Ambidexterity and the universality of finite spans
Yonatan Harpaz
Abstract
Pursuing the notions of ambidexterity and higher semiadditivity as developed by Hopkins and Lurie, we prove that the span ∞-category of m-finite spaces is the free m-semiadditive ∞-category generated by a single object. Passing to presentable ∞-categories we obtain a description of the free presentable m-semiadditive ∞-category in terms of a new notion of m-commutative monoids, which can be described as spaces in which families of points parameterized by m-finite spaces can be coherently summed. Such an abstract summation procedure can be used to give a formal ∞-categorical definition of the finite path integral described by Freed, Hopkins, Lurie and Teleman in the context of one-dimensional topological field theories.
Topics & Concepts
MathematicsUniversality (dynamical systems)Parameterized complexityTopological spacePure mathematicsAlgebra over a fieldAmbidexterityContext (archaeology)Field (mathematics)Path (computing)Discrete mathematicsSpan (engineering)Finite setCalculus (dental)SemilatticeSpace (punctuation)Homotopy and Cohomology in Algebraic TopologyLogic, programming, and type systemsAdvanced Operator Algebra Research