higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
For $A$ an ordinary associative algebra, its tangent complex is its module of derivations.
For $A$ a dg-algebra, its tangent complex is the essentially the value of the derived functor of the derivations-assigning functor on $A$. This is closely related to the automorphism ∞-Lie algebra of $A$.
tangent complex, André-Quillen cohomology, Hochschild cohomology
cotangent complex, André-Quillen homology, Hochschild homology
The concept goes back to
The tangent complex of an algebra over an operad in chain complexes is discussed in section 8 of
See also