have a natural transformation $$\mathcal{F}(U)\otimes \mathcal{G}(V)\to \mathcal{P}(U\times V)$$ (and here we mean natural in the sense of a pair of functors on the product of the topologies, i.e. \(\mathcal{T}(X)\times \mathcal{T}(Y)\)) then there is an induced morphism of sheaves
\begin{equation}
\label{universal-property-of-tensor-product}
\mathcal{F}\hat{\otimes} \mathcal{G}\to \mathcal{P}
\end{equation}Let us introduce \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\), where \(\mathcal{L}^{\bullet}\) is a cochain complex of sheaves over \(X\), and \(\mathcal{M}^{\bullet}\) is a cochain complex of sheaves of Abelian groups over \(Y\). $\mathcal{L}^{\bullet}$ and $\mathcal{M}^{\bullet}$ and all other complexes from here on out are concentrated in non-negative dimension.
Given \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\), we can construct over the product space \(X \times Y\) a double cochain complex of sheaves, concentrated in the first quadrant, which we denote $\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}$; we set
\begin{equation}
\label{eq:175}
(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet})^{p,q}=
\mathcal{L}^p\hat{\otimes} \mathcal{M}^q
\end{equation}
and the differentials are the obvious ones induced by the differentials of \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\), up to the standard alternating sign trick. We can collapse this down to a total tensor product complex of sheaves, a single complex which in dimension \(n\) is
\begin{equation}
(\mathcal{L}\hat{\otimes}\mathcal{M})^n
=\bigoplus_{i+j=n}\mathcal{L}^i\hat{\otimes}\mathcal{M}^j\label{eq:176}
\end{equation}
Of course, global sections being a right adjoint, it commutes with finite direct sums, and so the cochain complex of Abelian groups given by taking global sections is
\begin{equation}
\Gamma(\mathcal{L}\hat{\otimes}\mathcal{M})^n=\bigoplus_{p+q=n}
\Gamma(\mathcal{L}^p\hat{\otimes}\mathcal{M}^q)\label{eq:177}
\end{equation}
On the other hand, if we take the global sections of the complexes \(\mathcal{L}^{\bullet},\mathcal{M}^{\bullet}\) individually, we can take the total tensor product of the complexes of global sections, \(\Gamma(\mathcal{L}^{\bullet})\otimes \Gamma(\mathcal{M}^{\bullet})\). In light of \ref{eq:177}, the maps \ref{eq:173} assemble together to determine a chain map of cochain complexes of Abelian groups,
\begin{equation}
\label{eq:170}
\Gamma(\mathcal{L}^{\bullet})\otimes
\Gamma(\mathcal{M}^{\bullet})\to
\Gamma(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet})
\end{equation}
which in dimension \(n\) is the direct sum of maps
\begin{equation}
\bigoplus_{p+q=n}\Gamma(\mathcal{L}^p)\otimes
\Gamma(\mathcal{M}^q)\to
\bigoplus_{p+q=n}\Gamma(\mathcal{L}^p\hat{\otimes}\mathcal{M}^q)\label{eq:178}
\end{equation}
Recall from our previous blog post that there is a canonical "cross product" map in cohomology
\begin{equation}
H^p(\Gamma(\mathcal{L}^{\bullet}))\otimes
H^q(\Gamma(\mathcal{M}^{\bullet})\to
H^{p+q}(\Gamma(\mathcal{L}^{\bullet})\otimes
\Gamma(\mathcal{M}^{\bullet}))\label{eq:179}
\end{equation}
sending \([\sigma]\otimes [\tau]\) to \([\sigma\otimes\tau]\); it is elementary to check that this is well defined. Combining \ref{eq:178} and \ref{eq:179} gives
\begin{equation}
H^p(\Gamma(\mathcal{L}^{\bullet}))\otimes
H^q(\Gamma(\mathcal{M}^{\bullet}))\to
H^{p+q}(\Gamma(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}))\label{eq:180}
\end{equation}
which is starting to look a bit more like a cross product map.
Now we will specialize to the case where we have two fixed sheaves, \(\mathcal{A}\) a sheaf on \(X\) and \(\mathcal{B}\) a sheaf on \(Y\), and \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\) are resolutions of \(X\), \(Y\) respectively by flasque sheaves, so that the above equation specializes to
\begin{equation}
H^p(X;\mathcal{A})\otimes H^q(Y;\mathcal{B})\to
H^{p+q}(\Gamma(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}))\label{eq:172}
\end{equation}
We might now hope that the total complex of sheaves \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) is a flasque resolution of \(\mathcal{A}\hat{\otimes}\mathcal{B}\), so that the codomain of the above map is really \(H^{p+q}(X\times Y;\mathcal{A}\hat{\otimes}\mathcal{B})\). But unfortunately this will not be the case in general. First of all, the complex of sheaves \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) may not be exact; second, it is not clear that the sheaves of the resolution will be flasque. Neither of these are insurmountable. With regards to the first problem, Godement's canonical resolution of a sheaf by flasque sheaves (where we embed \(\mathcal{F}\) into \(\prod_{x\in X}\mathcal{F}_x\) and take the cokernel, iteratively) actually returns a complex of sheaves \(\mathcal{L}^\bullet\) for which the complex of stalks \(\mathcal{L}^{\bullet}_x\) is split exact, i.e. contractible. The complex of stalks at \((x,y)\) of the total tensor product \(\mathcal{L}^\bullet \hat{\otimes} \mathcal{M}^\bullet\) is the same as the total tensor product of the complexes of stalks \(\mathcal{L}^\bullet_x \otimes \mathcal{M}^\bullet_y\), and it is not hard to show that the total tensor product of contractible complexes is contractible; therefore, we can choose resolutions \(\mathcal{L},\mathcal{M}\) of \(\mathcal{A},\mathcal{B}\) so that \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) is exact, i.e. a resolution of \(\mathcal{A}\hat{\otimes}\mathcal{B}\).
