Cech cohomology and cohomology of a category : a cluster of questions.

Question

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a (poset-)category with values in a functor. I recall some definitions. In what follows $k$ is a (possibly commutative) unital ring, $X$ is a topological space and $\mathcal{F}$ is a (pre-)sheaf of $k$-modules on $X$.

1. Cohomology of a small category with values in a functor. Let $\mathcal{C}$ be a small category. The functor category $$k\mathcal{C}\mathrm{Mod}:=\mathrm{Fun}\big(\mathcal{C},k\mathrm{Mod}\big)$$ (with functors as objects, and natural transformations as morphisms) is an abelian category with enough injectives and projectives. Therefore functors $F,G:\mathcal{C}\to k\mathrm{Mod}$ admit projective and injective resolutions, one may define Ext groups $$\mathrm{Ext}^*_{k\mathcal{C}\mathrm{Mod}}(F,G)$$ by appliying either $\mathrm{Hom}_{k\mathcal{C}\mathrm{Mod}}(F,\:\:)$ to an injective resolution $0\to G\to I_*$ of $G$ and taking homology, or by appliying $\mathrm{Hom}_{k\mathcal{C}\mathrm{Mod}}(\:\:,G)$ to a projective resolution $P_*\to F\to 0$ of $F$ and, again, taking homology. The cohomology of $\mathcal{C}$ with values in $F$ is defined as $$\mathbf{H}^*(\mathcal{C},F):=\mathrm{Ext}^*(\underline{k},F)$$ where $\underline{k}$ is the constant functor equal to $k$ : it sends all objects of $\mathcal{C}$ to $k$, and all morphisms to the identity $\mathrm{id}:k\to k$.
2. Cohomology group of a sheaf relative to a cover. (I'm not certain that's the correct terminology.) Let $\mathcal{U}$ be an open cover of $X$. Usually when defining Cech cohomology this means "a set $I$ and a map $U:I\to\lbrace$ all open subsets of $X\:\rbrace=\tau_X$ such that $\bigcup_{i\in I}U_i=X$", but I shall mean "a self-indexed open cover", i.e. a subset $\mathcal{U}\subset\tau_X$ with $\bigcup\mathcal{U}=X$. There are cohomology groups $$H^*(\mathcal{U},\mathcal{F})$$ associated to the cover and the sheaf : these are the homology groups of the Cech complex $$\prod_{u_0\in\mathcal{U}}\mathcal{F}(u_0)\longrightarrow\prod_{(u_0,u_1)\in\mathcal{U}^2}\mathcal{F}(u_0\cap u_1)\longrightarrow\prod_{(u_0,u_1,u_2)\in\mathcal{U}^3}\mathcal{F}(u_0\cap u_1\cap u_2)\longrightarrow\cdots$$
3. Any poset $(P,\prec)$ may be considered a small category whith objects the points of $P$ and morphism sets defined as follows : for any objects $x,y\in P$ $$\mathrm{mor}(x,y)=\begin{cases}\lbrace * \rbrace & \text{if }x\preceq y\\ \emptyset & \text{if }x\npreceq y\end{cases}$$ An open cover (of the self-indexed kind) is a poset with the order relation $\prec$ defined as inclusion $\subset$.
4. The sheaf $\mathcal{F}$ induces a functor $$\mathcal{F}\big|_{\mathcal{U}}:\mathcal{U}^{\mathrm{op}}\to k\mathrm{Mod}$$ and we can define its cohomology groups $$\mathbf{H}^*\left(\mathcal{U}^{\mathrm{op}},\mathcal{F}\big|_{\mathcal{U}}\right)$$

The obvious question now is :

Question 1 : Are the cohomology groups $$H^*(\mathcal{U},\mathcal{F})\quad\text{and}\quad \mathbf{H}^*\left(\mathcal{U}^{\mathrm{op}},\mathcal{F}\big|_{\mathcal{U}}\right)$$ isomorphic?

If that is so, then presumably there is also a functor from the poset category $Cov(X)$ of (self-indexed) open covers of $X$ to that of graded-commutative algebras that takes an open cover $\mathcal{U}$ to $\mathbf{H}^*\left(\mathcal{U}^{\mathrm{op}},\mathcal{F}\big|_{\mathcal{U}}\right)$.

As in the case Cech cohomology, one would first have to define such a functor on the category $\widetilde{Cov}(X)$ of open covers where "morphisms = choice functions" (if a cover $\mathcal{V}$ is finer than a cover $\mathcal{V}$, there are choice functions from the finer cover to the coarser cover that take an open set from the fine cover and hand back an open set in the coarse cover that containing it).

Question 2 : Is the Cech cohomology isomorphic to the cohomology of the poset category of open cover with values in the functor $\mathcal{U}\mapsto \mathbf{H}(\mathcal{U}^{\mathrm{op}},\mathcal{F}\big|_{\mathcal{U}})$, i.e. is $$\check{H}^*(X,\mathcal{F})\simeq \mathbf{H}^*\Big(Cov(X),\mathbf{H}\big(\:?^{\mathrm{op}}\:,\mathcal{F}\big|_{\mathcal{\:?\:}}\big)\Big)$$

In some notes by Brian Conrad, it is shown that the augmented chain complex associated to the simplicial set $WI$ is acyclic, i.e. that it is a free resolution of $k$.

