Differential graded algebras in differential categories

Abstract: Differential categories, introduced in last week's talk by Jean-Simon Pacaud Lemay, provide a categorical framework for the algebraic foundations of differential calculus. Within this setting we can capture familiar notions such as derivations, Kähler differentials, differential algebras and de Rham cohomology. Along this line, in this talk, we will show how to define differential graded algebras in a differential category. In the case of polynomial differentiation, this construction recovers the classical commutative differential graded algebras, while for smooth functions it yields differential graded $C^\infty$-rings in the sense of Dmitri Pavlov. To further justify our definition, we will explain how the monad of a differential category can be lifted to its category of chain complexes and how the algebras of the lifted monad correspond precisely to differential graded algebras of the base category, with the free ones given by the de Rham complexes. Finally, we will discuss how the category of chain complexes of a differential category is itself a differential category, pointing towards the prospect of differential dg-categories. This is joint work with Jean-Simon Pacaud Lemay.

mathematical physicsalgebraic topologycategory theory

Audience: researchers in the topic

Topology and Geometry Seminar (Texas, Kansas)