Cohomological Coefficient Systems as Kan extensions
One method of extending the definitions of homology and cohomology to more general systems of coefficients is with (cohomological) coefficient systems, as they are called in Gelfand and Manin's book Methods of Homological Algebra and Godement's book Topologie Algebrique et Theorie des Faisceau, and in more modern writing, "local systems."
Given a simplicial set \(X : \Delta^{op}\to \mathbf{Sets} \) (and here, it is not essential that \(\Delta\) really be the standard simplex category; it seems that this notion would work reasonably well if we replaced it with the category of finite ordinals and strictly increasing maps between them, or, as Godement prefers, a skeleton of the category of finite sets) a cohomological coefficient system is a covariant functor from \(\mathbf{el}(X)\) to Ab (or R-mod; or your favorite working Abelian category to take coefficients in) where by \(\mathbf{el}(X)\) we mean the category of elements of \(X\) as a presheaf, the result of applying the Grothendieck construction. (See Sheaves in Geometry and Logic for a presentation of this construction.)
The standard example here is that if \(E\) is a topological space, and \(\left\{U_i\right\}_{i\in I}\) is an open cover of \(E\) (here, take the indexing set \(I\) of the cover to be equipped with a choice of total order, then we take \(X\) to be the nerve of the open cover, where \(X_n\) is the set of all ordered tuples \( (U_{i_0},U_{i_1},\dots, U_{i_n})\) where \( i_0\leq i_1\leq \dots\leq i_n\) such that the intersection \(\bigcap_{0\leq j\leq n} U_{i_j}\) is nonempty. Then for any presheaf \(\mathcal{F}\) valued in Ab, there is an associated cohomological coefficient system \( \mathscr{A}\), which is the functor which sends \( (U_{i_0},U_{i_1},\dots, U_{i_n})\) to \(\mathcal{F}(\bigcap_{0\leq j\leq n} U_{i_j})\). For a weakly monotonic map \( f : [n] \to [m] \) in the base category \(\Delta\), and each lift \(f'\) from \(X(f)(x)\to x\) between simplices in \(\mathbf{el}(X)\) above \(f\), we define \(\mathscr{A}(f')\) to be the restriction map of the presheaf \(\mathcal{F}\) associated to the inclusion \( \bigcap_{0\leq j\leq m} U_{i_j} \subset \bigcap_{0\leq j' \leq n} U_{i_{f(j')}}\). (The inclusion is valid because the latter intersection is indexed over a subset over the set of indices of the first intersection.)
Now, there is a canonical way, given a cohomological coefficient system of a sheaf, to induce a cosimplicial Abelian group which we denote \(C^{\bullet}(X;\mathscr{A})\) by associating to each simplex \([n]\) the product \(\prod_{x\in X_n} \mathscr{A}(x)\), and if \( f : [n]\to[m]\) is a weakly increasing (monotonic) map of simplices, then the induced map
$$ \prod_{x\in X_n} \mathscr{A}(x) \to \prod_{y\in X_m} \mathscr{A}(y) $$
is defined, on the \(y\)th component, by projecting onto the \(X(f)(y)\)-th coordinate of the product, and then applying the map \( \mathscr{A}(f) : \mathscr{A}(X(f)(y))\to\mathscr{A}(y)\).
What I have realized is that this construction is none other than the right Kan extension of \(\mathscr{A}: \mathbf{el}(X)\to \mathbf{Ab}\) along the obvious projection functor \(\pi : \mathbf{el}(X)\to \Delta\).
I have been trying to work out what this means, and I have found that many of the properties of coefficient systems can be proven using the properties of Kan extensions alone. I find this satisfying, as I often find myself asking "what is this thing really?" Who knew that the answer would turn out so frequently to be "It's a Kan extension." It has only been a few weeks or so since, while studying the classical Acyclic models theorem, I realized that the proof hinges on constructing, for a small discrete subcategory \(\mathcal{M}\) of a potentially large category \(\mathcal{C}\), left Kan extensions of functors \(\nu : \mathcal{M}\to\mathbf{Sets}\) along the inclusion functor \(i_{\mathcal{M}}: \mathcal{M}\to \mathcal{C}\). They really are everywhere!