$\Big[I$ is an arbitrary (non-empty) set, $WI$ is the simplicial set with $(WI)_0=I$, $(WI)_1=I\times I$, $(WI)_2=I\times I\times I$, $\dots$, with face maps $d_i:(WI)_n\to(WI)_{n-1}$ that cancel the $i$-th coordinate, degeneracies $s_i:(WI)_n\to(WI)_{n+1}$ that duplicate the $i$-th coordinate.$\Big]$

Question 3 : is this free resolution (when $I=\mathcal{U}$ is an open cover) somehow linked to a projective resolution of the constant functor $\underline{k}:\mathcal{U}^{\mathrm{op}}\to k\mathrm{Mod}$? If so, how?

Edit : to make things clearer, cohomology of a category with values in a functor is bold : $$\mathbf{H}^*(\mathrm{category},\mathrm{functor}).$$

Answer 1

Allow me address some general points instead of answering your question directly.

The cohomology of a small category already has an accepted definition: unfortunately, it is not the one you give. Rather, the cohomology of a small category $\mathbb{C}$ is the topos-theoretic cohomology of the presheaf topos $[\mathbb{C}^\mathrm{op}, \mathbf{Set}]$, or equivalently, the right derived functors of the global sections functor $\Gamma (1, -) : [\mathbb{C}^\mathrm{op}, \mathbf{Ab}] \to \mathbf{Ab}$ (which is represented by the constant presheaf with value $\mathbb{Z}$). In general, this is non-trivial; but usually $\mathbb{C}$ is a category with a terminal object, in which case $\Gamma (1, -)$ is an exact functor and has trivial right derived functors.

So instead we look at the right derived functors of the functor $\Gamma (U, -) : [\mathbb{C}^\mathrm{op}, \mathbf{Ab}] \to \mathbf{Ab}$, where $U$ is a presheaf on $\mathbb{C}$ (i.e. functor $\mathbb{C}^\mathrm{op} \to \mathbf{Set}$) and $\Gamma (U, \mathscr{F})$ is the set of all presheaf morphisms $U \to \mathscr{F}$ equipped with the abelian group structure induced by $\mathscr{F}$. This turns out to be a generalisation of Čech cohomology with respect to a cover.

Indeed, let $X$ be a topological space and let $\mathbb{C}$ be the poset of open subsets of $X$. Then any open cover $\{ U_i : i \in I \}$ of $X$ defines a presheaf $\mathfrak{U}$: we have $\mathfrak{U} (U) = 1$ if $U$ is contained in some $U_i$, and $\mathfrak{U} (U) = \emptyset$ otherwise. I claim that the right derived functors of $\Gamma (\mathfrak{U}, -)$ can be computed by Čech cohomology with respect to the cover $\mathfrak{U}$. Indeed, let $\mathbb{Z} \mathfrak{U}$ be the free abelian presheaf generated by $\mathfrak{U}$. Then the Čech chain complex $$\cdots \longrightarrow \bigoplus_{(i_0, i_1, i_2) \in I^3} \mathbb{Z} (U_{i_0} \cap U_{i_1} \cap U_{i_2}) \longrightarrow \bigoplus_{(i_0, i_1) \in I^2} \mathbb{Z} (U_{i_0} \cap U_{i_1}) \longrightarrow \bigoplus_{i \in I} \mathbb{Z} U_i$$ is a projective resolution of $\mathbb{Z} \mathfrak{U}$, where $\mathbb{Z} U$ denotes the abelian presheaf where $\mathbb{Z} U (V) = \mathbb{Z}$ if $V \subseteq U$ and $\mathbb{Z} U (V) = 0$ otherwise. It is clear that $\Gamma (\mathfrak{U}, -)$ is isomorphic to $\mathrm{Hom} (\mathbb{Z} \mathfrak{U}, -)$, and thus we have $$H^* (\mathfrak{U}, \mathscr{F}) \cong \mathrm{Ext}^* (\mathfrak{U}, \mathscr{F}) \cong R^* \Gamma (\mathfrak{U}, \mathscr{F})$$ as claimed.

The refined Čech cohomology groups, defined by taking the colimit over all covering families, can also be defined in terms of derived functors on the category of abelian presheaves. The details are explained here. However, I would not say that it is the cohomology of any particular category. (In fact – it cannot be, because $\check{H}{}^0 (X, -)$ does not preserve all limits, but $R^0 \Gamma (1, -)$ always preserves all limits.)

To get standard sheaf cohomology (instead of these other approximations!), one must pass to the category of (abelian) sheaves. But that rarely has enough projective objects, so it is not at all clear what one might mean by projective resolution.