Godement, Part V - Monads and Homological Algebra
In https://unity-of-opposites.blogspot.com/2020/06/godement-part-iv-definition-of-monads.html we introduced the Godement monad. To recap, if $X$ is any topological space, and we form the disjoint union of points $X^\delta = \coprod_{x\in X} x$ equipped with the discrete topology, there is a canonical map $i :X^\delta\to X$ which is the identity on points. This gives rise to two adjoint functors between the corresponding sheaf categories, the direct image functor and the inverse image functor. Their composition gives rise to an endofunctor on $Sh(X)$; this endofunctor is a monad, $T$. Its unit is the unit of the adjunction. Given any sheaf $\mathcal{F}$, the sequence of sheaves $T^{n+1}(\mathcal{F})$ is naturally endowed with the structure of a cosimplicial object, with augmentation map $\mathcal{F}\to T(\mathcal{F})$. By purely formal abstract nonsense, (and all the more beautiful for how trivial it is) the resulting cochain complex given by taking alternating sums of the coface maps is exact; so we have an exact resolution by flasque sheaves, which can be used to compute the sheaf cohomology. It is this resolution which we promised at the end of https://unity-of-opposites.blogspot.com/2020/06/godements-topologie-algebrique-et.html
Let us look a little more closely about what the elements of $\mathcal{F}^n=T^n(\mathcal{A})$ actually constitute. It is an inductive matter: a global section of $\mathcal{F}^n$ is represented by a function $\alpha$ which associates to each point $x_0\in X$ a point $\alpha(x_0)$ in the stalk $\mathcal{F}^{n-1}_{x_0}$ in a not-necessarily continuous way. In turn, $\alpha(x_0)$ is itself the germ at $x_0$ of the section of some not-necessarily-continuous section of $\mathcal{F}^{n-2}$, defined on some open neighborhood $U(x_0)$, which may depend on $x_0$. So $\alpha(x_0)$ may be represented as a function associating each $x_1$ in $U(x_0)$ to an element of $\mathcal{F}^{n-2}_{x_1}$, or, equivalently, we have a function $\alpha(x_0,x_1)$, where $x_0$ may vary across $X$ and $x_1$ may vary across $U(x_0)$. Continuing in this way we see that by iterating these construction, we end up with a function $\alpha$ whose inputs are $n+1$-tuples and where $\alpha(x_0,\dots, x_n)\in \mathcal{A}_{x_n}$, where $x_0$ varies over $X$, $x_1$ varies over $U(x_0)$, and $x_n$ varies over $U(x_0,x_1,\dots, x_{n-1})$.
On the converse, it is clear that any such function determines, without ambiguity, a global section of $\mathcal{F}^n$.
If we consider now the two sections of $\mathcal{F}$ over open sets $U,V$ respectively, coded by functions $\alpha(x_0,\dots,x_n), \beta(x_0,\dots, x_n)$ defined respectively according to domains $x_0\in U,x_1\in U(x_0),x_2\in U(x_0,x_1),\dots$ and $y_0\in V, y_1\in V(y_0), \dots$
In order that these two functions agree on their intersection $W=U\cap V$, it is necessary and sufficient that the relation $\alpha(x_0,\dots, x_n)=\beta(x_0,\dots, x_n)$ is satisfied on a set of the form consisting of all $x_0\in W,x_1\in W(x_0),\dots, x_n\in W(x_0,\dots, x_{n-1})$; for some nonempty open sets $W(x_0,\dots, x_i)$. This can be proved by induction on $n$. The base case is evident.
It is clear that for representing the sections of $\mathcal{F}^n(U)$, it suffices to confine ourselves to considering only the functions of the form $\alpha(x_0,\dots, x_n)\in \mathcal{A}_{x_n}$, defined on all of $U^{n+1}$, in which case the functions form an Abelian group in the evident way. The group of sections of $\mathcal{F}^n(U)$ is then the quotient group of the set of all functions $U^{n+1}\to \mathcal{A}_{x_n}$, quotiented out by all such functions defining the zero section, i.e. those functions satisfying $\alpha(x_0,\dots, x_n)=0$ on some subset of $U^{n+1}$ of the form $x_0\in U$, $x_1\in U(x_0),\dots, x_n\in U(x_0,\dots, x_{n-1})$. These are very similar to the Alexander-Spanier cochains.
