Godement, Part IV - definition of monads, examples
A monad is a monoid in a category of endofunctors.
In the previous post, we have established a very general notion of monoid which is suitable for many applications. For example, the category of chain complexes $R$-modules can be equipped with a monoidal product which associates to each pair $(C,C')$ of chain complexes the total complex $Tot(C\otimes C')$ of the double complex $(C\otimes C')_{i,j}= C_i\otimes C_j'$. The unit of this monoidal product is the complex with $R$ in dimension $0$ and $0$'s in all other dimension. A DG-algebra is precisely a monoid with respect to this monoidal product.
For a given category $\mathcal{C}$, the functor category $[\mathcal{C};\mathcal{C}]$ is naturally equipped with thestructure of a strict monoidal category, whose monoidal product $\otimes$ is precisely functor composition, $F\otimes G= F\circ G$; and whose unit $1$ is precisely the
identity functor $id_{\mathcal{C}}$. It is easily verified that this is functorial simultaneously in both
variables. (It would be good to work out explicitly as an exercise, if your category theory is at all rusty, what the functor $(S,T)\mapsto S\circ T$ does to a pair of morphisms $\tau : T_1\to T_2$ and $\sigma: S_1\to S_2$ and explicitly describe it.) It is obvious that it is associative and unital on the nose. Therefore, by the argument in the previous post, it is possible to formulate the notion of a monoid in the strict monoidal category $([\mathcal{C};\mathcal{C}],\circ, id_{\mathcal{C}})$. Such a monoid will be called a monad. For convenience, although this could be derived from the results of the previous post, we spell out the definition: a monad is a functor $T: \mathcal{C}\to \mathcal{C}$, together with a unit natural transformation $\eta : id_{\mathcal{C}}\to T$ and a multiplication natural transformation $\mu : T\circ T\to T$, such that $\mu\circ T\eta = \mu\circ \eta T = id_T$, and such that $\mu\circ T\mu = \mu \circ \mu T$. (The notation $\mu T$ means the natural transformation $T\circ T\circ T\to T\circ T$ which to each $X$ associates the map $\mu_{T(X)}: T^3(X)\to T^2(X)$; the notation $T\mu$ is a natural transformation between the same two functors but associates to $X$ the map $T(\mu_X) : T^3(X)\to T^2(X)$.)
As a corollary of the results of the previous post, each monad determines an augmented cosimplicial object $F^{\bullet}$ in the functor category $[\mathcal{C};\mathcal{C}]$, where $F^{-1} = id_{\mathcal{C}}$, $F^0 = T$, and $F^n= T^{n+1}$. The face maps $d_i: T^n\to T^{n+1}$ are given by $T^i\eta T^{n-i}$; the degeneracy maps $s_i: T^{n+1}\to T^{n}$ are given by $T^i\mu T^{n-i-1}$.
We are not generally interested in this augmented cosimplicial object $F^{\bullet}$ in its own right. Rather, we prefer to see $F^{\bullet}$ as a functor from $\mathcal{C}$ into the category of augmented cosimplicial objects in $\mathcal{C}$, which associates to each object $X$ the augmented cosimplicial object $F^{\bullet}(X)$ in $\mathcal{C}$, where $F^n(X)= T^{n+1}(X)$. Therefore we see that monads in $\mathcal{C}$ provide a systematic way of assigning cosimplicial ``resolutions'' $X\to T(X)\to T^2(X)\rightrightarrows T^3(X)\dots$ to every object in $\mathcal{C}$, in a functorial way.
If $\mathcal{C}$ happens to be an Abelian category, taking alternating sums of the face maps gives a cochain complex, which, if it happens to be exact, determines a resolution of $X$.
We will develop here some of the standard theory which gives sufficient conditions for this chain complex map to be exact. Even if $\mathcal{C}$ is not an Abelian category, nevertheless we may have available to us a certain functor $K: \mathcal{C}\to \mathbf{Ab}$, and so the cosimplical $\mathcal{C}$-object $F^{\bullet}(X)$ could be mapped, via $K$, to a cosimplicial Abelian group, where we can take alternating sums and compute its cohomology.
