Ambidexterity and the universality of finite spans


Salle Duhem M3
Yonatan Harpaz
Université Paris 13
Vendredi, 12 Janvier, 2018 - 14:00 - 15:00

Ambidexterity is a categorical phenomenon in which left and right adjoints coincide. Well-known particular cases include coincidence of initial and terminal objects (which are then known as zero-objects) and coincidence of products and coproducts (which are then known as biproducts). Somewhat unexpectedly, when such phenomena occur they endow the ambient category with a canonical additional structure. For example, categories with zero-objects (also known as pointed categories) are canonically enriched in pointed sets, while pointed categories which admit biproducts (also known as semiadditive categories) are canonically enriched in commutative monoids. In their recent and insightful paper, Hopkins and Lurie suggest viewing the previous two examples as the first two steps in a tower of ambidexterity properties, which can be considered as higher forms of semiadditivity. In particular, n-semiadditivity is the coincidence of limits and colimits indexed by n-truncated spaces which are $\pi$-finite (i.e., all whose homotopy groups are finite). In this talk we will describe recent work showing that the span $\infty$-category of $\pi$-finite n-truncated spaces is the free n-semiadditive $\infty$-category generated by a single object. This universal characterization also leads to a new notion of n-commutative monoids, which are spaces in which families of points parameterized by $\pi$-finite n-truncated spaces can be coherently summed. If time permits we will describe how these constructions can be used to study certain "finite path integrals" described by Freed, Hopkins, Lurie and Teleman in the context of 1-dimensional topological field theories.