Godement's "Topologie Algebrique et Theorie des Faisceaux", Part I
I am no expert in French, so my translation skills are pretty bad - it's about a toss up between whether Google Translate or myself will have a better translation. But I have started translating segments of Godement's "Algebraic Topology and Sheaf Theory" into English. It's an excellent exposition of homological algebra and sheaf theory. I particularly like how he introduces spectral sequences in the least degree of generality necessary to make efficient use out of them, it definitely made spectral sequences less intimidating for me. Now that I've read a good share of the book I think I have a good feel for how they can be used practically, as he gives lots of demonstrations.
I also like how he devotes a large section of the book, around a fifth, to the homology theory and homological algebra of simplicial Abelian groups.
Of course this book has its place in the history of category theory. The appendix is the first place where the definition of a monad was written down, and its basic properties in the context of homological algebra explained.
Here are my initial efforts.
https://www.math.wisc.edu/~nicodemus/Topologie_Algebrique_et_theorie_des_faisceaux.pdf
https://www.math.wisc.edu/~nicodemus/Topologie_Algebrique_et_theorie_des_faisceaux.tex
There are two functors \(K,L\), both from the category \(\mathbf{SSet}^2\) of pairs of simplicial sets, and valued in the category \(Ch(\mathbf{Ab})\)of chain complexes of Abelian groups.
\(K\) accepts a pair of simplicial sets \(X, Y\) and returns the chain complex which results from taking the pointwise Cartesian product of the two simplicial sets, and then taking free Abelian group on the pair \(X\times Y\), and then turning this simplicial Abelian group into a chain complex in the usual way by taking alternating sums of the face maps. So \(K(X,Y)_n = \mathbb{Z}(X_n\times Y_n)\).
\(L\) accepts a pair of simplicial sets \(X\) and \(Y\), takes the free Abelian group on each of them, turning them into simplicial Abelian groups, takes face maps of alternating sums on each to turn them into chain complexes, and then takes the total tensor product of the two complexes. So \(L(X,Y)_n = \bigoplus_{i+j=n} \mathbb{Z}(X_i) \otimes \mathbb{Z}(Y_j)\).
Eilenberg and Zilber proved that the functors \(K\) and \(L\) are naturally chain homotopy equivalent. The method they invented for this is a beautiful and powerful one called "acyclic models"; it is a hammer that can be used to hit many diverse nails. Find a PDF copy of Spanier's book and do a CTRL-F search for "acyclic models." There are a proliferation of closely related theorems that go under this name; in my research I have recently proved a new variant using comonads that subsumes a few existing ones.
As a corollary of the Eilenberg-Zilber theorem, it is possible to give a conceptually clean definition of the cross product and cup product in singular cohomology valid for an arbitrary topological space. Let \(X,Y\) be a pair of topological spaces, and \(S(X),S(Y)\) denote the corresponding simplicial sets of singular simplices valued in \(X,Y\) respectively; sometimes called the total singular complexes of \(X\) and \(Y\).
For any two chain complexes \(C,C'\) there are two "cross product" maps in homology and cohomology respectively. (Here, the tensor product \(C\otimes C'\) denotes the total tensor product of complexes.) They take the form of maps
$$H_p(C)\otimes H_q(C')\to H_{p+q}(C\otimes C')$$
which sends
\([c]\otimes [c']\) to \([c\otimes c']\) (it is not hard to check that this is well defined) and, for any two Abelian groups \(G,G'\) a map
$$H^p(C;G) \otimes H^q(C';G')\to H^{p+q}(C\otimes C', G\otimes G')$$
which send a pair of cohomology classes of maps \([f]\otimes [f']\) to the cohomology class of the map sending \(\sigma \otimes \sigma'\) in \(C\otimes C'\) to \(f(\sigma)\otimes f'(\sigma')\). Again, it is not hard to check that this is well-defined with respect to cohomology classes.