A comonad on $\mathcal{C}$ is a monad on $\mathcal{C}^{op}$; i.e. a monoid in the monoidal category $[\mathcal{C}^{op};\mathcal{C}^{op}]$. Explicitly it consists of a functor $G: \mathcal{C}\to \mathcal{C}$ together with natural transformations called the counit $\epsilon : G\to id_{\mathcal{C}}$ and comultiplication $\delta : G\to G\circ G$, satisfying the formal laws $G\epsilon\circ \delta = \epsilon G\circ\delta$, $G\delta\circ \delta = \delta G \circ \delta$. Every comonad on $\mathcal{C}$ determines a functor from $\mathcal{C}^{op}$ into the (opposite) category of cosimplicial objects in $\mathcal{C}^{op}$; or equivalently, a functor from $\mathcal{C}$ into the category of simplicial objects on $\mathcal{C}$; a systematic, functorial way of assigning to each object $X$ a simplicial resolution of $X$.
Here are three important classes of monads and comonads.\\
- Suppose that $\mathcal{C}$ is a monoidal category, $(\mathcal{C},\otimes,1)$. Suppose $(M,e: 1\to M, m: M\otimes M\to M)$ is a monoid in $\mathcal{C}$. Then the functor $X\mapsto M\otimes X$ is a monad, with unit natural transformation $\eta_X: X\to M\otimes X$ given by $(e\otimes id_X)\circ \rho : X\cong 1\otimes X\to M\otimes X$, and with multiplication $\mu : M\otimes (M\otimes X)\to M\otimes X$ given by $(m\otimes id_X)\circ \alpha_{M,M,X} : M\otimes (M\otimes X)\to (M\otimes M)\otimes X\to M\otimes X$. Similarly, it is not hard to see that if $(\mathcal{C},\otimes, 1)$ is a monoidal category, then $(\mathcal{C}^{op},\otimes, 1)$ is also a monoidal category in a natural way. If we call a monoid $S$ in $\mathcal{C}^{op}$ a ``comonoid'', then it is not hard to see that the functor $X\mapsto S\otimes X$ is a comonad on $\mathcal{C}$.
- Suppose that $\mathcal{C}, \mathcal{D}$ are two categories, and $F: \mathcal{C}\to \mathcal{D}, G: \mathcal{D}\to \mathcal{C}$ are adjoint, so that $F\dashv G$. Then the composition $G\circ F$ is a monad on $\mathcal{C}$ in a natural way; the unit map $\eta : id_{\mathcal{C}}\to G\circ F$ of the adjunction is also the unit of the monad, and the multiplication map $\mu : GFGF\to GF$ is $G \epsilon F$, where $\epsilon : FG\to id_{\mathcal{D}}$ is the counit of the adjunction. I omit the verification of the associativity and unit laws. Similarly, $FG : \mathcal{D}\to \mathcal{D}$ is a comonad on $\mathcal{D}$ in a natural way, where the counit map $FG\to id_{\mathcal{D}}$ is also the counit of the adjunction, and the comultiplication $FG\to FGFG$ is given by $F\eta G$, where $\eta$ is the unit of the adjunction.
- If $\left\{ M_i \right\}_{i\in I}$ is a family of set-theoretic monoids, it is not hard to see that $\prod_{i\in I} M_i$ can also be endowed with the structure of a monoid in a natural way. A similar result holds for monads; if $\mathcal{C}$ is a category with products, and $\left\{ T_i \right\}_{i\in I}$ is a family of monads on $\mathcal{C}$, then the functor $X\mapsto \prod_{i\in I}T_i(X)$ is a monad, with unit map $\eta:X\to \prod_{i\in I}T_i(X)$ given by the product of the maps $\eta_i: X\to T_i(X)$ and with multiplication $\mu : \prod_{i\in I}T_i(\prod_{i'\in I}T_{i'}(X))\to \prod_{j\in I}T_j(X)$ given on component $j$ by $\mu_j\circ T(\pi_j)\circ \pi_j$. Likewise, if $\mathcal{C}$ has coproducts, then the coproduct $\coprod_{i\in I}G_i$ of comonads $G_i$ is also a comonad in a natural way.
