We develop the notion of deformation of a morphism in a left-proper model category. As an application we provide a geometric/homotopic description of deformations of commutative (non-positively) graded differential algebras over a local DG-Artin ring.

This research is partially supported by Italian MIUR under PRIN project 2015ZWST2C "Moduli spaces and Lie theory". Both authors thank Ruggero Bandiera, Barbara Fantechi and Domenico Fiorenza for useful discussions on the subject of this paper. We are grateful to the referee for carefully reading the paper and for useful comments.