We have commented that $\mathcal{F}^\bullet$ is a cosimplicial sheaf augmented by $\mathcal{A}$, following from the fact that $T$ is a monad. This can be expressed by saying that the sections of $\mathcal{F}^{\bullet}$ over each open set $U$ form a cosimplicial Abelian group, with face and degeneracy maps that are natural with respect to $U$. If the sections of $\mathcal{F}^n(U)$ are represented by maps $\alpha : U^{n+1}\to \mathcal{A}_{x_n}$ as above, then for a weakly monotonic map $f: [n]\to [m]$ in the simplex category, the associated push-forward map $f_{\ast}$ acts on $\alpha$ as follows. In what follows, $\alpha(y_{f(0)},\dots, y_{f(n)})$ is, by definition, a germ of $\mathcal{A}_{y_{(f(n))}}$. Let $t$ be a choice function which associates to each point $x\in U$ and each germ $s$ in $\mathcal{A}_x$ a choice of section $t(x,s)\in \mathcal{A}(V)$ defined on some open subset $V\subset U$, where the germ of $t(x,s)$ at $x$ is exactly $s$. Then for any point $y\in V$, $t(x,s)(y)$ is defined, and is a germ of $\mathcal{A}_y$. Then we define $f_{\ast}(\alpha) = \beta$ by
\begin{equation}
\label{eq:186}
\beta(y_0,\dots, y_m)= t(y_{f(n)},\alpha(y_{f(0)},\dots, y_{f(n)})(y_m)
\end{equation}
There is a nice abstract-nonsense proof that the cosimplicial sheaf $\mathcal{F}^{\bullet}$ does define a resolution, i.e. is exact. Proofs of all these statements (or at least their duals) can be found in a textbook like Weibel. Let $F : \mathcal{C}\to \mathcal{D}$ and $U : \mathcal{D}\to \mathcal{C}$ be a pair of adjoint functors, and let $T$ be the monad $UF$. Fix $A$ in $\mathcal{C}$, and let $T^{\bullet}(A)$ be the cosimplicial resolution of $A$; then by a standard theorem about monads, the cosimplicial object $F(T^{\bullet}(A))$ in $\mathcal{D}$ is contractible, and thus has all trivial homotopy groups. If $\mathcal{D}$ is an Abelian category, the homotopy groups of a cosimplicial object are the same as the cohomology groups of the cochain complex given by taking alternating sums of the coface maps; so the chain complex $F(T^{\bullet}(A))$ is acyclic, i.e. exact. Therefore if the functor $F$ reflects exactness, the chain complex $T^{\bullet}(A)$ is exact as well, i.e. a resolution of $A$. In our situation, the functor $F$ is the inverse image functor with respect to the map $X^{\delta}\to X$, which easily preserves and reflects exactness, as exactness is determined at the level of stalks. So from this it follows that $\mathcal{F}^{\bullet}$ (as a cochain complex) is a resolution by flasque sheaves, and thus can be used to compute the cohomology of $\mathcal{A}$.
We also remind the reader that $T$ is an exact functor; from here it is easy to see (using the fact that the global sections functor is exact on a short exact sequence of flasque sheaves) that the resolutions associated to a short exact sequence $0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0$ of sheaves can be used to give a long exact sequence in sheaf cohomology.
We will construct now a natural map
\begin{equation}
\label{eq:187}
\mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\times} \mathcal{F}^{\bullet}(Y;\mathcal{B})\to \mathcal{F}^{\bullet}(X\times Y;\mathcal{A}\hat{\otimes} \mathcal{B})
\end{equation}
To this end it suffices to construct natural maps of chain complexes of Abelian groups
\begin{equation}
\label{eq:188}
F^{\bullet}(U;\mathcal{A})\times F^{\bullet}(V;\mathcal{B})\to F^{\bullet}(U\times V;\mathcal{A}\hat{\otimes}\mathcal{B})
\end{equation}
Using the representation we have established earlier of sections, in the form $\alpha(x_0,\dots, x_n)$ and $\beta(y_0,\dots, y_n)$, we send this pair of sections to the map $\alpha(x_0,\dots,x_n)\otimes \beta(y_0,\dots, y_n)$.