If one takes \(C,C'\) to be the singular homology complexes of two spaces \(X,Y\) respectively, it is then clear that these specialize to maps
$$H_p(X)\otimes H_q(Y)\to H_{p+q}(L(S(X),S(Y)))$$
and
$$H^p(X;G)\otimes H^q(Y,G')\to H^{p+q}(L(S(X),S(Y));G\otimes G')$$
It is clear that for spaces \(X\) and \(Y\), by the universal property of the product of spaces, there is a natural bijection \(S(X)_n\times S(Y)_n \cong S(X\times Y)_n\). Therefore, \(K(S(X),S(Y))\) is naturally isomorphic to the singular chain complex of the space \(X\times Y\).
Therefore, by Eilenberg and Zilber's chain homotopy result, the maps above can be replaced, up to isomorphism, with maps
$$H_p(X)\otimes H_q(Y)\to H_{p+q}(X\times Y)$$
and
$$H^p(X;G)\otimes H^q(Y;G')\to H^{p+q}(X\times Y;G\otimes G')$$
These maps are known as the cross products of singular homology and cohomology. In the case that \(X=Y\) and \(G = G' = R\) for some commutative ring \(R\), the cross product becomes
$$H^p(X;R)\otimes H^q(X;G')\to H^{p+q}(X\times X;R)$$
Using the contravariance of the cohomology functor, we can apply \(H^{p+q}(-;R)\) to the diagonal map \(\delta : X\to X\times X\) to get a map
$$ H^p(X;R)\otimes H^q(X;R) \to H^{p+q}(X;R)$$
It is a basic theorem of algebraic topology that this map makes the graded group \(H^{\bullet}(X;R)\) into a graded-commutative ring with unity. (The unit element is represented by the constant function in \(H^0(X;R)\) which sends every point in \(X\) to the unit element of \(R\).)
Godement realized that this had a significance beyond singular theory. In his book, he shows that Eilenberg and Zilber's theorem can be generalized as follows. We consider two functors \(K', L'\) from the category of pairs of simplicial Abelian groups \(\mathbf{SAb}^2\) to chain complexes of Abelian groups \(Ch(\mathbf{Ab})\); \(K'\) sends the pair of simplicial Abelian groups \(X,Y\) to the chain complex given by taking their pointwise tensor product and taking alternating sums of face maps; so \(K'(X,Y)_n = X_n\otimes Y_n\). On the other hand, \(L'\) converts both \(X\) and \(Y\) to to chain complexes by taking alternating sums of face maps, and then takes their total tensor product. In this way, Eilenberg and Zilber's result can be regarded as establishing a natural homotopy equivalence between \(K'\) and \(L'\) as restricted to a subcategory of \(\mathbf{SAb}^2\); namely, pairs of simplicial free Abelian groups, where the free group in each degree is equipped with a choice of basis, and where all face and degeneracy maps carry basis elements to basis elements. Godement proved that this chain homotopy equivalence could be extended to all of \(\mathbf{SAb}^2\), by a not-too-difficult Yoneda-like argument centering around the density of the free simplicial Abelian groups in \(\mathbf{SAb}\).
This dovetails nicely with another thread that Godement develops. Sheaf cohomology is a well-known idea that lets us take coefficients in a space \(X\) with "local" coefficients varying over \(X\). In particular, if \(p : E\to X\) is a continuous map of topological spaces, regarded as a bundle over \(X\), and the fibers of \(p\) are equipped with an Abelian group structure which continuously varies with the fibers, then we understand how to take the cohomology of \(X\) with coefficients in \(p\) - this is the sheaf cohomology of the sheaf of sections of the bundle. That is: for any Abelian group object in the slice category \(\mathbf{Top}/X\), there is a way to convert such a bundle to a functor \(\mathcal{T}(X)^{op} \to \mathbf{Ab}\) and, from this, extract cohomology groups.
