I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O\_X) such that Hom(X, Spec(A))= Hom(A,Gamma(O\_X)) for every affine scheme Spec(A)? Or does this require some more hypotheses. I have very much a hard time finding this out. 



question about higher geometric stacks
Yes -- affinization is defined (as you wrote) as the left adjoint to the inclusion of affine schemes into higher stacks.
This left adjoint exists by the ($\infty$-categorical) adjoint functor theorem, since the inclusion of affines into higher stacks preserves all limits (though it certainly changes colimits). Some references for this or closely related notions: Toen's Affine Stacks ([here](http://arxiv.org/abs/math/0012219)) and Lurie's DAG VIII (available [here](http://www.math.harvard.edu/~lurie/)), where the relevant notion is called "coaffine stacks". There's also a less professional and more informal discussion (in the derived context) in Section 3.2 [here](http://arxiv.org/abs/1002.3636).


