Abstract Alejo Lopez-Avila

Alejo Lopez-Avila will speak on:

Hermitian infinite loop space machines

Abstract: Algebraic \(K\)-theory started at the end of the 50s with the definition of the group completion, which means to add inverses to a commutative monoid in a universal way. In the late 60s and early 70s, there appeared different approaches to define the higher algebraic $K$-groups. Among others, there appeared the infinite loop space machines (ILSM). These machines follow the original idea of the group completion but up to homotopy and higher coherence. Instead of group completing a commutative monoid, they group complete a space with a commutative monoid structure up to homotopy and higher coherence, a so-called \(E_\infty\)-space, which arises naturally as the nerve of a symmetric monoidal category. The resulting group completion turns to be an infinite loop space, which can be delooped getting a spectrum representing the theory. For a long time, these machines were addressed using combinatorial constructions which involve explicit pairs of operads, but the development of quasicategories in the recent years enables us to tackle these machines from the higher category point of view. On the other hand, the Hermitian $K$-theory is the \(K\)-theory for categories with duality, taking into account this duality as data. At the start of the talk, we will see a review of the classical ILSMs and its version in the higher categorical setting. Then, the infinity categories with duality and a hermitian ILSM for these categories will be defined. Lastly, we will see some of the properties of this hermitian ILSM, like the preservation of second algebraic structure giving rise to an \(E_\infty\)-ring spectra after applying this machine.