Secondly, although the complex \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) may not be composed of flasque sheaves, it is a fundamental theorem of sheaf theory (the result of a spectral sequence argument) that for any resolution \(\mathcal{A}\to \mathcal{L}^\bullet\) of a sheaf \(\mathcal{A}\), there is a canonical map
\begin{equation}
H^p(\Gamma(\mathcal{L}^\bullet)\to
H^p(X;\mathcal{A})\label{fundamental-theorem}
\end{equation}
(One can then prove that this map is always an isomorphism when \(\mathcal{L}^\bullet\) is composed of flasque sheaves, which is how one proves that one can compute the sheaf cohomology by any resolution of flasque sheaves and not just a specific resolution, functorially given by a certain choice of functor.) By applying \ref{fundamental-theorem} in the case of the resolution \(\mathcal{A}\hat{\otimes}\mathcal{B}\to \mathcal{L}^\bullet\hat{\otimes}\mathcal{M}^\bullet\), we finally get the desired map $$H^p(X;\mathcal{A})\otimes H^q(Y;\mathcal{B})\to H^{p+q}(X\times Y;\mathcal{A}\hat{\otimes}\mathcal{B})$$ the cross product map in sheaf cohomology. (Godement actually says "Cartesian product", but I have not heard this terminology elsewhere.)
From here you may be able to guess the rest of the development if you have seen the development of much of singular cohomology theory. Sheaf cohomology is contravariantly functorial; if \(\mathcal{G}\) is a sheaf on \(Y\) and \(f: X\to Y\) is a continuous map of spaces, then there is an induced map \(H^{\bullet}(Y;\mathcal{G})\to H^{\bullet}(X,f^\ast \mathcal{G})\) . This is not hard to work out - if \(\mathcal{G}\to \mathcal{L}^\bullet\) is a flasque resolution of \(\mathcal{G}\), then \(f^{\ast}\mathcal{G}\to f^{\ast} \mathcal{L}^\bullet\) is a resolution of \(f^{\ast}\mathcal{G}\); the pullback functor is exact, which should be clear from its behavior on stalks. There is a map induced by \(f\) from global sections \(\Gamma (\mathcal{L}^{\bullet})\) to global sections \(\Gamma (f^{\ast}\mathcal{L}^{\bullet})\); thinking in terms of étalé spaces and the universal property of the pullback square, this is not hard to show. One easily checks that this is a complex map, which gives us a map \(H^{n}(Y;\mathcal{G})\to H^{n}(\Gamma(f^\ast \mathcal{L}^{\bullet}))\). The "fundamental theorem" \ref{fundamental-theorem} we cited earlier gives us a map \(H^n(\Gamma(f^\ast\mathcal{L}^{\bullet}))\to H^n(X;f^\ast \mathcal{G})\). The composition of these two is the desired cohomology map. (See 4.16 of Godement for details.)
Applying this in the present context, let \(\mathcal{A}\) be a sheaf of Abelian groups over \(X\). Then the cross product gives us a map \(H^p(X;\mathcal{A})\otimes H^q(X;\mathcal{A})\to H^{p+q}(X\times X;\mathcal{A}\hat{\otimes}\mathcal{A})\); one then pulls back the sheaf \(\mathcal{A}\hat{\otimes}\mathcal{A}\) along the diagonal \(\delta : X\to X\times X\) to give \(\mathcal{A}\otimes \mathcal{A}\), and we get by contravariance of sheaf cohomology a map \( \delta^{\ast} : H^{p+q}(X\times X; \mathcal{A}\hat{\otimes}\mathcal{A})\to H^{p+q}(X;\mathcal{A}\otimes\mathcal{A})\). If \(\mathcal{A}\) is in fact a sheaf of rings or algebras, then the multiplication over each open set gives rise to a natural transformation of sheaves \(\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}\). Sheaf cohomology is covariant as a functor in the Abelian argument, so we have an induced map \(H^{p+q}(X;\mathcal{A}\otimes\mathcal{A})\to H^{p+q}(X;\mathcal{A})\). The composition of these maps gives a cup product in sheaf cohomology for any sheaf of commutative rings \(\mathcal{A}\), $$H^{p}(X;\mathcal{A})\otimes H^q(X;\mathcal{A})\to H^{p+q}(X;\mathcal{A})$$ which endows the sheaf cohomology of \(X\) with the structure of an associative, graded-commutative ring with unity.
The development of this product closely parallels one method by which the cup product is developed in singular cohomology. But in computations, another presentation of the cup product in singular cohomology is most frequently used: one takes advantage of the Eilenberg-Zilber correspondence and the Alexander-Whitney diagonal map to give an equivalent presentation. This part of the analogy (with singular theory and Cech theory) is missing so far: in both of those cases, explaining the theory in terms of cohomological coefficient systems over a simplicial set, and using the Eilenberg-Zilber theorem or a generalization thereof, allowed us to get a deeper understanding of the cohomology theory. The presence of simplicial sets and simplicial Abelian groups in the background provide a common unifying framework for thinking about the multiplicative structure. This is Godement's task now: to convincingly root sheaf cohomology in simplicial theory.
A (co)simplicial sheaf on a topological space \(X\) is a (co)simplicial object in the category of sheaves of Abelian groups over \(X\).
