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We continue the explorations of derived \canal geometry started in [DAG-IX]and in http://arxiv.org/abs/1506.09042. We describe the category of $\mathcalO_X$-modules over a derived complex analytic space $X$ as the stabilization ofa suitable category of analytic algebras over $\mathcal O_X$. Finally, we applythis description to introduce the notion of analytic square-zero extension andprove a fundamental structure theorem for them.

Derived complex analytic geometry II: square-zero extensions