This, in turn, allows us to construct a notion of ``cross product'' which is given directly at the level of the simplicial structure. In a previous blog post, we constructed, for any pair of cosimplicial sheaves $\mathcal{L}^{\bullet}, \mathcal{M}^{\bullet}$, a map of cochain complexes
\begin{equation}
\label{eq:189}
\mathcal{L}^{\bullet}\hat{\otimes} \mathcal{M}^{\bullet}\to \mathcal{L}^{\bullet}\hat{\times} \mathcal{M}^{\bullet}
\end{equation}
which, in the case of the canonical resolutions we discuss here, takes the form
\begin{equation}
\label{eq:190}
\mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\otimes}\mathcal{F}^{\bullet}(Y;\mathcal{B})\to \mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\times}\mathcal{F}^{\bullet}(Y;\mathcal{B})
\end{equation}
Postcomposing with the map constructed in \ref{eq:187}, we get a map of the form
\begin{equation}
\label{eq:191}
\mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\otimes}\mathcal{F}^{\bullet}(Y;\mathcal{B})\to \mathcal{F}^{\bullet}(X\times Y,\mathcal{A}\hat{\otimes}\mathcal{B})
\end{equation}
which, taking global sections and precomposing with the natural map worked out in a previous post, gives
\begin{equation}
\label{eq:192}
F^{\bullet}(X;\mathcal{A})\hat{\otimes}F^{\bullet}(Y;\mathcal{B})\to F^{\bullet}(X\times Y; \mathcal{A}\hat{\otimes}\mathcal{B})
\end{equation}
Passing to cohomology and taking advantage of the map
\begin{equation}
H^p(F^{\bullet}(X;\mathcal{A}))\otimes H^q(F^{\bullet}(Y;\mathcal{B}))\to H^{p+q}(F^{\bullet}(X;\mathcal{A})\hat{\otimes}F^{\bullet}(X;\mathcal{A}))\label{eq:193}
\end{equation}
we get a map
\begin{equation}
\label{eq:194}
H^p(X;\mathcal{A})\otimes H^q(Y;\mathcal{B})\to H^{p+q}(X\times Y;\mathcal{A}\hat{\otimes} \mathcal{B})
\end{equation}
which is the same cross product map in sheaf cohomology which we previously derived, but this time by giving a formula for the cross product which can be explicitly computed from representatives thanks to the underlying cosimplicial structure of the resolutions that were used. For any two cohomology classes $\xi$ of degree $p$, choose a representative $\alpha(x_0,\dots, x_p)\in \mathcal{A}(x_p)$ as before; and if $\eta$ is a cohomology class of degree $q$ of $Y$, choose a representative function $\beta(y_0,\dots, y_q)\in \mathcal{B}(y_q)$; then the cross product of the classes $\xi\times \eta$ is represented by a function determined as a function of $\alpha,\beta$, namely
\begin{equation}
\label{eq:195}
\gamma((x_0,y_0),\dots, (x_{p+q},y_{p+q}))=\alpha(x_0,\dots, x_p)(x_{p+q})\otimes \beta(y_p,\dots, y_{p+q})(y_{p+q})
\end{equation}
It should be remarked how closely similar this is to the traditional Alexander-Whitney formula for computing the cross product of singular simplices! One comes to the conclusion that the notion of cross-product and sheaf theory is closely related to the singular cross product and the Cech cohomology cross product; they are unified under the umbrella of simplicial theory. The cup product in turn can be defined in terms of the cross product; and so a simplicial formula for it, too, can be given.
I think I will conclude this series of blog posts here as it has gone on very long. I hope you have gained some small inkling of appreciation for the historical purpose for which monads were invented, that helped to explain their role.
Let us look a little more closely about what the elements of $\mathcal{F}^n=T^n(\mathcal{A})$ actually constitute. It is an inductive matter: a global section of $\mathcal{F}^n$ is represented by a function $\alpha$ which associates to each point $x_0\in X$ a point $\alpha(x_0)$ in the stalk $\mathcal{F}^{n-1}_{x_0}$ in a not-necessarily continuous way. In turn, $\alpha(x_0)$ is itself the germ at $x_0$ of the section of some not-necessarily-continuous section of $\mathcal{F}^{n-2}$, defined on some open neighborhood $U(x_0)$, which may depend on $x_0$. So $\alpha(x_0)$ may be represented as a function associating each $x_1$ in $U(x_0)$ to an element of $\mathcal{F}^{n-2}_{x_1}$, or, equivalently, we have a function $\alpha(x_0,x_1)$, where $x_0$ may vary across $X$ and $x_1$ may vary across $U(x_0)$. Continuing in this way we see that by iterating these construction, we end up with a function $\alpha$ whose inputs are $n+1$-tuples and where $\alpha(x_0,\dots, x_n)\in \mathcal{A}_{x_n}$, where $x_0$ varies over $X$, $x_1$ varies over $U(x_0)$, and $x_n$ varies over $U(x_0,x_1,\dots, x_{n-1})$.
On the converse, it is clear that any such function determines, without ambiguity, a global section of $\mathcal{F}^n$.