Example: Let \(\mathfrak{U} = \left\{ U_i \right\}_{i\in I}\) be a cover of \(X\). Let \(N(\mathfrak{M})\) be the Cech nerve, as defined in the last post. Then let \(\mathscr{C}(\mathfrak{U};\mathcal{A})\) be the cosimplicial sheaf which in dimension \([n]\) returns the sheaf whose sections over an open set \(V\) are given by $$\mathscr{C}(\mathfrak{U};\mathcal{A})(V) =\prod_{i_0,\dots, i_n}\mathcal{A}(U_{i_0}\cap U_{i_1}\dots\cap U_{i_n}\cap V)$$ where \(i_0,\dots,i_n\) range over simplices of the Cech nerve. The map \(\mathscr{C}(\mathfrak{U};\mathcal{A})_{[n]}\to \mathscr{C}(\mathfrak{U};\mathcal{A})_{[m]}\) associated to \(f: [n]\to [m]\) is the product of the restriction maps $$\mathcal{A}(U_{i_{f(0)}}\cap U_{i_{f(1)}}\dots U_{i_{f(n)}}\cap V)\to \mathcal{A}(U_{i_{0}}\cap U_{i_{1}}\dots U_{i_{m}}\cap V)$$
Taking alternating sums gives you a cochain complex of sheaves whose global sections are exactly the Cech complex associated to the open cover \(\mathfrak{U}\). Similarly one could take the simplicial presheaf which in dimension \([n]\) and on an open set \(U\) returns the free Abelian group of singular \(n\)-chains in $U$; the face and degeneracy maps are clear. Sheafifying would give a simplicial sheaf.
We introduce two new pieces of notation. If $X,Y$ are two simplicial Abelian groups (or cosimplicial) then let $X\times Y = K'(X,Y)$, where $K'$ is the functor $\mathbf{SAb}^2\to \mathbf{Ch}(Ab)$ (respectively, cochain complexes) introduced in the previous post. That is,
\begin{equation}
\label{eq:181}
(X\times Y)_n = X_n\otimes Y_n
\end{equation}
Similarly, if \(X\) and \(Y\) are two spaces, and \(\mathcal{L}^{\bullet},\mathcal{M}^\bullet\) are a pair of cosimplicial sheaves over \(X\) and \(Y\) respectively, we let \(\mathcal{L}^\bullet \hat{\times}\mathcal{M}^\bullet\) denote the cosimplicial sheaf which in dimension \([n]\) contains the sheaf \(\mathcal{L}^n\hat{\otimes}\mathcal{M}^n\) over \(X\times Y\). (We need to introduce such a notation to differentiate from \(\mathcal{L}^\bullet\hat{\otimes}\mathcal{M}^\bullet\).) Taking alternating sums of the face maps of this cosimplicial sheaf gives a cochain complex of sheaves. This gives a functor \(K_{Sh} : \mathbf{S}(Sh(X))\times \mathbf{S}(Sh(Y)) \to Ch(Sh(X\times Y))\) sending \(\mathcal{L}^{\bullet},\mathcal{M}^{\bullet}\) to \(\mathcal{L}^\bullet\hat{\times}\mathcal{M}^\bullet\). $K_{Sh}$ is clearly closely related to the functor \(K' : \mathbf{S}(\mathbf{Ab})\times \mathbf{S}(\mathbf{Ab}) \to Ch(\mathbf{Ab})\) we studied in simplicial homology theory, which we now denote \(K_{Ab}\) to avoid confusion. It stands in contrast to the functor \(L_{Sh}: \mathbf{S}(Sh(X))\times \mathbf{S}(Sh(Y)) \to Ch(Sh(X\times Y))\) which sends a pair $(\mathcal{L}^{\bullet},\mathcal{M}^{\bullet})$ to $\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}$.
The maps \(\mathcal{F}(U)\otimes \mathcal{G}(V)\to (\mathcal{F}\hat{\otimes}G) (U\times V)\) of \ref{eq:174} are natural with respect to the sheaves \(\mathcal{F}\) and \(\mathcal{G}\); therefore, \(U,V\) being fixed, they determine maps of simplicial Abelian groups
\begin{equation}
K_{Ab}(\mathcal{L}^{\bullet}(U),\mathcal{M}^{\bullet}(V))\to
K_{Sh}(\mathcal{L}^\bullet
,\mathcal{M}^\bullet)(U\times V)\label{eq:182}
\end{equation}
or by the notation we have just introduced,
\begin{equation}\label{cartesian-product-map}
\mathcal{L}^{\bullet}(U)\times \mathcal{M}^{\bullet}(V)\to \mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet}(U\times V)
\end{equation}
We will not have any need to deal with the standard Cartesian product of Abelian groups or chain complexes of Abelian groups at any point here, so this choice of notation is safe.
There are two extreme cases of immediate interest: first, taking \(U=X,V=Y\), this determines a map between chain complexes of global sections,
\begin{equation}
\Gamma(\mathcal{L}^{\bullet})\times
\Gamma(\mathcal{M}^{\bullet})\to
\Gamma(\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})\label{eq:171}
\end{equation}
In the other direction, fixing a pair \((x,y)\) in \(X,Y\) and taking colimits of open neighborhoods \((U,V)\) of \((x,y)\), we get a map between the complexes at the stalks
$$\mathcal{L}^{\bullet}_x \times \mathcal{M}^\bullet_y \cong \mathcal{L}\hat{\times}\mathcal{M}_{(x,y)}$$ which is an isomorphism by construction of the total tensor product \(\hat{\otimes}\) of sheaves.