- Say that $\mathcal{C}$ is a category, and $T$ is a monad on $\mathcal{C}$. Then for any categories $\mathcal{B}$, there is an induced monad on the category $[\mathcal{B};\mathcal{C}]$ which sends $F : \mathcal{B}\to \mathcal{C}$ to $T\circ F$; and an induced monad on $[\mathcal{C};\mathcal{B}]$ which sends $J$ to $J\circ T$. The associated natural transformations are obvious. A similar result holds for comonads.
Monoids arise everywhere; so monads arise everywhere as a consequence. We pointed out in a precious blog post that the singleton space is a monoid with respect to the join operator in the category of topological spaces; it follows that the functor $X\mapsto \left\{ \bullet \right\}\ast X$ is a monad. It is easily checked that this is the cone functor $C$, and that the unit map $X\to C(X)$ is the canonical embedding of $X$ into the cone on $X$.
Likewise, the problem of finding ``comonoids'' is easier than it may seem at first glance. Although your first thought might be of Hopf algebras, they are more accessible than that. If $\mathcal{C}$ is a category with products, then every object $X$ in $\mathcal{C}$ is naturally a comonoid with respect to $\times$, with counit map $X\to 1$ the unique map into the terminal object, and with comultiplication $\delta :X\to X\times X$ the diagonal map. It follows from this that in any category with products, $1\leftarrow X \leftleftarrows X\times X\dots$ is an augmented simplicial object with $X_n = X^{n+1}$; with face maps $d_i$ the projection map $X^n\to X^{n-1}$ which omits the $i$-th index, and with degeneracy maps $s_i: X^n\to X^{n+1}$ the map which is the diagonal in the $i$th component and the identity everywhere else.
Adjunctions arise everywhere, so monads and comonads arise everywhere. An important example is the free-forgetful adjunction between $\mathbf{Sets}\dashv R-\mathbf{mod}$; the associated comonad adjunction sends each $R$-module $M$ to the free $R$-module on its underlying elements. Since $R$-mod is an Abelian category, the associated simplicial resolution of $M$ given by iteratively applying the comonad can be turned into a chain complex by taking alternating sums, giving a chain complex. This chain complex, we will prove later, is exact, which means that we have given a functorial simplicial free resolution of every $R$-module.
In sheaf cohomology theory, monads are abundant. Let $X$ be a topological space, and $x$ a point in $X$. Then there is an adjunction $Sh(X)\dashv \mathbf{Ab}$, whose left adjoint is the stalk functor $\mathcal{F}\mapsto \mathcal{F}_{x}$, and whose right adjoint is the skyscraper sheaf functor $A\mapsto x^{\ast}(A)$. It follows that the composition $\mathcal{F}\mapsto x^{\ast}(\mathcal{F}_x)$ is a monad on the category of sheaves. Since the product of monads is a monad, the functor $\mathcal{F}\mapsto \prod_{x\in X}x^{\ast}(\mathcal{F}_x)$ is a monad in the category of sheaves, which is called the Godement monad. By iteratively applying it, we get a simplicial resolution of $\mathcal{F}$ by flasque sheaves. Taking alternating sums gives a resolution which can be used for computing the cohomology. (We will prove later that this is exact.)
