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Suppose that we are given a nice space $X$ and a sheaf of abelian groups $F$ on $X$. Fix an integer $n$. Then We have a contravariant functor from nice spaces over $X$ to abelian groups; Namely, to a space $f: Y \to X$ we associate the abelian group $H^n (Y, f^{*}F)$ (Sheaf cohomology).

If $X$ is a point, Then this functor is represented, in the homotopy category, by the Eilenberg-Maclane space $K(F,n)$.

My questions are:

1) Can one formalize what will it mean for our functor to be representable, "homotopically"? I am not very sure, but I suspect that the most naive definition of homotopy category of spaces over $X$, and the requirement that the functor is representable in this category, are not right (and I did not check that the functor actually factors to this "homotopy" category).

2) Is it representable?

edited Jul 17 2010 at 15:54

Mariano Suárez-Alvarez
30.7k●8●69●155

asked Jul 17 2010 at 14:23

Sasha
1,957●3●15

Urs Schreiber · Accepted Answer · 2010-07-17 15:44:21Z UTC

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Sheaf cohomology over $X$ is representable in the homotopy category of oo-stacks over X / spaces over X, yes.

Details, links and references are at http://ncatlab.org/nlab/show/cohomology

answered Jul 17 2010 at 15:44

Urs Schreiber
6,735●14●22

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Urs -- maybe it's just me but I've taken a look at that page and I have no idea what the theorem is. Could you please state it explicitly: for any $Y$ over $X$ we have $H^{\ast}(Y,f^{-1}F)$ is the $Hom$ in the category such and such from $Y$ to the object such and such constructed from $X$ and $F$? – algori Jul 20 2010 at 23:47

The space Y itself is a sheaf over X, albeit not a sheaf of abelian groups, but a "nonabelian sheaf". You regard both Y as well as F as being oo-stacks over X (sheaves of simplicial sets, essentially): Y is just a sheaf of sets. for F you take the simplicial presheaf that you obtain under the Dold-Kan map from the chain complex of sheaves concentrated on F in the dired degree n. Call that B^n F. Then the cohomology in question is [Y,B^n F] in the homotopy category of the oo-stack oo-category. See the old article by Kenneth Brown that is linked at the entry. – Urs Schreiber Jul 21 2010 at 13:44

Thanks, Urs. This has clarified things a bit. – algori Jul 21 2010 at 19:10

Representing cohomology of a sheaf à la Eilenberg-Maclane

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