The generalized Eilenberg-Zilber theorem for simplicial Abelian groups gives us that \(\mathcal{L}^{\bullet}_x\otimes \mathcal{M}^{\bullet}_y\) is chain homotopy equivalent to \(\mathcal{L}^{\bullet}_x\times \mathcal{M}^{\bullet}_y\). By the isomorphism just established, \((\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})_{(x,y)}= \mathcal{L}^\bullet_x\times \mathcal{M}^\bullet_y\) is chain homotopy equivalent to the total complex \(\mathcal{L}_x\otimes \mathcal{M}_y = (\mathcal{L}^\bullet\hat{\otimes}\mathcal{M}^\bullet)_{(x,y)}\), and so they have the same homology. In particular, one is acyclic iff the other is. It follows that if \(\mathcal{L}^{\bullet} \) is a resolution of some sheaf \(\mathcal{A}\) over \(X\), and \(\mathcal{M}^{\bullet}\) is a resolution of some sheaf \(\mathcal{B}\) over \(Y\), then \(\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet}\) is a resolution of \(\mathcal{A}\hat{\otimes}\mathcal{B}\) over \(X\times Y\) iff \(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}\) is.
Let $X^{\bullet},Y^{\bullet}$ range across cosimplicial Abelian groups, we have two functors $(X^{\bullet},Y^{\bullet})\mapsto X^{\bullet}\otimes Y^{\bullet}$, and $(X^{\bullet},Y^{\bullet})\mapsto X^{\bullet}\times Y^{\bullet}$. The generalized Eilenberg-Zilber theorem (or rather its dual) gives us a natural transformation $T_{(X,Y)}: X^{\bullet}\otimes Y^{\bullet}\to X^{\bullet}\times Y^{\bullet}$; fix such a $T$ once and for all. Given two cosimplicial sheaves $\mathcal{L}^{\bullet},\mathcal{M}^{\bullet}$, over any pair of open sets $U,V$ we thus have a map
$$T_{U,V} : \mathcal{L}^{\bullet}(U)\otimes \mathcal{M}^{\bullet}(V)\to \mathcal{L}^{\bullet}(U)\times \mathcal{M}^{\bullet}(V)$$ natural in $U,V$. Composing this with the map \ref{cartesian-product-map} (again natural in $U,V$)
$$\mathcal{L}^{\bullet}(U)\times \mathcal{M}^{\bullet}(V)\to (\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})(U\times V)$$ we get a map
$$\mathcal{L}^{\bullet}(U)\otimes \mathcal{M}^{\bullet}(V)\to (\mathcal{L}^{\bullet}\hat{\times} \mathcal{M}^{\bullet})(U\times V)$$ By the universal property of the tensor product $\hat{\otimes}$ (\ref{universal-property-of-tensor-product}) this determines a canonical map of cochain complexes of sheaves over $X\times Y$,
$$\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}\to \mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet}$$
Passing to global sections and recalling the map \ref{eq:170} we get a map
\begin{equation}
\label{eq:169}
\Gamma(\mathcal{L}^{\bullet})\otimes
\Gamma(\mathcal{M}^{\bullet})\to
\Gamma(\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})
\end{equation}
It is clear, by the way, that this map \ref{eq:169} agrees with the composition of \(T: \Gamma(\mathcal{L}^{\bullet})\otimes \Gamma(\mathcal{M}^{\bullet})\to \Gamma(\mathcal{L}^{\bullet})\times \Gamma(\mathcal{M}^{\bullet})\) with \ref{eq:171}.
The natural transformation \ref{eq:169} allows now for the development of the cross product along the lines of that established for simplicial Abelian groups. In order to carry this out, we will first show that one can canonically construct, for each space $X$ and each sheaf $\mathcal{A}$ on $X$, a flasque resolution $\mathcal{F}^{\bullet}(X;A)$ - a cosimplicial resolution of $\mathcal{A}$ by flasque sheaves. We call this the canonical simplicial resolution of $\mathcal{A}$.
In the next post, we will introduce the notion of monad which makes this possible.
Let \(X,Y\) be a pair of spaces. We will work with sheaves of Abelian groups over \(X,Y\) and \(X\times Y\); everything we are saying of course is easily adaptable to \(A\)-modules for a commutative ring \(A\). If \(\mathcal{F}\) is a sheaf of Abelian groups over \(X\) and \(\mathcal{G}\) is a sheaf of Abelian groups (henceforth simply "sheaf") over \(Y\), then it is possible to construct over \(X\times Y\) a "total tensor product sheaf" \(\mathcal{F}\hat{\otimes}\mathcal{G}\). (We use the hat to distinguish it from a tensor product of sheaves over a single space.) The definition of this sheaf is by the standard process: we will define a bundle \(\pi : B\to X\times Y\), which will turn out to be espace étalé of the sheaf \(\mathcal{F}\hat{\otimes}\mathcal{G}\). We construct \(B\) by defining the fiber over \(x,y\) to be \(\mathcal{F}_x \otimes\mathcal{G}_y\), and we confer upon this space the weakest topology such that for any pair of open sets \(U\subset X\), \(V\subset Y\), and for any sections \(f_1,\dots, f_n \in \mathcal{F}(U)\),\(g_1 ,\dots, g_n \in \mathcal{F}(V)\), the section \(\sum_i f_i\otimes g_i\) of \(\pi\) over \(U\times V\) which sends \((x,y)\) to \(\sum_i f_i(x)\otimes g_i(y)\) is continuous. We then define \(\mathcal{F}\hat{\otimes}\mathcal{G}\) as the sheaf of sections of the bundle \(\pi\).