If $X$ is a topological space, let $K : \mathcal{T}(X)\to \mathbf{Sets}$ which sends every open set $U\in \mathcal{T}(X)$ to the set $U$. Let $Hom(-,-) : \mathbf{Sets}^{op}\times \mathbf{Ab}\to \mathbf{Ab}$ here denote the functor which sends a set $X$ and an Abelian group $A$ to the set of functions from $X$ to $A$ under pointwise multiplication. If $\mathcal{A}$ is any presheaf of Abelian groups $\mathcal{A}: \mathcal{T}(X)^{op}\to \mathbf{Ab}$, then $(K, \mathcal{A}) : \mathcal{T}(X)^{op}\to \mathbf{Sets}^{op}\times \mathbf{Ab}$, and $Hom(K(-),\mathcal{A}(-)) : \mathcal{T}(X)^{op}\to \mathbf{Ab}$. Furthermore the assignment
\begin{equation}
\label{eq:183}
\mathcal{A}\mapsto Hom(K(-),\mathcal{A}(-))
\end{equation}
is functorial in $\mathcal{A}$ as a functor from presheaves on $X$ to presheaves on $X$. I claim that this is in fact a monad on the presheaf category.
The unit natural transformation $\eta_A: \mathcal{A}\to Hom(K(-),\mathcal{A}(-))$ over the open set $U$ is the map $\mathcal{A}(U)\to Hom(U,\mathcal{A}(U))$ which sends every section of $\mathcal{A}(U)$ to the constant map $U\to \mathcal{A}(U)$ at that section.
The multiplication map $\mu: Hom(K(-),Hom(K(-),\mathcal{A}(-)))\to Hom(K(-),\mathcal{A}(-))$ over an open set $U$ is given as follows. $Hom(U,Hom(U,\mathcal{A}(U)))\cong Hom(U\times U,\mathcal{A}(U))$, so the map $Hom(U\times U, \mathcal{A}(U))\to Hom(U,\mathcal{A}(U))$ is given by precomposition with the diagonal $\delta : U\to U\times U$. I call this the Alexander-Spanier monad, as the cosimplicial presheaf resolution that this monad associates to any preheaf can be sheafified to give a cosimplicial sheaf resolution; this sheaf resolution computes the Alexander-Spanier cohomology theory.
If $X$ is a topological space, and $M$ is a subset of $X$, then there is an adjunction between sheaves over $X$ and sheaves over $M$ given by the pushforward/pullback adjunction (i.e. the direct image/inverse image adjunction). By composing the left and right adjoint, we get a monad $T_M$ on the sheaf category $Sh(X)$; if $\mathcal{A}$ is a sheaf on $X$, then $T_M(\mathcal{A})(V) = A(M\cap V)$. If, moreover, $\left\{ M_i \right\}_{i\in I}$ is a cover of $X$, then each set $M_i$ induces a monad $T_i$ on $Sh(X)$ by taking inverse images, then direct images; the poduct $\prod T_i$ of these monads is itself a monad $T$ on the sheaf category; it associates to each sheaf $\mathcal{A}$ the sheaf $T(\mathcal{A})(U) = \prod_{i\in I}\mathcal{A}(M_i\cap U)$. Iterating $T$ twice, we get $T(T(\mathcal{A}))(U) = T(\prod_{i\in I}\mathcal{A}(M_i\cap -))(U) = \prod_{j\in J}\prod_{i\in I}\mathcal{A}(M_i\cap M_j\cap U)$; iterating it $n$ times, we get $T^n(\mathcal{A})(U) = \prod_{i_1,i_2,\dots, i_n\in I}\mathcal{A}(M_{i_1}\cap M_{i_2}\cap\dots M_{i_n}\cap U)$, where without loss of generality we can take the product only over those $i_0,\dots, i_n$ for which $M_{i_1}\cap \dots\cap M_{i_n}\neq \emptyset$, as $\mathcal{A}(\emptyset) =0$ by the sheaf axioms. So we get a cosimplicial resolution of $\mathcal{A}$ whose global sections are precisely the Cech cochain complex of $\mathcal{A}$ with respect to the cover $\left\{ M_i \right\}$.
