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Hello,

I have noticed some fact about cohomologie : when i have some kind of strucutre in a topos (for example the $G$-object for $G$ a group object in the topos) and a particular object $X$ model of this strucutre such that the sheaf of automorphism of $X$ is a sheaf of commutative group, Then there is (at least on a lot of examples) a bijection between isomorphism class of model of this structure which are locally isomorphic to $X$ and $H^1(T,G)$.

Examples I have in mind are the representation of dimension 1 of a group $G$ over a field $k$ which correspond in one hand to one dimensional $k$-vector space in the topos of $G$-set and on the other hand to the cohomology group $H^1(G-set,k^*) $. Or the principal bundle over a topological space $X$ which corresponds to some $H^1(X,G)$ too.

Is there a "general explication" to those facts ? I mean by that a result valid on an arbitrary topos who gave a bijection between a $H^1(T,G)$ and isomorphism class of objects internally isomorph.

And Is there "higher dimensional" generalization ? ( I am working on an example which seem to involve a 2-category of object inside a topos $T$ and where "equivalence class" of objects "localy equivalent" seem to be classified by some $H^2$ group in a way that i don't understand yet... )

Thank you !

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The classical fact that $H^1(X, \mathscr{O}_X^*)$ classifies the isomorphism classes of locally free vector bundles of rank $1$, for $X$ a sufficiently nice topological space, is much more plausible when you think about it in terms of Čech cohomology than in terms of derived functor cohomology. Fortunately, for nice spaces $X$, these coincide. –  Zhen Lin Mar 15 '12 at 7:54
    
@Zhen: Arbitrary ringed space. –  Martin Brandenburg Mar 15 '12 at 8:35

1 Answer 1

up vote 5 down vote accepted

It is a general fact that if you consider an abelian group $A$ in topos $T$, then equivalence classes of $A$-torsors in $T$ are classified by cohomology group $H^1(T;A)=Ext^1_T(\mathbb{Z},A)$. Here $\mathbb{Z}$ is a free abelian group generated by final object $1\in T$ and ext-group can be defined in a classical way using injective resolutions. $A$-torsor is an object $X$ equipped with an action $\alpha: X\times A\to X$, such that $X\to 1$ is epimorphism and $< \alpha, \pi_1 >: X\times A \to X \times X$ is isomorphism. See P.T. Johnstone, "Topos theory", chapter 8.

One-dimensional representations of $G$ over $\mathbb{k}$ are just $\mathbb{k}^*$-torsors in a category of $G$-sets, so general theory applies in your example.

I am not familiar with higher classification theory, although I am sure it exists. In p.8.3 of Johnstone's book there's some talk on higher classifying spaces in toposes and associated cohomology theories. There is also Jacob Lurie's book "Higher topos theory", which, I assume, deals with higher cohomology theories, although I didn't read it yet.

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Thank you. The Johnstone's book actually mention that this theory as an analogue in Higher dimension due to J.W.Duskin, give a references on the subject and even explain a construction of the correspondence in the $H^2$ case (which is the one I was interested in ). Thank you ! –  Simon Henry Mar 15 '12 at 16:03

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