As Kan had recently established the beginnings of homotopy theory for simplicial sets; it must have seemed natural to see whether cohomology with coefficients in a bundle could be imported to that category. Indeed, if \(X\) is a simplicial set, one can easily formulate the analogous notion of a bundle of Abelian groups fibered over \(X\) - one asks exactly for an Abelian group object in the slice category \(\mathbf{SSet}/X\), i.e. a simplicial set \(E\) together with a map of simplicial sets \(p : E \to B\) such that for any element \(x_n\) of \(X_n\), the fiber \(p^{-1}(x_n)\) is equipped with the structure of an Abelian group, and the face and degeneracy maps of \(E\) are homomorphisms when restricted to each fiber. We call such a group object a homological coefficient system for the simplicial set X. (See, for example, Manin and Gelfand's Methods of Homological Algebra.)
There is also an analogous notion to a sheaf of Abelian groups over a simplicial set. Rather than a sheaf being a functor with domain category given by the lattice of opens, for us the domain category is \(\mathbf{el}(X)\), the category of elements of \(X\) (see here, https://en.wikipedia.org/wiki/Category_of_elements or a book on category theory like Sheaves in Geometry and Logic by Mac Lane and Moerdijk); it has as its objects the individual elements \(x_n\) of the sets \(X_n\); and for its morphisms, the set of morphisms from \(x_n \in X_n\) to \(x_m \in X_m\) is defined to be the subset of the nondecreasing maps from \([n]\) to \([m]\) consisting of those maps \(f\) for which \(X(f)(x_m)=x_n\). The category of elements of any simplicial set \(X\) is always naturally equipped with a projection functor \(\pi : \mathbf{el}(X)\to \Delta\), where here \(\Delta\) represents the simplex category of finite nonempty ordinals and monotonically increasing maps between them. If \(x_n\) is an element of \(X_n\), then we define \( \pi(x_n)= [n]\); since we have defined the hom-set between \(x_n\) and \(x_m\) to be a subset of \(Hom([n],[m])\) in \(\Delta\) to begin with, the behavior of the functor \(\pi\) on hom-sets is simply the inclusion map.
Define a cohomological coefficient system associated to a simplicial set \(X\) to be a (covariant) functor \(P : \mathbf{el}(X)\to \mathbf{Ab}\), although the notion would work to take coefficients in any Abelian category with limits. If \(P\) is a cohomological coefficient system, the right Kan extension of \(P\) along \(\pi\) exists and is uniquely characterized up to a canonical isomorphism according to its universal property; this functor \(Ran_P : \Delta \to \mathbf{Ab}\) is, simply by virtue of its domain and codomain, a cosimplicial Abelian group. Thus, by taking alternating sums of face maps, one easily turns this functor into a cochain complex of Abelian groups concentrated in nonnegative degree, which we denote \(C^{\bullet}(X;P)\); its cohomology is called the cohomology of \(X\) with coefficients in \(P\). (In some recent notes I have worked out that many of the crucial properties of this cohomology theory can be derived purely from the universal property of the Kan extension, suggesting a generalization coefficients in other categories is possible; in standard texts like Godement they are justified by computations in Abelian groups. I may discuss this in another post sometime.)
A few examples may help to clarify. Applying the computational formulas for the Kan extension given in Categories for the Working Mathematician or another text on category theory, one sees that
$$ Ran_P([n]) = \prod_{x\in X_n} P(x)$$
This is not hard to check; an algebra exercise.
If \(X\) is a topological space, and \(\mathfrak{M}=\{M_i \}_{i\in I} \) a cover by open sets, then there is associated to the cover a simplicial set \(N(\mathfrak{M})\) called the nerve of the cover, which in degree \(n\) contains all finite sequences \((i_0,\dots, i_n)\) of indices of sets \(M_i\) such that the intersection \(M_{i_0}\cap \dots\cap M_{i_n}\) is nonempty. There are some nice theorems about the nerve telling us that when \(\{M_i\}_{i\in I}\) is a good cover by opens, then the nerve is a pretty good combinatorial approximation to \(X\), and can serve as a decent proxy for \(X\) in the category of simplicial sets.