In the previous post, we have established a very general notion of monoid which is suitable for many applications. For example, the category of chain complexes $R$-modules can be equipped with a monoidal product which associates to each pair $(C,C')$ of chain complexes the total complex $Tot(C\otimes C')$ of the double complex $(C\otimes C')_{i,j}= C_i\otimes C_j'$. The unit of this monoidal product is the complex with $R$ in dimension $0$ and $0$'s in all other dimension. A DG-algebra is precisely a monoid with respect to this monoidal product.
For a given category $\mathcal{C}$, the functor category $[\mathcal{C};\mathcal{C}]$ is naturally equipped with thestructure of a strict monoidal category, whose monoidal product $\otimes$ is precisely functor composition, $F\otimes G= F\circ G$; and whose unit $1$ is precisely the
identity functor $id_{\mathcal{C}}$. It is easily verified that this is functorial simultaneously in both
variables. (It would be good to work out explicitly as an exercise, if your category theory is at all rusty, what the functor $(S,T)\mapsto S\circ T$ does to a pair of morphisms $\tau : T_1\to T_2$ and $\sigma: S_1\to S_2$ and explicitly describe it.) It is obvious that it is associative and unital on the nose. Therefore, by the argument in the previous post, it is possible to formulate the notion of a monoid in the strict monoidal category $([\mathcal{C};\mathcal{C}],\circ, id_{\mathcal{C}})$. Such a monoid will be called a monad. For convenience, although this could be derived from the results of the previous post, we spell out the definition: a monad is a functor $T: \mathcal{C}\to \mathcal{C}$, together with a unit natural transformation $\eta : id_{\mathcal{C}}\to T$ and a multiplication natural transformation $\mu : T\circ T\to T$, such that $\mu\circ T\eta = \mu\circ \eta T = id_T$, and such that $\mu\circ T\mu = \mu \circ \mu T$. (The notation $\mu T$ means the natural transformation $T\circ T\circ T\to T\circ T$ which to each $X$ associates the map $\mu_{T(X)}: T^3(X)\to T^2(X)$; the notation $T\mu$ is a natural transformation between the same two functors but associates to $X$ the map $T(\mu_X) : T^3(X)\to T^2(X)$.)
As a corollary of the results of the previous post, each monad determines an augmented cosimplicial object $F^{\bullet}$ in the functor category $[\mathcal{C};\mathcal{C}]$, where $F^{-1} = id_{\mathcal{C}}$, $F^0 = T$, and $F^n= T^{n+1}$. The face maps $d_i: T^n\to T^{n+1}$ are given by $T^i\eta T^{n-i}$; the degeneracy maps $s_i: T^{n+1}\to T^{n}$ are given by $T^i\mu T^{n-i-1}$.
We are not generally interested in this augmented cosimplicial object $F^{\bullet}$ in its own right. Rather, we prefer to see $F^{\bullet}$ as a functor from $\mathcal{C}$ into the category of augmented cosimplicial objects in $\mathcal{C}$, which associates to each object $X$ the augmented cosimplicial object $F^{\bullet}(X)$ in $\mathcal{C}$, where $F^n(X)= T^{n+1}(X)$. Therefore we see that monads in $\mathcal{C}$ provide a systematic way of assigning cosimplicial ``resolutions'' $X\to T(X)\to T^2(X)\rightrightarrows T^3(X)\dots$ to every object in $\mathcal{C}$, in a functorial way.
If $\mathcal{C}$ happens to be an Abelian category, taking alternating sums of the face maps gives a cochain complex, which, if it happens to be exact, determines a resolution of $X$.
We will develop here some of the standard theory which gives sufficient conditions for this chain complex map to be exact. Even if $\mathcal{C}$ is not an Abelian category, nevertheless we may have available to us a certain functor $K: \mathcal{C}\to \mathbf{Ab}$, and so the cosimplical $\mathcal{C}$-object $F^{\bullet}(X)$ could be mapped, via $K$, to a cosimplicial Abelian group, where we can take alternating sums and compute its cohomology.
