In recent years, a combination of ideas from category theory, algebraic geometry,
and mathematical logic has led to the development of an area that may be called commutative 2-algebra,
in which the standard notions used in commutative algebra are replaced by appropriate category-theoretic
counterparts. For example, commutative monoids are replaced by symmetric monoidal categories.
The aim of this talk is to explain the analogy between standard commutative algebra and commutative 2-algebra,
and to outline how it can be used to develop a counterpart of some basic aspects of algebraic geometry.
In particular, I will describe some joint work with Mathieu Anel and Andre’ Joyal on operads and analytic
functors in this context.