# Isomorphism class of locally trivial object classified by some $H^1$ ?

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

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.
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