A comonad on $\mathcal{C}$ is a monad on $\mathcal{C}^{op}$; i.e. a monoid in the monoidal category $[\mathcal{C}^{op};\mathcal{C}^{op}]$. Explicitly it consists of a functor $G: \mathcal{C}\to \mathcal{C}$ together with natural transformations called the counit $\epsilon : G\to id_{\mathcal{C}}$ and comultiplication $\delta : G\to G\circ G$, satisfying the formal laws $G\epsilon\circ \delta = \epsilon G\circ\delta$, $G\delta\circ \delta = \delta G \circ \delta$. Every comonad on $\mathcal{C}$ determines a functor from $\mathcal{C}^{op}$ into the (opposite) category of cosimplicial objects in $\mathcal{C}^{op}$; or equivalently, a functor from $\mathcal{C}$ into the category of simplicial objects on $\mathcal{C}$; a systematic, functorial way of assigning to each object $X$ a simplicial resolution of $X$.
Here are three important classes of monads and comonads.\\
- Suppose that $\mathcal{C}$ is a monoidal category, $(\mathcal{C},\otimes,1)$. Suppose $(M,e: 1\to M, m: M\otimes M\to M)$ is a monoid in $\mathcal{C}$. Then the functor $X\mapsto M\otimes X$ is a monad, with unit natural transformation $\eta_X: X\to M\otimes X$ given by $(e\otimes id_X)\circ \rho : X\cong 1\otimes X\to M\otimes X$, and with multiplication $\mu : M\otimes (M\otimes X)\to M\otimes X$ given by $(m\otimes id_X)\circ \alpha_{M,M,X} : M\otimes (M\otimes X)\to (M\otimes M)\otimes X\to M\otimes X$. Similarly, it is not hard to see that if $(\mathcal{C},\otimes, 1)$ is a monoidal category, then $(\mathcal{C}^{op},\otimes, 1)$ is also a monoidal category in a natural way. If we call a monoid $S$ in $\mathcal{C}^{op}$ a ``comonoid'', then it is not hard to see that the functor $X\mapsto S\otimes X$ is a comonad on $\mathcal{C}$.
- Suppose that $\mathcal{C}, \mathcal{D}$ are two categories, and $F: \mathcal{C}\to \mathcal{D}, G: \mathcal{D}\to \mathcal{C}$ are adjoint, so that $F\dashv G$. Then the composition $G\circ F$ is a monad on $\mathcal{C}$ in a natural way; the unit map $\eta : id_{\mathcal{C}}\to G\circ F$ of the adjunction is also the unit of the monad, and the multiplication map $\mu : GFGF\to GF$ is $G \epsilon F$, where $\epsilon : FG\to id_{\mathcal{D}}$ is the counit of the adjunction. I omit the verification of the associativity and unit laws. Similarly, $FG : \mathcal{D}\to \mathcal{D}$ is a comonad on $\mathcal{D}$ in a natural way, where the counit map $FG\to id_{\mathcal{D}}$ is also the counit of the adjunction, and the comultiplication $FG\to FGFG$ is given by $F\eta G$, where $\eta$ is the unit of the adjunction.
- If $\left\{ M_i \right\}_{i\in I}$ is a family of set-theoretic monoids, it is not hard to see that $\prod_{i\in I} M_i$ can also be endowed with the structure of a monoid in a natural way. A similar result holds for monads; if $\mathcal{C}$ is a category with products, and $\left\{ T_i \right\}_{i\in I}$ is a family of monads on $\mathcal{C}$, then the functor $X\mapsto \prod_{i\in I}T_i(X)$ is a monad, with unit map $\eta:X\to \prod_{i\in I}T_i(X)$ given by the product of the maps $\eta_i: X\to T_i(X)$ and with multiplication $\mu : \prod_{i\in I}T_i(\prod_{i'\in I}T_{i'}(X))\to \prod_{j\in I}T_j(X)$ given on component $j$ by $\mu_j\circ T(\pi_j)\circ \pi_j$. Likewise, if $\mathcal{C}$ has coproducts, then the coproduct $\coprod_{i\in I}G_i$ of comonads $G_i$ is also a comonad in a natural way.