If we consider now the two sections of $\mathcal{F}$ over open sets $U,V$ respectively, coded by functions $\alpha(x_0,\dots,x_n), \beta(x_0,\dots, x_n)$ defined respectively according to domains $x_0\in U,x_1\in U(x_0),x_2\in U(x_0,x_1),\dots$ and $y_0\in V, y_1\in V(y_0), \dots$
In order that these two functions agree on their intersection $W=U\cap V$, it is necessary and sufficient that the relation $\alpha(x_0,\dots, x_n)=\beta(x_0,\dots, x_n)$ is satisfied on a set of the form consisting of all $x_0\in W,x_1\in W(x_0),\dots, x_n\in W(x_0,\dots, x_{n-1})$; for some nonempty open sets $W(x_0,\dots, x_i)$. This can be proved by induction on $n$. The base case is evident.
It is clear that for representing the sections of $\mathcal{F}^n(U)$, it suffices to confine ourselves to considering only the functions of the form $\alpha(x_0,\dots, x_n)\in \mathcal{A}_{x_n}$, defined on all of $U^{n+1}$, in which case the functions form an Abelian group in the evident way. The group of sections of $\mathcal{F}^n(U)$ is then the quotient group of the set of all functions $U^{n+1}\to \mathcal{A}_{x_n}$, quotiented out by all such functions defining the zero section, i.e. those functions satisfying $\alpha(x_0,\dots, x_n)=0$ on some subset of $U^{n+1}$ of the form $x_0\in U$, $x_1\in U(x_0),\dots, x_n\in U(x_0,\dots, x_{n-1})$. These are very similar to the Alexander-Spanier cochains.
We have commented that $\mathcal{F}^\bullet$ is a cosimplicial sheaf augmented by $\mathcal{A}$, following from the fact that $T$ is a monad. This can be expressed by saying that the sections of $\mathcal{F}^{\bullet}$ over each open set $U$ form a cosimplicial Abelian group, with face and degeneracy maps that are natural with respect to $U$. If the sections of $\mathcal{F}^n(U)$ are represented by maps $\alpha : U^{n+1}\to \mathcal{A}_{x_n}$ as above, then for a weakly monotonic map $f: [n]\to [m]$ in the simplex category, the associated push-forward map $f_{\ast}$ acts on $\alpha$ as follows. In what follows, $\alpha(y_{f(0)},\dots, y_{f(n)})$ is, by definition, a germ of $\mathcal{A}_{y_{(f(n))}}$. Let $t$ be a choice function which associates to each point $x\in U$ and each germ $s$ in $\mathcal{A}_x$ a choice of section $t(x,s)\in \mathcal{A}(V)$ defined on some open subset $V\subset U$, where the germ of $t(x,s)$ at $x$ is exactly $s$. Then for any point $y\in V$, $t(x,s)(y)$ is defined, and is a germ of $\mathcal{A}_y$. Then we define $f_{\ast}(\alpha) = \beta$ by
\begin{equation}
\label{eq:186}
\beta(y_0,\dots, y_m)= t(y_{f(n)},\alpha(y_{f(0)},\dots, y_{f(n)})(y_m)
\end{equation}
There is a nice abstract-nonsense proof that the cosimplicial sheaf $\mathcal{F}^{\bullet}$ does define a resolution, i.e. is exact. Proofs of all these statements (or at least their duals) can be found in a textbook like Weibel. Let $F : \mathcal{C}\to \mathcal{D}$ and $U : \mathcal{D}\to \mathcal{C}$ be a pair of adjoint functors, and let $T$ be the monad $UF$. Fix $A$ in $\mathcal{C}$, and let $T^{\bullet}(A)$ be the cosimplicial resolution of $A$; then by a standard theorem about monads, the cosimplicial object $F(T^{\bullet}(A))$ in $\mathcal{D}$ is contractible, and thus has all trivial homotopy groups. If $\mathcal{D}$ is an Abelian category, the homotopy groups of a cosimplicial object are the same as the cohomology groups of the cochain complex given by taking alternating sums of the coface maps; so the chain complex $F(T^{\bullet}(A))$ is acyclic, i.e. exact. Therefore if the functor $F$ reflects exactness, the chain complex $T^{\bullet}(A)$ is exact as well, i.e. a resolution of $A$. In our situation, the functor $F$ is the inverse image functor with respect to the map $X^{\delta}\to X$, which easily preserves and reflects exactness, as exactness is determined at the level of stalks. So from this it follows that $\mathcal{F}^{\bullet}$ (as a cochain complex) is a resolution by flasque sheaves, and thus can be used to compute the cohomology of $\mathcal{A}$.
We also remind the reader that $T$ is an exact functor; from here it is easy to see (using the fact that the global sections functor is exact on a short exact sequence of flasque sheaves) that the resolutions associated to a short exact sequence $0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0$ of sheaves can be used to give a long exact sequence in sheaf cohomology.