If \(X\) is equipped with a presheaf \(\mathcal{A}\) of Abelian groups, moreover, one can construct a cohomological coefficient system over \(N(\mathfrak{M})\) which associates to the simplex \(\{i_0,\dots, i_n \}\) the Abelian group \(\mathcal{A}(M_{i_0}\cap \dots\cap M_{i_n})\). The homomorphisms are given by restriction maps. Taking the right Kan extension along the projection functor \(\mathbf{el}(N(\mathfrak{M}))\) and taking alternating face maps; we get the Čech complex \(\check{C}^{\bullet}(\mathfrak{M};\mathcal{A})\).
Let \(X\) be a topological space, and let \(S(X)\) be the total singular complex; then if \(P\) is the constant functor which associates to every simplex the same Abelian group \(G\), then \(Ran_P([n]) = \prod_{x\in X_n}G = Hom(S(X)_n;G)\); clearly the associated cohomology is the singular cohomology of \(X\) with coefficients in \(G\).
Thus, the theory of singular cohomology and Cech cohomology can both be seen as attempts to convert a topological space into a simplicial set and employ the cohomology theory of simplicial sets with a system of coefficients; ultimately, both constructions yield a cosimplicial Abelian group before they yield a cochain complex; suggesting that this notion is conceptually prior. Furthermore, the theorem of Eilenberg-Zilber shows that the cup product arises entirely at the level of simplicial sets and simplicial Abelian groups - the cup product is not a feature of topological spaces per se. Godement realized that, if \(X, Y\) are two cosimplicial Abelian groups, his extension to the Eilenberg-Zilber theorem gives a cross product map
$$H^p(X)\otimes H^q(Y)\to H^{p+q}( K'(X,Y))$$
and that if there is any "diagonal approximation" map \(K'(X,X)\to X\), we can use this to construct a cup product for the cohomology of \(H^{\bullet}(X)\) which is very structurally similar to the cup product of singular homology. Moreover it is inessential that we consider simplicial Abelian groups; in a suitable Abelian category \(\mathcal{A}\) with tensor products and countable direct sums, it would be possible to formulate analogous functors $$K,L: \mathbf{S}(\mathcal{A})\times\mathbf{S}(\mathcal{A})\to Ch(\mathcal{A})$$ and we could then attempt to establish an analogue or generalization of the Eilenberg-Zilber theorem.
In a next blog post I will turn to the application of these ideas to constructing products in sheaf cohomology; and in particular the relevance of the simplicial perspective.
I also like how he devotes a large section of the book, around a fifth, to the homology theory and homological algebra of simplicial Abelian groups.
Of course this book has its place in the history of category theory. The appendix is the first place where the definition of a monad was written down, and its basic properties in the context of homological algebra explained.
Here are my initial efforts.
https://www.math.wisc.edu/~nicodemus/Topologie_Algebrique_et_theorie_des_faisceaux.pdf
https://www.math.wisc.edu/~nicodemus/Topologie_Algebrique_et_theorie_des_faisceaux.tex
There are two functors \(K,L\), both from the category \(\mathbf{SSet}^2\) of pairs of simplicial sets, and valued in the category \(Ch(\mathbf{Ab})\)of chain complexes of Abelian groups.
\(K\) accepts a pair of simplicial sets \(X, Y\) and returns the chain complex which results from taking the pointwise Cartesian product of the two simplicial sets, and then taking free Abelian group on the pair \(X\times Y\), and then turning this simplicial Abelian group into a chain complex in the usual way by taking alternating sums of the face maps. So \(K(X,Y)_n = \mathbb{Z}(X_n\times Y_n)\).
\(L\) accepts a pair of simplicial sets \(X\) and \(Y\), takes the free Abelian group on each of them, turning them into simplicial Abelian groups, takes face maps of alternating sums on each to turn them into chain complexes, and then takes the total tensor product of the two complexes. So \(L(X,Y)_n = \bigoplus_{i+j=n} \mathbb{Z}(X_i) \otimes \mathbb{Z}(Y_j)\).
