Title: The Categorical Algebra of Rigs and Kähler Differential Functor for Rigs
Abstract: The theory of rigs (rings without negatives), also known as semirings in the literature, is an interesting and important field of algebra with applications to theoretical computer science, logic, economics (through the use of tropical geometry and the tropical semiring), and also to arithmetic geometry (in the styles of Deitmar, or of Toën and Vasquie, or of Lorscheid — all of these are built to discuss a theory of schemes over the “field with one element”).
In this talk, based on joint work in progress with Robin Cockett, I will introduce the category of commutative rigs and indicate a careful and precise construction of the ways in which the categories of commutative rig algebras and modules over commutative rigs interact. More precisely, I will show that there are fibrations associated to both the commutative algebra and module constructions and that the underlying module/symmetric algebra functors sit as fibre-wise adjoints between the categories of said fibrations. Afterwards, depending on time, I will discuss some combination and/or permuation of the following topics: what localizations of rigs are, what the module of Kähler differentials are, how they arise as a functor into the module fibration, and also how the module of Kähler differentials interacts with localizations and tensor products.