In the case where \(X = Y\), then we can pull back the sheaf \(\mathcal{F}\hat{\otimes} \mathcal{G}\) along the diagonal \(\delta: X\to X\times X\) to give a tensor product of sheaves over \(X\); this sheaf, \( \delta^\ast (\mathcal{F}\hat{\otimes}\mathcal{G})\), we refer to simply as \(\mathcal{F}\otimes\mathcal{G}\) (without the hat.) This is what is more commonly known as the tensor product of sheaves.
It is clear that for each pair of open sets \(U \subset X, V\subset Y\), there is a canonical bilinear map of Abelian groups
\begin{equation}
\mathcal{F}(U)\otimes \mathcal{G}(V)\to
(\mathcal{F}\hat{\otimes}\mathcal{G})(U\times
V)\label{eq:174}
\end{equation}
This bilinear map is natural with respect to pairs \((U,V)\) in the product of the topologies. In particular we will be interested in the case \(U=X,V=Y\), where we get a bilinear map
\begin{equation}
\Gamma(\mathcal{F})\otimes\Gamma(\mathcal{G})\to
\Gamma(\mathcal{F}\hat{\otimes}\mathcal{G})\label{eq:173}
\end{equation}
Indeed, as with the standard tensor product of Abelian groups, this total tensor product of sheaves is characterized by a universal property with respect to bilinear maps: whenever \(\mathcal{P}\) is any sheaf on \(X\times Y\), and if we have a natural transformation $$\mathcal{F}(U)\otimes \mathcal{G}(V)\to \mathcal{P}(U\times V)$$ (and here we mean natural in the sense of a pair of functors on the product of the topologies, i.e. \(\mathcal{T}(X)\times \mathcal{T}(Y)\)) then there is an induced morphism of sheaves
\begin{equation}
\label{universal-property-of-tensor-product}
\mathcal{F}\hat{\otimes} \mathcal{G}\to \mathcal{P}
\end{equation}Let us introduce \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\), where \(\mathcal{L}^{\bullet}\) is a cochain complex of sheaves over \(X\), and \(\mathcal{M}^{\bullet}\) is a cochain complex of sheaves of Abelian groups over \(Y\). $\mathcal{L}^{\bullet}$ and $\mathcal{M}^{\bullet}$ and all other complexes from here on out are concentrated in non-negative dimension.
Given \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\), we can construct over the product space \(X \times Y\) a double cochain complex of sheaves, concentrated in the first quadrant, which we denote $\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}$; we set
\begin{equation}
\label{eq:175}
(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet})^{p,q}=
\mathcal{L}^p\hat{\otimes} \mathcal{M}^q
\end{equation}
and the differentials are the obvious ones induced by the differentials of \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\), up to the standard alternating sign trick. We can collapse this down to a total tensor product complex of sheaves, a single complex which in dimension \(n\) is
\begin{equation}
(\mathcal{L}\hat{\otimes}\mathcal{M})^n
=\bigoplus_{i+j=n}\mathcal{L}^i\hat{\otimes}\mathcal{M}^j\label{eq:176}
\end{equation}
Of course, global sections being a right adjoint, it commutes with finite direct sums, and so the cochain complex of Abelian groups given by taking global sections is
\begin{equation}
\Gamma(\mathcal{L}\hat{\otimes}\mathcal{M})^n=\bigoplus_{p+q=n}
\Gamma(\mathcal{L}^p\hat{\otimes}\mathcal{M}^q)\label{eq:177}
\end{equation}
On the other hand, if we take the global sections of the complexes \(\mathcal{L}^{\bullet},\mathcal{M}^{\bullet}\) individually, we can take the total tensor product of the complexes of global sections, \(\Gamma(\mathcal{L}^{\bullet})\otimes \Gamma(\mathcal{M}^{\bullet})\). In light of \ref{eq:177}, the maps \ref{eq:173} assemble together to determine a chain map of cochain complexes of Abelian groups,
\begin{equation}
\label{eq:170}
\Gamma(\mathcal{L}^{\bullet})\otimes
\Gamma(\mathcal{M}^{\bullet})\to
\Gamma(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet})
\end{equation}
which in dimension \(n\) is the direct sum of maps
\begin{equation}
\bigoplus_{p+q=n}\Gamma(\mathcal{L}^p)\otimes
\Gamma(\mathcal{M}^q)\to
\bigoplus_{p+q=n}\Gamma(\mathcal{L}^p\hat{\otimes}\mathcal{M}^q)\label{eq:178}
\end{equation}
Recall from our previous blog post that there is a canonical "cross product" map in cohomology
\begin{equation}
H^p(\Gamma(\mathcal{L}^{\bullet}))\otimes
H^q(\Gamma(\mathcal{M}^{\bullet})\to
H^{p+q}(\Gamma(\mathcal{L}^{\bullet})\otimes
\Gamma(\mathcal{M}^{\bullet}))\label{eq:179}
\end{equation}
sending \([\sigma]\otimes [\tau]\) to \([\sigma\otimes\tau]\); it is elementary to check that this is well defined. Combining \ref{eq:178} and \ref{eq:179} gives
\begin{equation}
H^p(\Gamma(\mathcal{L}^{\bullet}))\otimes
H^q(\Gamma(\mathcal{M}^{\bullet}))\to
H^{p+q}(\Gamma(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}))\label{eq:180}
\end{equation}
which is starting to look a bit more like a cross product map.