Eilenberg and Zilber proved that the functors \(K\) and \(L\) are naturally chain homotopy equivalent. The method they invented for this is a beautiful and powerful one called "acyclic models"; it is a hammer that can be used to hit many diverse nails. Find a PDF copy of Spanier's book and do a CTRL-F search for "acyclic models." There are a proliferation of closely related theorems that go under this name; in my research I have recently proved a new variant using comonads that subsumes a few existing ones.
As a corollary of the Eilenberg-Zilber theorem, it is possible to give a conceptually clean definition of the cross product and cup product in singular cohomology valid for an arbitrary topological space. Let \(X,Y\) be a pair of topological spaces, and \(S(X),S(Y)\) denote the corresponding simplicial sets of singular simplices valued in \(X,Y\) respectively; sometimes called the total singular complexes of \(X\) and \(Y\).
For any two chain complexes \(C,C'\) there are two "cross product" maps in homology and cohomology respectively. (Here, the tensor product \(C\otimes C'\) denotes the total tensor product of complexes.) They take the form of maps
$$H_p(C)\otimes H_q(C')\to H_{p+q}(C\otimes C')$$
which sends
\([c]\otimes [c']\) to \([c\otimes c']\) (it is not hard to check that this is well defined) and, for any two Abelian groups \(G,G'\) a map
$$H^p(C;G) \otimes H^q(C';G')\to H^{p+q}(C\otimes C', G\otimes G')$$
which send a pair of cohomology classes of maps \([f]\otimes [f']\) to the cohomology class of the map sending \(\sigma \otimes \sigma'\) in \(C\otimes C'\) to \(f(\sigma)\otimes f'(\sigma')\). Again, it is not hard to check that this is well-defined with respect to cohomology classes.
If one takes \(C,C'\) to be the singular homology complexes of two spaces \(X,Y\) respectively, it is then clear that these specialize to maps
$$H_p(X)\otimes H_q(Y)\to H_{p+q}(L(S(X),S(Y)))$$
and
$$H^p(X;G)\otimes H^q(Y,G')\to H^{p+q}(L(S(X),S(Y));G\otimes G')$$
It is clear that for spaces \(X\) and \(Y\), by the universal property of the product of spaces, there is a natural bijection \(S(X)_n\times S(Y)_n \cong S(X\times Y)_n\). Therefore, \(K(S(X),S(Y))\) is naturally isomorphic to the singular chain complex of the space \(X\times Y\).
Therefore, by Eilenberg and Zilber's chain homotopy result, the maps above can be replaced, up to isomorphism, with maps
$$H_p(X)\otimes H_q(Y)\to H_{p+q}(X\times Y)$$
and
$$H^p(X;G)\otimes H^q(Y;G')\to H^{p+q}(X\times Y;G\otimes G')$$
These maps are known as the cross products of singular homology and cohomology. In the case that \(X=Y\) and \(G = G' = R\) for some commutative ring \(R\), the cross product becomes
$$H^p(X;R)\otimes H^q(X;G')\to H^{p+q}(X\times X;R)$$
Using the contravariance of the cohomology functor, we can apply \(H^{p+q}(-;R)\) to the diagonal map \(\delta : X\to X\times X\) to get a map
$$ H^p(X;R)\otimes H^q(X;R) \to H^{p+q}(X;R)$$
It is a basic theorem of algebraic topology that this map makes the graded group \(H^{\bullet}(X;R)\) into a graded-commutative ring with unity. (The unit element is represented by the constant function in \(H^0(X;R)\) which sends every point in \(X\) to the unit element of \(R\).)
