Derived category
Overview
Category of chain complexes with morphisms up to homotopy. Verdier (Grothendieck's student) formalized it. Enables derived functors (Ext, Tor) and homological algebra.
Era: 1960s
Proposed at: Paris, France
Formal structure
Domain: Chain complexes C^• = (C^n, d^n); morphisms chain maps
D(A) = K(A)[Q^{-1}]; Q = quasi-isomorphisms; Hom_D(X,Y) = Hom_K(X,Y')
Constraints: Triangulated structure; distinguished triangles; t-structure
Objective: Derived functors Rf_*, Lf^*; spectral sequences