Now we will specialize to the case where we have two fixed sheaves, \(\mathcal{A}\) a sheaf on \(X\) and \(\mathcal{B}\) a sheaf on \(Y\), and \(\mathcal{L}^{\bullet}\) and \(\mathcal{M}^{\bullet}\) are resolutions of \(X\), \(Y\) respectively by flasque sheaves, so that the above equation specializes to
\begin{equation}
H^p(X;\mathcal{A})\otimes H^q(Y;\mathcal{B})\to
H^{p+q}(\Gamma(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}))\label{eq:172}
\end{equation}
We might now hope that the total complex of sheaves \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) is a flasque resolution of \(\mathcal{A}\hat{\otimes}\mathcal{B}\), so that the codomain of the above map is really \(H^{p+q}(X\times Y;\mathcal{A}\hat{\otimes}\mathcal{B})\). But unfortunately this will not be the case in general. First of all, the complex of sheaves \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) may not be exact; second, it is not clear that the sheaves of the resolution will be flasque. Neither of these are insurmountable. With regards to the first problem, Godement's canonical resolution of a sheaf by flasque sheaves (where we embed \(\mathcal{F}\) into \(\prod_{x\in X}\mathcal{F}_x\) and take the cokernel, iteratively) actually returns a complex of sheaves \(\mathcal{L}^\bullet\) for which the complex of stalks \(\mathcal{L}^{\bullet}_x\) is split exact, i.e. contractible. The complex of stalks at \((x,y)\) of the total tensor product \(\mathcal{L}^\bullet \hat{\otimes} \mathcal{M}^\bullet\) is the same as the total tensor product of the complexes of stalks \(\mathcal{L}^\bullet_x \otimes \mathcal{M}^\bullet_y\), and it is not hard to show that the total tensor product of contractible complexes is contractible; therefore, we can choose resolutions \(\mathcal{L},\mathcal{M}\) of \(\mathcal{A},\mathcal{B}\) so that \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) is exact, i.e. a resolution of \(\mathcal{A}\hat{\otimes}\mathcal{B}\).
Secondly, although the complex \(\mathcal{L}^\bullet \hat{\otimes}\mathcal{M}^{\bullet}\) may not be composed of flasque sheaves, it is a fundamental theorem of sheaf theory (the result of a spectral sequence argument) that for any resolution \(\mathcal{A}\to \mathcal{L}^\bullet\) of a sheaf \(\mathcal{A}\), there is a canonical map
\begin{equation}
H^p(\Gamma(\mathcal{L}^\bullet)\to
H^p(X;\mathcal{A})\label{fundamental-theorem}
\end{equation}
(One can then prove that this map is always an isomorphism when \(\mathcal{L}^\bullet\) is composed of flasque sheaves, which is how one proves that one can compute the sheaf cohomology by any resolution of flasque sheaves and not just a specific resolution, functorially given by a certain choice of functor.) By applying \ref{fundamental-theorem} in the case of the resolution \(\mathcal{A}\hat{\otimes}\mathcal{B}\to \mathcal{L}^\bullet\hat{\otimes}\mathcal{M}^\bullet\), we finally get the desired map $$H^p(X;\mathcal{A})\otimes H^q(Y;\mathcal{B})\to H^{p+q}(X\times Y;\mathcal{A}\hat{\otimes}\mathcal{B})$$ the cross product map in sheaf cohomology. (Godement actually says "Cartesian product", but I have not heard this terminology elsewhere.)
From here you may be able to guess the rest of the development if you have seen the development of much of singular cohomology theory. Sheaf cohomology is contravariantly functorial; if \(\mathcal{G}\) is a sheaf on \(Y\) and \(f: X\to Y\) is a continuous map of spaces, then there is an induced map \(H^{\bullet}(Y;\mathcal{G})\to H^{\bullet}(X,f^\ast \mathcal{G})\) . This is not hard to work out - if \(\mathcal{G}\to \mathcal{L}^\bullet\) is a flasque resolution of \(\mathcal{G}\), then \(f^{\ast}\mathcal{G}\to f^{\ast} \mathcal{L}^\bullet\) is a resolution of \(f^{\ast}\mathcal{G}\); the pullback functor is exact, which should be clear from its behavior on stalks. There is a map induced by \(f\) from global sections \(\Gamma (\mathcal{L}^{\bullet})\) to global sections \(\Gamma (f^{\ast}\mathcal{L}^{\bullet})\); thinking in terms of étalé spaces and the universal property of the pullback square, this is not hard to show. One easily checks that this is a complex map, which gives us a map \(H^{n}(Y;\mathcal{G})\to H^{n}(\Gamma(f^\ast \mathcal{L}^{\bullet}))\). The "fundamental theorem" \ref{fundamental-theorem} we cited earlier gives us a map \(H^n(\Gamma(f^\ast\mathcal{L}^{\bullet}))\to H^n(X;f^\ast \mathcal{G})\). The composition of these two is the desired cohomology map. (See 4.16 of Godement for details.)
Applying this in the present context, let \(\mathcal{A}\) be a sheaf of Abelian groups over \(X\). Then the cross product gives us a map \(H^p(X;\mathcal{A})\otimes H^q(X;\mathcal{A})\to H^{p+q}(X\times X;\mathcal{A}\hat{\otimes}\mathcal{A})\); one then pulls back the sheaf \(\mathcal{A}\hat{\otimes}\mathcal{A}\) along the diagonal \(\delta : X\to X\times X\) to give \(\mathcal{A}\otimes \mathcal{A}\), and we get by contravariance of sheaf cohomology a map \( \delta^{\ast} : H^{p+q}(X\times X; \mathcal{A}\hat{\otimes}\mathcal{A})\to H^{p+q}(X;\mathcal{A}\otimes\mathcal{A})\). If \(\mathcal{A}\) is in fact a sheaf of rings or algebras, then the multiplication over each open set gives rise to a natural transformation of sheaves \(\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}\). Sheaf cohomology is covariant as a functor in the Abelian argument, so we have an induced map \(H^{p+q}(X;\mathcal{A}\otimes\mathcal{A})\to H^{p+q}(X;\mathcal{A})\). The composition of these maps gives a cup product in sheaf cohomology for any sheaf of commutative rings \(\mathcal{A}\), $$H^{p}(X;\mathcal{A})\otimes H^q(X;\mathcal{A})\to H^{p+q}(X;\mathcal{A})$$ which endows the sheaf cohomology of \(X\) with the structure of an associative, graded-commutative ring with unity.
