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Presenter: Samuel Silver

Faculty Sponsor: Martina Rovelli

School: UMass Amherst

Research Area: Mathematics and Statistics

Session: Poster Session 4, 2:15 PM - 3:00 PM, Auditorium, A64

ABSTRACT

Rings are structures in mathematics which encapsulate the most general framework in which you can add and multiply. Two rings are "isomorphic" if there is a one-to-one correspondence between them respecting addition/multiplication. In the 1950s, Kiiti Morita defined a weaker notion of equivalence of rings than isomorphism, and proved some powerful structural theorems about it.

Modules are structures on which rings can "act" by multiplication. A category is another mathematical structure, in which one has "objects" and "morphisms" (thought of as arrows between objects) such that consecutive morphisms can be composed, just as we can compose functions. For any ring R, there is a category R-Mod of modules over R. We say two rings R and S are Morita equivalent if their categories of modules R-Mod ≃ S-Mod are equivalent. Part of the structure theorem effectively states that Morita equivalence is equivalence in a certain bicategory (like a category, but with additional structure of "morphisms between morphisms").

Monoidal categories are categories equipped with a "multiplication" for objects and morphisms. Rings are monoids in a certain monoidal category (abelian groups with tensor product), and it turns out we can carry out Morita's construction for any monoidal category (with some higher-categorical complications, which we avoid by working with some mild assumptions). 

We'll begin by providing some background on basic category theory. Then we'll define the basic concepts of monoidal category theory, monoids, and modules. We then display our main result: an explicit construction of the Morita bicategory under our simplifying assumptions.