# Sheaf Cohomology and Homotopy

In classical algebraic topology, we have the following important relationship between cohomology and homotopy:

$H^{n}(X,G)\approx[X,K(G,n)]$

where $K(G,n)$ is an Eilenberg-Maclane space, characterized by the property that only its nth homotopy group is nontrivial and it is isomorphic to $G$.

How does this generalize to other cohomology theories, in particular sheaf cohomology? I have found "Abstract Homotopy and Generalized Sheaf Cohomology" by Brown, which states that

$H^{n}(X,F)\approx[\mathbf{Z},F_{(n)}]_{\text{Ho }\mathcal{C}(X)}$

where $\mathbf{Z}$ and $F_{(n)}$ are complexes of sheaves, with $\mathbf{Z}$ concentrated in dimension zero, consisting of the constant sheaf with stalk equal to the group of integers., while $F_{(n)}$ is the sheaf $F$ concentrated in dimension $-n$.

I looked at the article "cohomology" from the nLab, which states what is possibly the most general definition of cohomology, but I do not yet grasp how $\mathbf{Z}$ and $F_{(n)}$ in the category of complexes of abelian sheaves are analogous respectively to $X$ and $K(G,n)$ in the category of topological spaces.

• The statement you found in Brown seems, to me, like just a derived category statement--I don't understand what $\mathcal{C}(X)$ is though. But, in the derived category if you look at Hom's from $\mathbb{Z}$ to $F_(n)$ you get $\text{Ext}^n(\mathbb{Z},F)$ which is just $H^n(X,F)$. So, perhaps this Brown paper is just doing some sort of derived version of the standard theory? Also, can you make precise what you're after. So, if $G$ is a group (let's say discrete for now) then the functor $X\mapsto H^n(X,G)$ is represented in the homotopy category by $K(G,n)$. Are you starting with some fixed – Alex Youcis Nov 17 '16 at 13:40
• space $X$ with a fixed sheaf $\mathcal{F}$ on $X$ and trying to represent the functor $Y\mapsto H^n(Y,f^{-1}\mathcal{F})$ on the homotopy category over $X$? @Qiaochu Would probably know this. – Alex Youcis Nov 17 '16 at 13:41
• $\mathcal{C}(X)$ is the category of complexes of abelian sheaves. I am not very familiar with derived categories (I know only about derived functors for sheaf cohomology from Hartshorne). I do know that Ext is the derived functor of Hom. – Anton Hilado Nov 18 '16 at 3:23
• But I can take a guess as to how the cohomology is $\text{Ext}^{n}(\mathbb{Z},F)$: It comes from singular homology with integer coefficients. Cohomology is the dual of homology, so I guess that is where Ext comes in, as the derived functor of Hom. I guess this also answers in a way my own question of why the constant sheaf with integers as the stalk takes the place of the topological space $X$ in the category of complexes of abelian sheaves. Please do tell me if this insight is incorrect. – Anton Hilado Nov 18 '16 at 3:31
• I am also not yet very familiar with the Eilenberg-Maclane space as representing the cohomology functor, but I will look into this. – Anton Hilado Nov 18 '16 at 3:32