The development of this product closely parallels one method by which the cup product is developed in singular cohomology. But in computations, another presentation of the cup product in singular cohomology is most frequently used: one takes advantage of the Eilenberg-Zilber correspondence and the Alexander-Whitney diagonal map to give an equivalent presentation. This part of the analogy (with singular theory and Cech theory) is missing so far: in both of those cases, explaining the theory in terms of cohomological coefficient systems over a simplicial set, and using the Eilenberg-Zilber theorem or a generalization thereof, allowed us to get a deeper understanding of the cohomology theory. The presence of simplicial sets and simplicial Abelian groups in the background provide a common unifying framework for thinking about the multiplicative structure. This is Godement's task now: to convincingly root sheaf cohomology in simplicial theory.
A (co)simplicial sheaf on a topological space \(X\) is a (co)simplicial object in the category of sheaves of Abelian groups over \(X\).
Example: Let \(\mathfrak{U} = \left\{ U_i \right\}_{i\in I}\) be a cover of \(X\). Let \(N(\mathfrak{M})\) be the Cech nerve, as defined in the last post. Then let \(\mathscr{C}(\mathfrak{U};\mathcal{A})\) be the cosimplicial sheaf which in dimension \([n]\) returns the sheaf whose sections over an open set \(V\) are given by $$\mathscr{C}(\mathfrak{U};\mathcal{A})(V) =\prod_{i_0,\dots, i_n}\mathcal{A}(U_{i_0}\cap U_{i_1}\dots\cap U_{i_n}\cap V)$$ where \(i_0,\dots,i_n\) range over simplices of the Cech nerve. The map \(\mathscr{C}(\mathfrak{U};\mathcal{A})_{[n]}\to \mathscr{C}(\mathfrak{U};\mathcal{A})_{[m]}\) associated to \(f: [n]\to [m]\) is the product of the restriction maps $$\mathcal{A}(U_{i_{f(0)}}\cap U_{i_{f(1)}}\dots U_{i_{f(n)}}\cap V)\to \mathcal{A}(U_{i_{0}}\cap U_{i_{1}}\dots U_{i_{m}}\cap V)$$
Taking alternating sums gives you a cochain complex of sheaves whose global sections are exactly the Cech complex associated to the open cover \(\mathfrak{U}\). Similarly one could take the simplicial presheaf which in dimension \([n]\) and on an open set \(U\) returns the free Abelian group of singular \(n\)-chains in $U$; the face and degeneracy maps are clear. Sheafifying would give a simplicial sheaf.
We introduce two new pieces of notation. If $X,Y$ are two simplicial Abelian groups (or cosimplicial) then let $X\times Y = K'(X,Y)$, where $K'$ is the functor $\mathbf{SAb}^2\to \mathbf{Ch}(Ab)$ (respectively, cochain complexes) introduced in the previous post. That is,
\begin{equation}
\label{eq:181}
(X\times Y)_n = X_n\otimes Y_n
\end{equation}
Similarly, if \(X\) and \(Y\) are two spaces, and \(\mathcal{L}^{\bullet},\mathcal{M}^\bullet\) are a pair of cosimplicial sheaves over \(X\) and \(Y\) respectively, we let \(\mathcal{L}^\bullet \hat{\times}\mathcal{M}^\bullet\) denote the cosimplicial sheaf which in dimension \([n]\) contains the sheaf \(\mathcal{L}^n\hat{\otimes}\mathcal{M}^n\) over \(X\times Y\). (We need to introduce such a notation to differentiate from \(\mathcal{L}^\bullet\hat{\otimes}\mathcal{M}^\bullet\).) Taking alternating sums of the face maps of this cosimplicial sheaf gives a cochain complex of sheaves. This gives a functor \(K_{Sh} : \mathbf{S}(Sh(X))\times \mathbf{S}(Sh(Y)) \to Ch(Sh(X\times Y))\) sending \(\mathcal{L}^{\bullet},\mathcal{M}^{\bullet}\) to \(\mathcal{L}^\bullet\hat{\times}\mathcal{M}^\bullet\). $K_{Sh}$ is clearly closely related to the functor \(K' : \mathbf{S}(\mathbf{Ab})\times \mathbf{S}(\mathbf{Ab}) \to Ch(\mathbf{Ab})\) we studied in simplicial homology theory, which we now denote \(K_{Ab}\) to avoid confusion. It stands in contrast to the functor \(L_{Sh}: \mathbf{S}(Sh(X))\times \mathbf{S}(Sh(Y)) \to Ch(Sh(X\times Y))\) which sends a pair $(\mathcal{L}^{\bullet},\mathcal{M}^{\bullet})$ to $\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}$.