Godement realized that this had a significance beyond singular theory. In his book, he shows that Eilenberg and Zilber's theorem can be generalized as follows. We consider two functors \(K', L'\) from the category of pairs of simplicial Abelian groups \(\mathbf{SAb}^2\) to chain complexes of Abelian groups \(Ch(\mathbf{Ab})\); \(K'\) sends the pair of simplicial Abelian groups \(X,Y\) to the chain complex given by taking their pointwise tensor product and taking alternating sums of face maps; so \(K'(X,Y)_n = X_n\otimes Y_n\). On the other hand, \(L'\) converts both \(X\) and \(Y\) to to chain complexes by taking alternating sums of face maps, and then takes their total tensor product. In this way, Eilenberg and Zilber's result can be regarded as establishing a natural homotopy equivalence between \(K'\) and \(L'\) as restricted to a subcategory of \(\mathbf{SAb}^2\); namely, pairs of simplicial free Abelian groups, where the free group in each degree is equipped with a choice of basis, and where all face and degeneracy maps carry basis elements to basis elements. Godement proved that this chain homotopy equivalence could be extended to all of \(\mathbf{SAb}^2\), by a not-too-difficult Yoneda-like argument centering around the density of the free simplicial Abelian groups in \(\mathbf{SAb}\).
This dovetails nicely with another thread that Godement develops. Sheaf cohomology is a well-known idea that lets us take coefficients in a space \(X\) with "local" coefficients varying over \(X\). In particular, if \(p : E\to X\) is a continuous map of topological spaces, regarded as a bundle over \(X\), and the fibers of \(p\) are equipped with an Abelian group structure which continuously varies with the fibers, then we understand how to take the cohomology of \(X\) with coefficients in \(p\) - this is the sheaf cohomology of the sheaf of sections of the bundle. That is: for any Abelian group object in the slice category \(\mathbf{Top}/X\), there is a way to convert such a bundle to a functor \(\mathcal{T}(X)^{op} \to \mathbf{Ab}\) and, from this, extract cohomology groups.
As Kan had recently established the beginnings of homotopy theory for simplicial sets; it must have seemed natural to see whether cohomology with coefficients in a bundle could be imported to that category. Indeed, if \(X\) is a simplicial set, one can easily formulate the analogous notion of a bundle of Abelian groups fibered over \(X\) - one asks exactly for an Abelian group object in the slice category \(\mathbf{SSet}/X\), i.e. a simplicial set \(E\) together with a map of simplicial sets \(p : E \to B\) such that for any element \(x_n\) of \(X_n\), the fiber \(p^{-1}(x_n)\) is equipped with the structure of an Abelian group, and the face and degeneracy maps of \(E\) are homomorphisms when restricted to each fiber. We call such a group object a homological coefficient system for the simplicial set X. (See, for example, Manin and Gelfand's Methods of Homological Algebra.)
There is also an analogous notion to a sheaf of Abelian groups over a simplicial set. Rather than a sheaf being a functor with domain category given by the lattice of opens, for us the domain category is \(\mathbf{el}(X)\), the category of elements of \(X\) (see here, https://en.wikipedia.org/wiki/Category_of_elements or a book on category theory like Sheaves in Geometry and Logic by Mac Lane and Moerdijk); it has as its objects the individual elements \(x_n\) of the sets \(X_n\); and for its morphisms, the set of morphisms from \(x_n \in X_n\) to \(x_m \in X_m\) is defined to be the subset of the nondecreasing maps from \([n]\) to \([m]\) consisting of those maps \(f\) for which \(X(f)(x_m)=x_n\). The category of elements of any simplicial set \(X\) is always naturally equipped with a projection functor \(\pi : \mathbf{el}(X)\to \Delta\), where here \(\Delta\) represents the simplex category of finite nonempty ordinals and monotonically increasing maps between them. If \(x_n\) is an element of \(X_n\), then we define \( \pi(x_n)= [n]\); since we have defined the hom-set between \(x_n\) and \(x_m\) to be a subset of \(Hom([n],[m])\) in \(\Delta\) to begin with, the behavior of the functor \(\pi\) on hom-sets is simply the inclusion map.