We will construct now a natural map
\begin{equation}
\label{eq:187}
\mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\times} \mathcal{F}^{\bullet}(Y;\mathcal{B})\to \mathcal{F}^{\bullet}(X\times Y;\mathcal{A}\hat{\otimes} \mathcal{B})
\end{equation}
To this end it suffices to construct natural maps of chain complexes of Abelian groups
\begin{equation}
\label{eq:188}
F^{\bullet}(U;\mathcal{A})\times F^{\bullet}(V;\mathcal{B})\to F^{\bullet}(U\times V;\mathcal{A}\hat{\otimes}\mathcal{B})
\end{equation}
Using the representation we have established earlier of sections, in the form $\alpha(x_0,\dots, x_n)$ and $\beta(y_0,\dots, y_n)$, we send this pair of sections to the map $\alpha(x_0,\dots,x_n)\otimes \beta(y_0,\dots, y_n)$.
This, in turn, allows us to construct a notion of ``cross product'' which is given directly at the level of the simplicial structure. In a previous blog post, we constructed, for any pair of cosimplicial sheaves $\mathcal{L}^{\bullet}, \mathcal{M}^{\bullet}$, a map of cochain complexes
\begin{equation}
\label{eq:189}
\mathcal{L}^{\bullet}\hat{\otimes} \mathcal{M}^{\bullet}\to \mathcal{L}^{\bullet}\hat{\times} \mathcal{M}^{\bullet}
\end{equation}
which, in the case of the canonical resolutions we discuss here, takes the form
\begin{equation}
\label{eq:190}
\mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\otimes}\mathcal{F}^{\bullet}(Y;\mathcal{B})\to \mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\times}\mathcal{F}^{\bullet}(Y;\mathcal{B})
\end{equation}
Postcomposing with the map constructed in \ref{eq:187}, we get a map of the form
\begin{equation}
\label{eq:191}
\mathcal{F}^{\bullet}(X;\mathcal{A})\hat{\otimes}\mathcal{F}^{\bullet}(Y;\mathcal{B})\to \mathcal{F}^{\bullet}(X\times Y,\mathcal{A}\hat{\otimes}\mathcal{B})
\end{equation}
which, taking global sections and precomposing with the natural map worked out in a previous post, gives
\begin{equation}
\label{eq:192}
F^{\bullet}(X;\mathcal{A})\hat{\otimes}F^{\bullet}(Y;\mathcal{B})\to F^{\bullet}(X\times Y; \mathcal{A}\hat{\otimes}\mathcal{B})
\end{equation}
Passing to cohomology and taking advantage of the map
\begin{equation}
H^p(F^{\bullet}(X;\mathcal{A}))\otimes H^q(F^{\bullet}(Y;\mathcal{B}))\to H^{p+q}(F^{\bullet}(X;\mathcal{A})\hat{\otimes}F^{\bullet}(X;\mathcal{A}))\label{eq:193}
\end{equation}
we get a map
\begin{equation}
\label{eq:194}
H^p(X;\mathcal{A})\otimes H^q(Y;\mathcal{B})\to H^{p+q}(X\times Y;\mathcal{A}\hat{\otimes} \mathcal{B})
\end{equation}
which is the same cross product map in sheaf cohomology which we previously derived, but this time by giving a formula for the cross product which can be explicitly computed from representatives thanks to the underlying cosimplicial structure of the resolutions that were used. For any two cohomology classes $\xi$ of degree $p$, choose a representative $\alpha(x_0,\dots, x_p)\in \mathcal{A}(x_p)$ as before; and if $\eta$ is a cohomology class of degree $q$ of $Y$, choose a representative function $\beta(y_0,\dots, y_q)\in \mathcal{B}(y_q)$; then the cross product of the classes $\xi\times \eta$ is represented by a function determined as a function of $\alpha,\beta$, namely
\begin{equation}
\label{eq:195}
\gamma((x_0,y_0),\dots, (x_{p+q},y_{p+q}))=\alpha(x_0,\dots, x_p)(x_{p+q})\otimes \beta(y_p,\dots, y_{p+q})(y_{p+q})
\end{equation}
It should be remarked how closely similar this is to the traditional Alexander-Whitney formula for computing the cross product of singular simplices! One comes to the conclusion that the notion of cross-product and sheaf theory is closely related to the singular cross product and the Cech cohomology cross product; they are unified under the umbrella of simplicial theory. The cup product in turn can be defined in terms of the cross product; and so a simplicial formula for it, too, can be given.
I think I will conclude this series of blog posts here as it has gone on very long. I hope you have gained some small inkling of appreciation for the historical purpose for which monads were invented, that helped to explain their role.
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