The maps \(\mathcal{F}(U)\otimes \mathcal{G}(V)\to (\mathcal{F}\hat{\otimes}G) (U\times V)\) of \ref{eq:174} are natural with respect to the sheaves \(\mathcal{F}\) and \(\mathcal{G}\); therefore, \(U,V\) being fixed, they determine maps of simplicial Abelian groups
\begin{equation}
K_{Ab}(\mathcal{L}^{\bullet}(U),\mathcal{M}^{\bullet}(V))\to
K_{Sh}(\mathcal{L}^\bullet
,\mathcal{M}^\bullet)(U\times V)\label{eq:182}
\end{equation}
or by the notation we have just introduced,
\begin{equation}\label{cartesian-product-map}
\mathcal{L}^{\bullet}(U)\times \mathcal{M}^{\bullet}(V)\to \mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet}(U\times V)
\end{equation}
We will not have any need to deal with the standard Cartesian product of Abelian groups or chain complexes of Abelian groups at any point here, so this choice of notation is safe.
There are two extreme cases of immediate interest: first, taking \(U=X,V=Y\), this determines a map between chain complexes of global sections,
\begin{equation}
\Gamma(\mathcal{L}^{\bullet})\times
\Gamma(\mathcal{M}^{\bullet})\to
\Gamma(\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})\label{eq:171}
\end{equation}
In the other direction, fixing a pair \((x,y)\) in \(X,Y\) and taking colimits of open neighborhoods \((U,V)\) of \((x,y)\), we get a map between the complexes at the stalks
$$\mathcal{L}^{\bullet}_x \times \mathcal{M}^\bullet_y \cong \mathcal{L}\hat{\times}\mathcal{M}_{(x,y)}$$ which is an isomorphism by construction of the total tensor product \(\hat{\otimes}\) of sheaves.
The generalized Eilenberg-Zilber theorem for simplicial Abelian groups gives us that \(\mathcal{L}^{\bullet}_x\otimes \mathcal{M}^{\bullet}_y\) is chain homotopy equivalent to \(\mathcal{L}^{\bullet}_x\times \mathcal{M}^{\bullet}_y\). By the isomorphism just established, \((\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})_{(x,y)}= \mathcal{L}^\bullet_x\times \mathcal{M}^\bullet_y\) is chain homotopy equivalent to the total complex \(\mathcal{L}_x\otimes \mathcal{M}_y = (\mathcal{L}^\bullet\hat{\otimes}\mathcal{M}^\bullet)_{(x,y)}\), and so they have the same homology. In particular, one is acyclic iff the other is. It follows that if \(\mathcal{L}^{\bullet} \) is a resolution of some sheaf \(\mathcal{A}\) over \(X\), and \(\mathcal{M}^{\bullet}\) is a resolution of some sheaf \(\mathcal{B}\) over \(Y\), then \(\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet}\) is a resolution of \(\mathcal{A}\hat{\otimes}\mathcal{B}\) over \(X\times Y\) iff \(\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}\) is.
Let $X^{\bullet},Y^{\bullet}$ range across cosimplicial Abelian groups, we have two functors $(X^{\bullet},Y^{\bullet})\mapsto X^{\bullet}\otimes Y^{\bullet}$, and $(X^{\bullet},Y^{\bullet})\mapsto X^{\bullet}\times Y^{\bullet}$. The generalized Eilenberg-Zilber theorem (or rather its dual) gives us a natural transformation $T_{(X,Y)}: X^{\bullet}\otimes Y^{\bullet}\to X^{\bullet}\times Y^{\bullet}$; fix such a $T$ once and for all. Given two cosimplicial sheaves $\mathcal{L}^{\bullet},\mathcal{M}^{\bullet}$, over any pair of open sets $U,V$ we thus have a map
$$T_{U,V} : \mathcal{L}^{\bullet}(U)\otimes \mathcal{M}^{\bullet}(V)\to \mathcal{L}^{\bullet}(U)\times \mathcal{M}^{\bullet}(V)$$ natural in $U,V$. Composing this with the map \ref{cartesian-product-map} (again natural in $U,V$)
$$\mathcal{L}^{\bullet}(U)\times \mathcal{M}^{\bullet}(V)\to (\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})(U\times V)$$ we get a map
$$\mathcal{L}^{\bullet}(U)\otimes \mathcal{M}^{\bullet}(V)\to (\mathcal{L}^{\bullet}\hat{\times} \mathcal{M}^{\bullet})(U\times V)$$ By the universal property of the tensor product $\hat{\otimes}$ (\ref{universal-property-of-tensor-product}) this determines a canonical map of cochain complexes of sheaves over $X\times Y$,
$$\mathcal{L}^{\bullet}\hat{\otimes}\mathcal{M}^{\bullet}\to \mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet}$$
Passing to global sections and recalling the map \ref{eq:170} we get a map
\begin{equation}
\label{eq:169}
\Gamma(\mathcal{L}^{\bullet})\otimes
\Gamma(\mathcal{M}^{\bullet})\to
\Gamma(\mathcal{L}^{\bullet}\hat{\times}\mathcal{M}^{\bullet})
\end{equation}
It is clear, by the way, that this map \ref{eq:169} agrees with the composition of \(T: \Gamma(\mathcal{L}^{\bullet})\otimes \Gamma(\mathcal{M}^{\bullet})\to \Gamma(\mathcal{L}^{\bullet})\times \Gamma(\mathcal{M}^{\bullet})\) with \ref{eq:171}.
The natural transformation \ref{eq:169} allows now for the development of the cross product along the lines of that established for simplicial Abelian groups. In order to carry this out, we will first show that one can canonically construct, for each space $X$ and each sheaf $\mathcal{A}$ on $X$, a flasque resolution $\mathcal{F}^{\bullet}(X;A)$ - a cosimplicial resolution of $\mathcal{A}$ by flasque sheaves. We call this the canonical simplicial resolution of $\mathcal{A}$.
In the next post, we will introduce the notion of monad which makes this possible.
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