We will give an introduction to the deformation of algebraic varieties, and its relationship with differential graded Lie algebras (in characteristic zero). This can only be made precise through the language of higher category theory, replacing sets of deformations with spaces of deformations and rings with animated rings. We will conclude by introducing new results decomposing differential graded Lie algebras into simpler pieces corresponding to the Koszul dual of the truncations of the commutative operad. This has applications in studying nilpotent algebras and deformation theory in condensed mathematics.