- Say that $\mathcal{C}$ is a category, and $T$ is a monad on $\mathcal{C}$. Then for any categories $\mathcal{B}$, there is an induced monad on the category $[\mathcal{B};\mathcal{C}]$ which sends $F : \mathcal{B}\to \mathcal{C}$ to $T\circ F$; and an induced monad on $[\mathcal{C};\mathcal{B}]$ which sends $J$ to $J\circ T$. The associated natural transformations are obvious. A similar result holds for comonads.
Monoids arise everywhere; so monads arise everywhere as a consequence. We pointed out in a precious blog post that the singleton space is a monoid with respect to the join operator in the category of topological spaces; it follows that the functor $X\mapsto \left\{ \bullet \right\}\ast X$ is a monad. It is easily checked that this is the cone functor $C$, and that the unit map $X\to C(X)$ is the canonical embedding of $X$ into the cone on $X$.
Likewise, the problem of finding ``comonoids'' is easier than it may seem at first glance. Although your first thought might be of Hopf algebras, they are more accessible than that. If $\mathcal{C}$ is a category with products, then every object $X$ in $\mathcal{C}$ is naturally a comonoid with respect to $\times$, with counit map $X\to 1$ the unique map into the terminal object, and with comultiplication $\delta :X\to X\times X$ the diagonal map. It follows from this that in any category with products, $1\leftarrow X \leftleftarrows X\times X\dots$ is an augmented simplicial object with $X_n = X^{n+1}$; with face maps $d_i$ the projection map $X^n\to X^{n-1}$ which omits the $i$-th index, and with degeneracy maps $s_i: X^n\to X^{n+1}$ the map which is the diagonal in the $i$th component and the identity everywhere else.
Adjunctions arise everywhere, so monads and comonads arise everywhere. An important example is the free-forgetful adjunction between $\mathbf{Sets}\dashv R-\mathbf{mod}$; the associated comonad adjunction sends each $R$-module $M$ to the free $R$-module on its underlying elements. Since $R$-mod is an Abelian category, the associated simplicial resolution of $M$ given by iteratively applying the comonad can be turned into a chain complex by taking alternating sums, giving a chain complex. This chain complex, we will prove later, is exact, which means that we have given a functorial simplicial free resolution of every $R$-module.
In sheaf cohomology theory, monads are abundant. Let $X$ be a topological space, and $x$ a point in $X$. Then there is an adjunction $Sh(X)\dashv \mathbf{Ab}$, whose left adjoint is the stalk functor $\mathcal{F}\mapsto \mathcal{F}_{x}$, and whose right adjoint is the skyscraper sheaf functor $A\mapsto x^{\ast}(A)$. It follows that the composition $\mathcal{F}\mapsto x^{\ast}(\mathcal{F}_x)$ is a monad on the category of sheaves. Since the product of monads is a monad, the functor $\mathcal{F}\mapsto \prod_{x\in X}x^{\ast}(\mathcal{F}_x)$ is a monad in the category of sheaves, which is called the Godement monad. By iteratively applying it, we get a simplicial resolution of $\mathcal{F}$ by flasque sheaves. Taking alternating sums gives a resolution which can be used for computing the cohomology. (We will prove later that this is exact.)