Define a cohomological coefficient system associated to a simplicial set \(X\) to be a (covariant) functor \(P : \mathbf{el}(X)\to \mathbf{Ab}\), although the notion would work to take coefficients in any Abelian category with limits. If \(P\) is a cohomological coefficient system, the right Kan extension of \(P\) along \(\pi\) exists and is uniquely characterized up to a canonical isomorphism according to its universal property; this functor \(Ran_P : \Delta \to \mathbf{Ab}\) is, simply by virtue of its domain and codomain, a cosimplicial Abelian group. Thus, by taking alternating sums of face maps, one easily turns this functor into a cochain complex of Abelian groups concentrated in nonnegative degree, which we denote \(C^{\bullet}(X;P)\); its cohomology is called the cohomology of \(X\) with coefficients in \(P\). (In some recent notes I have worked out that many of the crucial properties of this cohomology theory can be derived purely from the universal property of the Kan extension, suggesting a generalization coefficients in other categories is possible; in standard texts like Godement they are justified by computations in Abelian groups. I may discuss this in another post sometime.)
A few examples may help to clarify. Applying the computational formulas for the Kan extension given in Categories for the Working Mathematician or another text on category theory, one sees that
$$ Ran_P([n]) = \prod_{x\in X_n} P(x)$$
This is not hard to check; an algebra exercise.
If \(X\) is a topological space, and \(\mathfrak{M}=\{M_i \}_{i\in I} \) a cover by open sets, then there is associated to the cover a simplicial set \(N(\mathfrak{M})\) called the nerve of the cover, which in degree \(n\) contains all finite sequences \((i_0,\dots, i_n)\) of indices of sets \(M_i\) such that the intersection \(M_{i_0}\cap \dots\cap M_{i_n}\) is nonempty. There are some nice theorems about the nerve telling us that when \(\{M_i\}_{i\in I}\) is a good cover by opens, then the nerve is a pretty good combinatorial approximation to \(X\), and can serve as a decent proxy for \(X\) in the category of simplicial sets.
If \(X\) is equipped with a presheaf \(\mathcal{A}\) of Abelian groups, moreover, one can construct a cohomological coefficient system over \(N(\mathfrak{M})\) which associates to the simplex \(\{i_0,\dots, i_n \}\) the Abelian group \(\mathcal{A}(M_{i_0}\cap \dots\cap M_{i_n})\). The homomorphisms are given by restriction maps. Taking the right Kan extension along the projection functor \(\mathbf{el}(N(\mathfrak{M}))\) and taking alternating face maps; we get the Čech complex \(\check{C}^{\bullet}(\mathfrak{M};\mathcal{A})\).
Let \(X\) be a topological space, and let \(S(X)\) be the total singular complex; then if \(P\) is the constant functor which associates to every simplex the same Abelian group \(G\), then \(Ran_P([n]) = \prod_{x\in X_n}G = Hom(S(X)_n;G)\); clearly the associated cohomology is the singular cohomology of \(X\) with coefficients in \(G\).
Thus, the theory of singular cohomology and Cech cohomology can both be seen as attempts to convert a topological space into a simplicial set and employ the cohomology theory of simplicial sets with a system of coefficients; ultimately, both constructions yield a cosimplicial Abelian group before they yield a cochain complex; suggesting that this notion is conceptually prior. Furthermore, the theorem of Eilenberg-Zilber shows that the cup product arises entirely at the level of simplicial sets and simplicial Abelian groups - the cup product is not a feature of topological spaces per se. Godement realized that, if \(X, Y\) are two cosimplicial Abelian groups, his extension to the Eilenberg-Zilber theorem gives a cross product map
$$H^p(X)\otimes H^q(Y)\to H^{p+q}( K'(X,Y))$$
and that if there is any "diagonal approximation" map \(K'(X,X)\to X\), we can use this to construct a cup product for the cohomology of \(H^{\bullet}(X)\) which is very structurally similar to the cup product of singular homology. Moreover it is inessential that we consider simplicial Abelian groups; in a suitable Abelian category \(\mathcal{A}\) with tensor products and countable direct sums, it would be possible to formulate analogous functors $$K,L: \mathbf{S}(\mathcal{A})\times\mathbf{S}(\mathcal{A})\to Ch(\mathcal{A})$$ and we could then attempt to establish an analogue or generalization of the Eilenberg-Zilber theorem.
In a next blog post I will turn to the application of these ideas to constructing products in sheaf cohomology; and in particular the relevance of the simplicial perspective.
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