Given a simplicial set \(X : \Delta^{op}\to \mathbf{Sets} \) (and here, it is not essential that \(\Delta\) really be the standard simplex category; it seems that this notion would work reasonably well if we replaced it with the category of finite ordinals and strictly increasing maps between them, or, as Godement prefers, a skeleton of the category of finite sets) a cohomological coefficient system is a covariant functor from \(\mathbf{el}(X)\) to Ab (or R-mod; or your favorite working Abelian category to take coefficients in) where by \(\mathbf{el}(X)\) we mean the category of elements of \(X\) as a presheaf, the result of applying the Grothendieck construction. (See Sheaves in Geometry and Logic for a presentation of this construction.)
The standard example here is that if \(E\) is a topological space, and \(\left\{U_i\right\}_{i\in I}\) is an open cover of \(E\) (here, take the indexing set \(I\) of the cover to be equipped with a choice of total order, then we take \(X\) to be the nerve of the open cover, where \(X_n\) is the set of all ordered tuples \( (U_{i_0},U_{i_1},\dots, U_{i_n})\) where \( i_0\leq i_1\leq \dots\leq i_n\) such that the intersection \(\bigcap_{0\leq j\leq n} U_{i_j}\) is nonempty. Then for any presheaf \(\mathcal{F}\) valued in Ab, there is an associated cohomological coefficient system \( \mathscr{A}\), which is the functor which sends \( (U_{i_0},U_{i_1},\dots, U_{i_n})\) to \(\mathcal{F}(\bigcap_{0\leq j\leq n} U_{i_j})\). For a weakly monotonic map \( f : [n] \to [m] \) in the base category \(\Delta\), and each lift \(f'\) from \(X(f)(x)\to x\) between simplices in \(\mathbf{el}(X)\) above \(f\), we define \(\mathscr{A}(f')\) to be the restriction map of the presheaf \(\mathcal{F}\) associated to the inclusion \( \bigcap_{0\leq j\leq m} U_{i_j} \subset \bigcap_{0\leq j' \leq n} U_{i_{f(j')}}\). (The inclusion is valid because the latter intersection is indexed over a subset over the set of indices of the first intersection.)
Now, there is a canonical way, given a cohomological coefficient system of a sheaf, to induce a cosimplicial Abelian group which we denote \(C^{\bullet}(X;\mathscr{A})\) by associating to each simplex \([n]\) the product \(\prod_{x\in X_n} \mathscr{A}(x)\), and if \( f : [n]\to[m]\) is a weakly increasing (monotonic) map of simplices, then the induced map
$$ \prod_{x\in X_n} \mathscr{A}(x) \to \prod_{y\in X_m} \mathscr{A}(y) $$
is defined, on the \(y\)th component, by projecting onto the \(X(f)(y)\)-th coordinate of the product, and then applying the map \( \mathscr{A}(f) : \mathscr{A}(X(f)(y))\to\mathscr{A}(y)\).
What I have realized is that this construction is none other than the right Kan extension of \(\mathscr{A}: \mathbf{el}(X)\to \mathbf{Ab}\) along the obvious projection functor \(\pi : \mathbf{el}(X)\to \Delta\).
I have been trying to work out what this means, and I have found that many of the properties of coefficient systems can be proven using the properties of Kan extensions alone. I find this satisfying, as I often find myself asking "what is this thing really?" Who knew that the answer would turn out so frequently to be "It's a Kan extension." It has only been a few weeks or so since, while studying the classical Acyclic models theorem, I realized that the proof hinges on constructing, for a small discrete subcategory \(\mathcal{M}\) of a potentially large category \(\mathcal{C}\), left Kan extensions of functors \(\nu : \mathcal{M}\to\mathbf{Sets}\) along the inclusion functor \(i_{\mathcal{M}}: \mathcal{M}\to \mathcal{C}\). They really are everywhere!
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