# Ambidexterity and the universality of finite spans

## TL;DR

This paper establishes that the span $mbda$-category of $m$-finite spaces is the free $m$-semiadditive $mbda$-category generated by a single object, connecting higher semiadditivity with topological field theories.

## Contribution

It introduces a new perspective on $m$-semiadditivity and defines $m$-commutative monoids, linking abstract summation to topological path integrals.

## Key findings

- Characterization of the span $mbda$-category as free $m$-semiadditive category
- Introduction of $m$-commutative monoids as spaces with coherent summation
- Application to formalize finite path integrals in topological field theories

## Abstract

Pursuing the notions of ambidexterity and higher semiadditivity as developed by Hopkins and Lurie, we prove that the span $\infty$-category of $m$-finite spaces is the free $m$-semiadditive $\infty$-category generated by a single object. Passing to presentable $\infty$-categories we obtain a description of the free presentable $m$-semiadditive $\infty$-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 $\infty$-categorical definition of the finite path integral described by Freed, Hopkins, Lurie and Teleman in the context of 1-dimensional topological field theories.