If $X$ is a topological space, let $K : \mathcal{T}(X)\to \mathbf{Sets}$ which sends every open set $U\in \mathcal{T}(X)$ to the set $U$. Let $Hom(-,-) : \mathbf{Sets}^{op}\times \mathbf{Ab}\to \mathbf{Ab}$ here denote the functor which sends a set $X$ and an Abelian group $A$ to the set of functions from $X$ to $A$ under pointwise multiplication. If $\mathcal{A}$ is any presheaf of Abelian groups $\mathcal{A}: \mathcal{T}(X)^{op}\to \mathbf{Ab}$, then $(K, \mathcal{A}) : \mathcal{T}(X)^{op}\to \mathbf{Sets}^{op}\times \mathbf{Ab}$, and $Hom(K(-),\mathcal{A}(-)) : \mathcal{T}(X)^{op}\to \mathbf{Ab}$. Furthermore the assignment
\begin{equation}
\label{eq:183}
\mathcal{A}\mapsto Hom(K(-),\mathcal{A}(-))
\end{equation}
is functorial in $\mathcal{A}$ as a functor from presheaves on $X$ to presheaves on $X$. I claim that this is in fact a monad on the presheaf category.
The unit natural transformation $\eta_A: \mathcal{A}\to Hom(K(-),\mathcal{A}(-))$ over the open set $U$ is the map $\mathcal{A}(U)\to Hom(U,\mathcal{A}(U))$ which sends every section of $\mathcal{A}(U)$ to the constant map $U\to \mathcal{A}(U)$ at that section.
The multiplication map $\mu: Hom(K(-),Hom(K(-),\mathcal{A}(-)))\to Hom(K(-),\mathcal{A}(-))$ over an open set $U$ is given as follows. $Hom(U,Hom(U,\mathcal{A}(U)))\cong Hom(U\times U,\mathcal{A}(U))$, so the map $Hom(U\times U, \mathcal{A}(U))\to Hom(U,\mathcal{A}(U))$ is given by precomposition with the diagonal $\delta : U\to U\times U$. I call this the Alexander-Spanier monad, as the cosimplicial presheaf resolution that this monad associates to any preheaf can be sheafified to give a cosimplicial sheaf resolution; this sheaf resolution computes the Alexander-Spanier cohomology theory.
If $X$ is a topological space, and $M$ is a subset of $X$, then there is an adjunction between sheaves over $X$ and sheaves over $M$ given by the pushforward/pullback adjunction (i.e. the direct image/inverse image adjunction). By composing the left and right adjoint, we get a monad $T_M$ on the sheaf category $Sh(X)$; if $\mathcal{A}$ is a sheaf on $X$, then $T_M(\mathcal{A})(V) = A(M\cap V)$. If, moreover, $\left\{ M_i \right\}_{i\in I}$ is a cover of $X$, then each set $M_i$ induces a monad $T_i$ on $Sh(X)$ by taking inverse images, then direct images; the poduct $\prod T_i$ of these monads is itself a monad $T$ on the sheaf category; it associates to each sheaf $\mathcal{A}$ the sheaf $T(\mathcal{A})(U) = \prod_{i\in I}\mathcal{A}(M_i\cap U)$. Iterating $T$ twice, we get $T(T(\mathcal{A}))(U) = T(\prod_{i\in I}\mathcal{A}(M_i\cap -))(U) = \prod_{j\in J}\prod_{i\in I}\mathcal{A}(M_i\cap M_j\cap U)$; iterating it $n$ times, we get $T^n(\mathcal{A})(U) = \prod_{i_1,i_2,\dots, i_n\in I}\mathcal{A}(M_{i_1}\cap M_{i_2}\cap\dots M_{i_n}\cap U)$, where without loss of generality we can take the product only over those $i_0,\dots, i_n$ for which $M_{i_1}\cap \dots\cap M_{i_n}\neq \emptyset$, as $\mathcal{A}(\emptyset) =0$ by the sheaf axioms. So we get a cosimplicial resolution of $\mathcal{A}$ whose global sections are precisely the Cech cochain complex of $\mathcal{A}$ with respect to the cover $\left\{ M_i \right\}$.
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