higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The tangent complex of an algebro-geometric object is meant to behave as the (sheaf of) sections of the tangent bundle.
At least in the generality of derived algebraic stacks $X$, the tangent complex is equivalently (up to a shift in degree) the module of sections of the infinitesimal disk bundle of $X$ (the formal completion of the diagonal $\Delta_X \colon X \to X \times X$) (Hennion 13, theorem 1).
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
Jonathan Block, A. Lazarev, André-Quillen cohomology and rational homotopy of function spaces , Advances in mathematics, 193 (2005) (pdf)
Benjamin Hennion, Tangent Lie algebra of derived Artin stacks, Journal für die reine und angewandte Mathematik (Crelles Journal), December 2015 (arXiv:1312.3167, DOI:10.1515/crelle-2015-0065)
Sam Raskin, p. 2,3 of The cotangent stack (pdf)