Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
Differential categories use category theory to provide the foundations of differential calculus. In this talk, I will give you guided tour of the world of differential categories. We will see (1) differential categories, which give the algebraic foundations of differentiation; Cartesian differential categories, which give the foundations of multivariable differential calculus; and (3) tangent categories, which give the foundations of differential geometry. In particular we will look at the map of differential categories and see how these three concepts relate to each other. Moreover, the theory of differential categories has been successful in formalising various important concepts related to differentiation. In particular, this talk will set the table for next week’s talk, where Chiara Sava will explain how differential categories capture differential graded algebras.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.There are two oftentimes unspoken truths in measure theory. 1) Practically all useful measures in practice are given by Radon measures. 2) One does not really care so much about the sigma-algebra of measurable sets, but rather about its quotient by the ideal of null sets.
The quotient of measurable sets by null sets is, in the case of a given Radon measure, an example of what is called a measurable locale, and can be treated like a (usually point-free) space. We argue that this measurable locale can be constructed directly from a Grothendieck topology on the poset of compact sets. This opens the door to a purely sheaf-theoretic perspective on measure theory. As an application, we show that the locale of sublocales of a given Hausdorff space X equipped with a Radon measure can be equipped with a natural extension of the measure, invariant under measure preserving homeomorphisms.