U. Haifa Topology & Geometry seminar: Tuesday, December 16, 2025, 17:00. Speaker: Arpon Raksit (Northwestern). Title: “Derived categories via spectral algebraic theories”.
Many categories of algebraic structures, like abelian groups and commutative rings, can be presented in terms of “algebraic theories”, a precise term which was given a categorical formulation by Lawvere. Quillen, using the simplex category, developed a construction of “nonabelian derived categories” in this general setting, which more recently has been reinterpreted as a universal construction in the framework of infinity categories, and dubbed “animation”. Animated abelian groups are equivalent to connective complexes of abelian groups, and animated commutative rings analogously form a “connective derived category” of commutative rings (corresponding to affine derived schemes). In this talk, I will review these ideas and then discuss a variant of them that can be used to produce the full derived category of complexes of abelian groups (i.e. not necessarily connective) and the category of derived commutative rings, an analogous enlargement of animated commutative rings (which remains closely tied to derived algebraic geometry), as well as other analogous constructions of a more homotopy-theoretic flavor.
Recording:
